The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 0
Subject Introduction
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Subject Introduction
Welcome to Declarative Programming
Lecturer: Peter Schachte
Contact information is available from the LMS.
There will be two pre-recorded one-hour lectures per week, plus one live
one-hour practical meeting for questions, discussion, and demonstrations.
There will be eleven one-hour workshops (labs), starting in week 2.
You should have already been allocated a workshop. Please check your
personal timetable after the lecture.
COMP90048 Declarative Programming Lecture 0 – 1 / 16
Subject Introduction
Grok
We use Grok to provide added self-paced instructional material, exercises,
and self-assessment for both Haskell and Prolog.
You can access Grok by following the link from the subject LMS page.
If you are unable to access Grok or find that it is not working correctly,
please email
Grok University Support
from your university email account and explain the problem.
If you have questions regarding the Grok lessons or exercises, please post a
message to the subject LMS discussion forum.
COMP90048 Declarative Programming Lecture 0 – 2 / 16
Subject Introduction
Workshops
The workshops will reinforce the material from lectures, partly by asking
you to apply it to small scale programming tasks.
To get the most out of each workshop, you should read and attempt the
exercises before your workshop. You are encouraged to ask questions,
discuss, and actively engage in workshops. The more you put into
workshops, the more you will get out of them.
Workshop exercises will be available through Grok, so they can be
undertaken even if you are not present in Australia. Sample solutions for
each set of workshop exercises will also be available through Grok.
Most programming questions have more than one correct answer; your
answer may be correct even if it differs from the sample solution.
NOTE
If your laptop can access the building’s wireless network, you will be able
to use Grok, giving you access to Prolog and Haskell. If your laptop
cannot access the building’s wireless network, then you will be able to test
your Haskell or Prolog code if you install the implementations of those
languages on your machine yourself. For both languages this is typically
fast and simple.
COMP90048 Declarative Programming Lecture 0 – 3 / 16
Subject Introduction
Resources
The lecture notes contain copies of the slides presented in lectures, plus
some additional material.
All subject materials (lecture notes, workshop exercises, project
specifications etc) will be available online through the LMS.
The recommended text (which is available online)is
Bryan O’Sullivan, John Goerzen and Don Stewart: Real world Haskell.
Other recommended resources are listed on the LMS.
COMP90048 Declarative Programming Lecture 0 – 4 / 16
Subject Introduction
Academic Integrity
All assessment for this subject is individual; what you submit for
assessment must be your work and your work alone.
It is important to distinguish project work (which is assessed) from
tutorials and other unassessed exercises.
We are well aware that there are many online sources of material for
subjects like this one; you are encouraged to learn from any online sources,
and from other students, but do not submit for assessment anything that
is not your work alone.
Do not provide or show your project work to any other student.
Do not store your project work in a public Github or other repository.
We use sophisticated software to find code that is similar to other
submissions this year or in past years. Students who submit another
person’s work as their own or provide their work for another student to
submit in whole or in part will be subject to disciplinary action.
COMP90048 Declarative Programming Lecture 0 – 5 / 16
Subject Introduction
How to succeed
Declarative programming is substantially different from imperative
programming.
Even after you can understand declarative code, it can take a while before
you can master writing your own.
If you have been writing imperative code all your programming life, you will
probably try to write imperative code even in a declarative language. This
often does not work, and when it does work, it usually does not work well.
Writing declarative code requires a different mindset, which takes a while
to acquire.
This is why attending the workshops, and practicing, practicing and
practicing some more are essential for passing the subject.
COMP90048 Declarative Programming Lecture 0 – 6 / 16
Subject Introduction
Sources of help
During contact hours:
Ask me during or after a lecture (not before).
Ask the demonstrator in your workshop.
Outside contact hours:
The LMS discussion board (preferred: everyone can see it)
Email (if not of interest to everyone)
Attend my consultation hours (see LMS for schedule)
Email to schedule an appointment
Subject announcements will be made on the LMS.
Please monitor the LMS for announcements, and the discussion forum for
detailed information. Read the discussion forum before asking questions;
questions that have already been answered will not be answered again.
COMP90048 Declarative Programming Lecture 0 – 7 / 16
Subject Introduction
Objectives
On completion of this subject, students should be able to:
apply declarative programming techniques;
write medium size programs in a declarative language;
write programs in which different components use different languages;
select appropriate languages for each component task in a project.
These objectives are not all of equal weight; we will spend almost all of
our time on the first two objectives.
COMP90048 Declarative Programming Lecture 0 – 8 / 16
Subject Introduction
Content
Introduction to logic programming and Prolog
Introduction to constraint programming
Introduction to functional programming and Haskell
Declarative programming techniques
Tools for declarative programming, such as debuggers
Interfacing to imperative language code
This subject will teach you Haskell and Prolog, with an emphasis on
Haskell.
For logistical reasons, we will begin with Prolog.
COMP90048 Declarative Programming Lecture 0 – 9 / 16
Subject Introduction
Why Declarative Programming
Declarative programming languages are quite different from imperative and
object oriented languages.
They give you a different perspective: a focus on what is to be done,
rather than how.
They work at a higher level of abstraction.
They make it easier to use some powerful programming techniques.
Their clean semantics means you can do things with declarative
programs that you can’t do with conventional programs.
The ultimate objective of this subject is to widen your horizons and thus
to make you better programmers, and not just when using declarative
programming languages.
COMP90048 Declarative Programming Lecture 0 – 10 / 16
Subject Introduction
Imperative vs logic vs functional programming
Imperative languages are based on commands, in the form of instructions
and statements.
Commands are executed.
Commands have an effect, such as to update the computation state,
and later code may depend on this update.
Logic programming languages are based on finding values that satisfy a set
of constraints.
Constraints may have multiple solutions or none at all.
Constraints do not have an effect.
Functional languages are based on evaluating expressions.
Expressions are evaluated.
Expressions do not have an effect.
COMP90048 Declarative Programming Lecture 0 – 11 / 16
Subject Introduction
Side effects
Code is said to have a side effect if, in addition to producing a value, it
also modifies some state or has an observable interaction with calling
functions or the outside world. For example, a function might
modify a global or a static variable,
modify one of its arguments,
raise an exception (e.g. divide by zero),
write data to a display, file or network,
read data from a keyboard, mouse, file or network, or
call other side-effecting functions.
NOTE
Reading from a file is a side effect because it moves the current position in
the file being read, so that the next read from that file will get something
else.
COMP90048 Declarative Programming Lecture 0 – 12 / 16
Subject Introduction
An example: destructive update
In imperative languages, the natural way to insert a new entry into a table
is to modify the table in place: a side-effect. This effectively destroys the
old table.
In declarative languages, you would instead create a new version of the
table, but the old version (without the new entry) would still be there.
The price is that the language implementation has to work harder to
recover memory and to ensure efficiency.
The benefit is that you don’t need to worry what other code will be
affected by the change. It also allows you to keep previous versions, for
purposes of comparison, or for implementing undo.
The immutability of data structures also makes parallel programming
much easier. Some people think that programming the dozens of cores
that CPUs will have in future is the killer application of declarative
programming languages.
COMP90048 Declarative Programming Lecture 0 – 13 / 16
Subject Introduction
Guarantees
If you pass a pointer to a data structure to a function, can you
guarantee that the function does not update the data structure?
If not, you will need to make a copy of the data structure, and pass a
pointer to that.
You add a new field to a structure. Can you guarantee that every
piece of code that handles the structure has been updated to handle
the new field?
If not, you will need many more test cases, and will need to find and
fix more bugs.
Can you guarantee that this function does not read or write global
variables? Can you guarantee that this function does no I/O?
If the answer to either question is “no”, you will have much more
work to do during testing and debugging, and parallelising the
program will be a lot harder.
COMP90048 Declarative Programming Lecture 0 – 14 / 16
Subject Introduction
Some uses of declarative languages
In a US Navy study in which several teams wrote code for the same
task in several languages, declarative languages like Haskell were
much more productive than imperative languages.
Mission Critical used Mercury to build an insurance application in one
third the time and cost of the next best quote (which used Java).
Ericsson, one of the largest manufacturers of phone network switches,
uses Erlang in some of their switches.
The statistical machine learning algorithms behind Bing’s advertising
system are written in F#.
Facebook used Haskell to build the system they use to fight spam.
Haskell allowed them to increase power and performance over their
previous system.
NOTE
Erlang, F# and Lisp are of course declarative languages.
For a whole bunch of essays about programming, including some about the
use of Lisp in Yahoo! Store, see paulgraham.com.
COMP90048 Declarative Programming Lecture 0 – 15 / 16
Subject Introduction
The Blub paradox
Consider Blub, a hypothetical average programming language right in the
middle of the power continuum.
When a Blub programmer looks down the power continuum, he knows he
is looking down. Languages below Blub are obviously less powerful,
because they are missing some features he is used to.
But when a Blub programmer looks up the power continuum, he does not
realize he is looking up. What he sees are merely weird languages. He
thinks they are about equivalent in power to Blub, but with some extra
hairy stuff. Blub is good enough for him, since he thinks in Blub.
When we switch to the point of view of a programmer using a language
higher up the power continuum, however, we find that she in turn looks
down upon Blub, because it is missing some things she is used to.
Therefore understanding the differences in power between languages
requires understanding the most powerful ones.
NOTE
This slide is itself paraphrased from one of Paul Graham’s essays. (The full
quotation is too big to fit on one slide.)
The least abstract and therefore least powerful language is machine code.
One step above that is assembler, and one step above that are the lowest
level imperative languages, like C and Fortran. Everyone agrees on that.
Most people (but not all) would also agree that modern object-oriented
languages like Java and C#, scripting languages like awk and Perl and
multi-paradigm languages like Python and Ruby are more abstract and
more powerful than C and Fortran, but there is no general agreement on
their relative position on the continuum. However, almost everyone who
knows declarative programming languages agrees that they are more
abstract and more powerful than Java, C#, awk, Perl, Python and Ruby.
A large part of that extra power is the ability to offer many more
guarantees.
COMP90048 Declarative Programming Lecture 0 – 16 / 16
Introduction to Logic Programming
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 1
Introduction to
Logic Programming
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Introduction to Logic Programming
Logic programming
Imperative programming languages are based on the machine architecture
of John von Neumann, which executes a set of instructions step by step.
Functional programming languages are based on the lambda calculus of
Alonzo Church, in which functions map inputs to outputs.
Logic programming languages are based on the predicate calculus of
Gottlob Frege and the concept of a relation, which captures a relationship
among a number of individuals, and the predicate that relates them.
A function is a special kind of relation that can only be used in one
direction (inputs to outputs), and can only have one result. Relations do
not have these limitations.
While the first functional programming language was Lisp, implemented by
John McCarthy’s group at MIT in 1958, the first logic programming
language was Prolog, implemented by Alain Colmerauer’s group at
Marseille in 1971.
NOTE
Since the early 1980s, the University of Melbourne has been one of the
world’s top centers for research in logic programming.
Lee Naish designed and implemented MU-Prolog, and led the development
of its successor, NU-Prolog.
Zoltan Somogyi led the development of Mercury, and was one of its main
implementors.
NOTE
The name “Prolog” was chosen by Philippe Roussel as an abbreviation for
“programmation en logique”, which is French for “programming in logic”.
MU-Prolog and NU-Prolog are two closely-related dialects of Prolog.
There are many others, since most centers of logic programming research
in the 1980s implemented their own versions of the language.
The other main centers of logic programming research are in Leuven,
Belgium; Uppsala, Sweden; Madrid, Spain; and Las Cruces, New Mexico,
USA.
COMP90048 Declarative Programming Lecture 1 – 1 / 20
Introduction to Logic Programming
Relations
A relation specifies a relationship; for example, a family relationship. In
Prolog syntax,
parent(queen_elizabeth, prince_charles).
specifies (a small part of the) parenthood relation, which relates parents to
their children. This says that Queen Elizabeth is a parent of Prince
Charles.
The name of a relation is called a predicate. Predicates have no
directionality: it makes just as much sense to ask of whom is Queen
Elizabeth a parent as to ask who is Prince Charles’s parent. There is also
no promise that there is a unique answer to either of these questions.
COMP90048 Declarative Programming Lecture 1 – 2 / 20
Introduction to Logic Programming
Facts
A statement such as:
parent(queen_elizabeth, prince_charles).
is called a fact. It may take many facts to define a relation:
% (A small part of) the British Royal family
parent(queen_elizabeth, prince_charles).
parent(prince_philip, prince_charles).
parent(prince_charles, prince_william).
parent(prince_charles, prince_harry).
parent(princess_diana, prince_william).
parent(princess_diana, prince_harry).
...
Text between a percent sign (%) and end-of-line is treated as a comment.
COMP90048 Declarative Programming Lecture 1 – 3 / 20
Introduction to Logic Programming
Using Prolog
Most Prolog systems have an environment similar to GHCi. A file
containing facts like this should be written in a file whose name begins
with a lower-case letter and contains only letters, digits, and underscores,
and ends with “.pl”.
A source file can be loaded into Prolog by typing its filename (without the
.pl extension) between square brackets at the Prolog prompt (?-). Prolog
prints a message to say the file was compiled, and true to indicate it was
successful (user input looks like this):
?- [royals].
% royals compiled 0.00 sec, 8 clauses
true.
?-
NOTE
Some Prolog GUI environments provide other, more convenient, ways to
load code, such as menu items or drag-and-drop.
COMP90048 Declarative Programming Lecture 1 – 4 / 20
Introduction to Logic Programming
Queries
Once your code is loaded, you can use or test it by issuing queries at the
Prolog prompt. A Prolog query looks just like a fact. When written in a
source file and loaded into Prolog, it is treated as a true statement. At the
Prolog prompt, it is treated as a query, asking if the statement is true.
?- parent(prince_charles, prince_william).
true .
?- parent(prince_william, prince_charles).
false.
COMP90048 Declarative Programming Lecture 1 – 5 / 20
Introduction to Logic Programming
Variables
Each predicate argument may be a variable, which in Prolog begins with a
capital letter or underscore and follows with letters, digits, and underscores.
A query containing a variable asks if there exists a value for that variable
that makes that query true, and prints the value that makes it true.
If there is more than one answer to the query, Prolog prints them one at a
time, pausing to see if more solutions are wanted. Typing semicolon asks
for more solutions; just hitting enter (return) finishes without more
solutions.
This query asks: of whom Prince Charles is a parent?
?- parent(prince_charles, X).
X = prince_william ;
X = prince_harry.
COMP90048 Declarative Programming Lecture 1 – 6 / 20
Introduction to Logic Programming
Multiple modes
The same parenthood relation can be used just as easily to ask who is a
parent of Prince Charles or even who is a parent of whom. Each of these is
a different mode, based on which arguments are bound (inputs;
non-variables) and which are unbound (outputs; variables).
?- parent(X, prince_charles).
X = queen_elizabeth ;
X = prince_philip.
?- parent(X, Y).
X = queen_elizabeth,
Y = prince_charles ;
X = prince_philip,
Y = prince_charles ;
...
COMP90048 Declarative Programming Lecture 1 – 7 / 20
Introduction to Logic Programming
Compound queries
Queries may use multiple predicate applications (called goals in Prolog and
atoms in predicate logic). The simplest way to combine multiple goals is
to separate them with a comma. This asks Prolog for all bindings for the
variables that satisfy both (or all) of the goals. The comma can be read as
“and”. In relational algebra, this is called an inner join (but do not worry if
you do not know what that is).
?- parent(queen_elizabeth, X), parent(X, Y).
X = prince_charles,
Y = prince_william ;
X = prince_charles,
Y = prince_harry.
COMP90048 Declarative Programming Lecture 1 – 8 / 20
Introduction to Logic Programming
Rules
Predicates can be defined using rules as well as facts. A rule has the form
Head :- Body,
where Head has the form of a fact and Body has the form of a (possibly
compound) query. The :- is read “if”, and the clause means that the
Head is true if the Body is. For example
grandparent(X,Z) :- parent(X, Y), parent(Y, Z).
means “X is grandparent of Z if X is parent of Y and Y is parent of Z .”
Rules and facts are the two different kinds of clauses. A predicate can be
defined with any number of clauses of either or both kinds, intermixed in
any order.
COMP90048 Declarative Programming Lecture 1 – 10 / 20
Introduction to Logic Programming
Recursion
Rules can be recursive. Prolog has no looping constructs, so recursion is
widely used. Prolog does not have a well-developed a library of
higher-order operations, so recursion is used more in Prolog than in
Haskell, as you will see.
A person’s ancestors are their parents and the ancestors of their parents.
ancestor(Anc, Desc) :-
parent(Anc, Desc).
ancestor(Anc, Desc) :-
parent(Parent, Desc),
ancestor(Anc, Parent).
COMP90048 Declarative Programming Lecture 1 – 11 / 20
Introduction to Logic Programming
Equality
Equality in Prolog, written “=” and used as an infix operator, can be used
both to bind variables and to check for equality. Prolog is a
single-assignment language: once bound, a variable cannot be reassigned.
?- X = 7.
X = 7.
?- a = b.
false.
?- X = 7, X = a.
false.
?- X = 7, Y = 8, X = Y.
false.
COMP90048 Declarative Programming Lecture 1 – 12 / 20
Introduction to Logic Programming
Disjunction
Goals can be combined with disjunction (or) as well as conjunction (and).
Disjunction is written “;” and used as an infix operator. Conjunction
(“,”) has higher precedence (binds tighter) than disjunction, but
parentheses can be used to achieve the desired precedence.
Who are the children of Queen Elizabeth or Princess Diana?
?- parent(queen_elizabeth, X) ; parent(princess_diana, X).
X = prince_charles ;
X = prince_william ;
X = prince_harry.
COMP90048 Declarative Programming Lecture 1 – 13 / 20
Introduction to Logic Programming
Negation
Negation in Prolog is written “\+” and used as a prefix operator. Negation
has higher (tighter) precedence than both conjunction and disjunction. Be
sure to leave a space between the \+ and an open parenthesis.
Who are the parents of Prince William other than Prince Charles?
?- parent(X, prince_william), \+ X = prince_charles.
X = princess_diana.
Disequality in Prolog is written as an infix “\=”. So X \= Y is the same as
\+ X = Y.
?- parent(X, prince_william), X \= prince_charles.
X = princess_diana.
COMP90048 Declarative Programming Lecture 1 – 14 / 20
Introduction to Logic Programming
The Closed World Assumption
Prolog assumes that all true things can be derived from the program. This
is called the closed world assumption. Of course, this is not true for our
parent relation (that would require tens of billions of clauses!).
?- \+ parent(queen_elizabeth, princess_anne).
true.
but Princess Anne is a daughter of Queen Elizabeth. Our program simply
does not know about her.
So use negation with great care on predicates that are not complete, such
as parent.
COMP90048 Declarative Programming Lecture 1 – 16 / 20
Introduction to Logic Programming
Negation as failure
Prolog executes \+ G by first trying to prove G. If this fails, then \+ G
succeeds; if it succeeds, then \+ G fails. This is called negation as failure.
In Prolog, failing goals can never bind variables, so any variable bindings
made in solving G are thrown away when \+ G fails. Therefore, \+ G cannot
solve for any variables, and goals such as these cannot work properly.
Is there anyone of whom Queen Elizabeth is not a parent?
Is there anyone who is not Queen Elizabeth?
?- \+ parent(queen_elizabeth, X).
false.
?- X \= queen_elizabeth.
false.
COMP90048 Declarative Programming Lecture 1 – 17 / 20
Introduction to Logic Programming
Execution Order
The solution to this problem is simple: ensure all variables in a negated
goal are bound before the goal is executed.
Prolog executes goals in a query (and the body of a clause) from first to
last, so put the goals that will bind the variables in a negation before the
negation (or \=).
In this case, we can generate all people who are either parents or children,
and ask whether any of them is different from Queen Elizabeth.
?- (parent(X,_) ; parent(_,X)), X \= queen_elizabeth.
X = prince_philip ;
...
COMP90048 Declarative Programming Lecture 1 – 18 / 20
Introduction to Logic Programming
Datalog
The fragment of Prolog discussed so far, which omits data structures, is
called Datalog. It is a generalisation of what is provided by relational
databases. Many modern databases now provide Datalog features or use
Datalog implementation techniques.
capital(australia, canberra).
capital(france, paris).
...
continent(australia, australia).
continent(france, europe).
...
population(australia, 22_680_000).
population(france, 65_700_000).
...
COMP90048 Declarative Programming Lecture 1 – 19 / 20
Introduction to Logic Programming
Datalog Queries
What is the capital of France?
?- capital(france, Capital).
Capital = paris.
What are capitals of European countries?
?- continent(Country, europe), capital(Country, Capital).
Country = france,
Capital = paris.
What European countries have populations > 50,000,000?
?- continent(Country, europe), population(Country, Pop),
| Pop > 50_000_000.
Country = france,
Pop = 65700000.
COMP90048 Declarative Programming Lecture 1 – 20 / 20
Beyond Datalog
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 2
Beyond Datalog
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Beyond Datalog
Terms
In Prolog, all data structures are called terms. A term can be atomic or
compound, or it can be a variable. Datalog has only atomic terms and
variables.
Atomic terms include integers and floating point numbers, written as you
would expect, and atoms.
An atom begins with a lower case letter and follows with letters, digits and
underscores, for example a, queen_elizabeth, or banana.
An atom can also be written beginning and ending with a single quote,
and have any intervening characters. The usual character escapes can be
used, for example \n for newline, \t for tab, and \’ for a single quote.
For example: ’Queen Elizabeth’ or ’Hello, World!\n’.
COMP90048 Declarative Programming Lecture 2 – 1 / 21
Beyond Datalog
Compound Terms
In the syntax of Prolog, each compound term is a functor (sometimes
called function symbol) followed by zero or more arguments; if there are
any arguments, they are shown in parentheses, separated by commas.
Functors are Prolog’s equivalent of what Haskell calls data constructors,
and have the same syntax as atoms.
For example, a small tree with 1 at the root, an empty left branch, and 2
in the right branch, could be written as:
node(leaf, 1, node(leaf, 2, leaf))
Because Prolog is dynamically typed, each argument of a term can be any
term, and there is no need to declare types.
Prolog has special syntax for some functors, such as infix notation.
COMP90048 Declarative Programming Lecture 2 – 2 / 21
Beyond Datalog
Variables
A variable is also a term. It denotes a single unknown term.
A variable name begins with an upper case letter or underscore, followed
by any number of letters, digits, and underscores.
A single underscore _ is special: it specifies a different variable each time it
appears.
Prolog is a single-assignment language: a variable can only be bound
(assigned) once.
Because the arguments of a compound term can be any terms, and
variables are terms, variables can appear in terms.
For example f(A,A) denotes a term whose functor is f and whose two
arguments can be anything, as long as they are the same; f(_,_) denotes
a term whose functor is f and has any two arguments.
COMP90048 Declarative Programming Lecture 2 – 3 / 21
Beyond Datalog
List syntax
Prolog has a special syntax for lists.
The empty list is denoted by [].
A non-empty list is denoted by [H | T], where H is the head element of
the list and T is the tail.
For example, the list containing 1, 2 and 3 can be denoted by
[1 | [2 | [3 | []]]].
If multiple head elements are known, this can be shortened to
[1, 2 | [3 | []]].
If all elements are known, then this can be shortened to [1, 2, 3]
COMP90048 Declarative Programming Lecture 2 – 4 / 21
Beyond Datalog
Ground vs nonground terms
A term is a ground term if it contains no variables, and it is a nonground
term if it contains at least one variable.
3 and f(a, b) are ground terms.
Since Name and f(a, X) each contain at least one variable, they are
nonground terms.
If a variable occurs more than once in a term, then, just as in Algebra,
each occurrance must be bound to an identical term.
COMP90048 Declarative Programming Lecture 2 – 5 / 21
Beyond Datalog
Substitutions
A substitution is a mapping from variables to terms.
Applying a substitution to a term means consistently replacing all
occurrences of each variable in the map with the term it is mapped to.
Note that a substitution only replaces variables, never atomic or
compound terms.
For example, applying the substitution {X1 7→ leaf, X2 7→ 1, X3 7→ leaf}
to the term node(X1,X2,X3) yields the term node(leaf,1,leaf).
Since you can get node(leaf,1,leaf) from node(X1,X2,X3) by
applying a substitution to it, node(leaf,1,leaf) is an instance of
node(X1,X2,X3).
Any ground Prolog term has only one instance, while a nonground Prolog
terms has an infinite number of instances.
COMP90048 Declarative Programming Lecture 2 – 6 / 21
Beyond Datalog
Unification
The term that results from applying a substitution θ to a term t is
denoted tθ.
A term u is therefore an instance of term t if there is some substitution θ
such that u = tθ.
A substitution θ unifies two terms t and u if tθ = uθ.
Consider the terms f(X, b) and f(a, Y).
Applying a substitution {X 7→ a} to those two terms yields f(a, b) and
f(a, Y), which are not syntactically identical, so this substitution is not a
unifier.
On the other hand, applying the substitution {X 7→ a, Y 7→ b} to those
terms yields f(a, b) in both cases, so this substitution is a unifier.
COMP90048 Declarative Programming Lecture 2 – 7 / 21
Beyond Datalog
Recognising proper lists
A proper list is either empty ([]) or not ([X|Y]), in which case, the tail of
the list must be a proper list. We can define a predicate to recognise these.
proper_list([]).
proper_list([Head|Tail]) :-
proper_list(Tail).
?- [list].
Warning: list.pl:3:
Singleton variables: [Head]
% list compiled 0.00 sec, 2 clauses
true.
COMP90048 Declarative Programming Lecture 2 – 9 / 21
Beyond Datalog
Detour: singleton variables
Warning: list.pl:3:
Singleton variables: [Head]
The variable Head appears only once in this clause:
proper_list([Head|Tail]) :-
proper_list(Tail).
This often indicates a typo in the source code. For example, if Tail were
spelled Tial in one place, this would be easy to miss. But Prolog’s
singleton warning would alert us to the problem.
COMP90048 Declarative Programming Lecture 2 – 10 / 21
Beyond Datalog
Detour: singleton variables
In this case, there is no problem; to avoid the warning, we should begin
the variable name Head with an underscore, or just name the variable _.
proper_list([]).
proper_list([_Head|Tail]) :-
proper_list(Tail).
?- [list].
% list compiled 0.00 sec, 2 clauses
true.
General programming advice: always fix compiler warnings (if possible).
Some warnings may indicate a real problem, and you will not see them if
they’re lost in a sea of unimportant warnings. It is easier to fix a problem
when the compiler points it out than when you have to find it yourself.
COMP90048 Declarative Programming Lecture 2 – 11 / 21
Beyond Datalog
Append
Appending two lists is a common operation in Prolog. This is a built in
predicate in most Prolog systems, but could easily be implemented as:
append([], C, C).
append([A|B], C, [A|BC]) :-
append(B, C, BC).
?- append([a,b,c],[d,e],List).
List = [a, b, c, d, e].
COMP90048 Declarative Programming Lecture 2 – 12 / 21
Beyond Datalog
append is like proper_list
Compare the code for proper_list to the code for append:
proper_list([]).
proper_list([Head|Tail]) :-
proper_list(Tail).
append([], C, C).
append([A|B], C, [A|BC]) :-
append(B, C, BC).
This is common: code for a predicate that handles a term often follows
the structure of that term (as we will see in Haskell).
While the proper_list predicate is not very useful itself, it was worth
designing, as it gives a hint at the structure of other code that traverses
lists. Since types are not declared in Prolog, predicates like proper_list
can serve to indicate the notional type.
COMP90048 Declarative Programming Lecture 2 – 13 / 21
Beyond Datalog
Appending backwards
Unlike ++ in Haskell, append in Prolog can work in other modes:
?- append([1,2,3], Rest, [1,2,3,4,5]).
Rest = [4, 5].
?- append(Front, [3,4], [1,2,3,4]).
Front = [1, 2] ;
false.
?- append(Front,Back,[a,b,c]).
Front = [],
Back = [a, b, c] ;
Front = [a],
Back = [b, c] ;
Front = [a, b],
Back = [c] ;
Front = [a, b, c],
Back = [] ;
false.
COMP90048 Declarative Programming Lecture 2 – 14 / 21
Beyond Datalog
Length
The length/2 built-in predicate relates a list to its length:
?- length([a,b,c], Len).
Len = 3.
?- length(List, 3).
List = [_2956, _2962, _2968].
The _. . . terms are how Prolog prints out unbound variables. The number
reflects when the variable was created; because these variables are all
printed differently, we can tell they are all distinct variables.
[_2956, _2962, _2968] is a list of three distinct unbound variables, and
each unbound variable can be any term, so this can be any three-element
list, as specified by the query.
COMP90048 Declarative Programming Lecture 2 – 15 / 21
Beyond Datalog
Putting them together
How would we implement a predicate to take the front n elements of a
list in Prolog?
take(N,List,Front) should hold if Front is the first N elements of
List. So length(Front,N) should hold.
Also, append(Front, _, List) should hold. Then:
take(N, List, Front) :-
length(Front,N),
append(Front, _, List).
Prolog coding hint: think about checking if the result is correct rather
than computing it. That is, think of what instead of how.
Then you need to think about whether your code will work the ways you
want it to. We will return to that.
COMP90048 Declarative Programming Lecture 2 – 16 / 21
Beyond Datalog
Member
Here is list membership, two ways:
member1(Elt, List) :- append(_,[Elt|_], List).
member2(Elt, [Elt|_]).
member2(Elt, [_|Rest]) :- member2(Elt, Rest).
These behave the same, but the second is a bit more efficient because the
first builds and ignores the list of elements before Elt in List, and the
second does not.
Note the recursive version does not exactly match the structure of our
earlier proper_list predicate. This is because Elt is never a member of
the empty list, so we do not need a clause for []. In Prolog, we do not
need to specify when a predicate should fail; only when it should succeed.
We also have two cases to consider when the list is non-empty (like
Haskell in this respect).
COMP90048 Declarative Programming Lecture 2 – 18 / 21
Beyond Datalog
Arithmetic
In Prolog, terms like 6 * 7 are just data structures, and = does not
evaluate them, it just unifies them.
The built-in predicate is/2 (an infix operator) evaluates expressions.
?- X = 6 * 7.
X = 6*7.
?- X is 6 * 7.
X = 42.
COMP90048 Declarative Programming Lecture 2 – 19 / 21
Beyond Datalog
Arithmetic modes
Use is/2 to evaluate expression
square(N, N2) :- N2 is N * N.
Unfortunately, square only works when the first argument is bound. This
is because is/2 only works if its second argument is ground.
?- square(5, X).
X = 25.
?- square(X, 25).
ERROR: is/2: Arguments are not sufficiently instantiated
?- 25 is X * X.
ERROR: is/2: Arguments are not sufficiently instantiated
Later we shall see how to write code to do arithmetic in different modes.
COMP90048 Declarative Programming Lecture 2 – 20 / 21
Beyond Datalog
Arithmetic
Prolog provides the usual arithmetic operators, including:
+ - * add, subtract, multiply
/ division (may return a float)
// integer division (rounds toward 0)
mod modulo (result has same sign as second argument)
- unary minus (negation)
integer float coersions (not operators)
More arithmetic predicates (infix operators; both arguments must be
ground expressions):
< =< less, less or equal (note!)
> >= greater, greater or equal
=:= =\= equal, not equal (only numbers)
COMP90048 Declarative Programming Lecture 2 – 21 / 21
Logic and Resolution
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 3
Logic and Resolution
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Logic and Resolution
Interpretations
In the mind of the person writing a logic program,
each constant (atomic term) stands for an entity in the “domain of
discourse” (world of the program);
each functor (function symbol of arity n where n > 0) stands for a
function from n entities to one entity in the domain of discourse; and
each predicate of arity n stands for a particular relationship between n
entities in the domain of discourse.
This mapping from the symbols in the program to the world of the
program (which may be the real world or some imagined world) is called
an interpretation.
The obvious interpretation of the atomic formula
parent(queen_elizabeth, prince_charles) is that Queen Elizabeth
II is a parent of Prince Charles, but other interpretations are also possible.
NOTE
Another interpretation would use the function symbol queen_elizabeth
to refer to George W Bush, the function symbol prince_charles to refer
to Barack Obama, and the predicate symbol parent to refer to the notion
“succeeded by as US President”. However, any programmer using this
interpretation, or pretty much any interpretation other than the obvious
one, would be guilty of using a horribly misleading programming style.
Terms using non-meaningful names such as f(g, h) do not lead readers
to expect a particular interpretation, so these can have many different
non-misleading interpretations.
COMP90048 Declarative Programming Lecture 3 – 1 / 21
Logic and Resolution
Two views of predicates
As the name implies, the main focus of the predicate calculus is on
predicates.
You can think of a predicate with n arguments in two equivalent ways.
You can view the predicate as a function from all possible
combinations of n terms to a truth value (i.e. true or false).
You can view the predicate as a set of tuples of n terms. Every tuple
in this set is implicitly mapped to true, while every tuple not in this
set is implicitly mapped to false.
The task of a predicate definition is to define the mapping in the first
view, or equivalently, to define the set of tuples in the second view.
COMP90048 Declarative Programming Lecture 3 – 2 / 21
Logic and Resolution
The meaning of clauses
The meaning of the clause
grandparent(A, C) :- parent(A, B), parent(B, C).
is: for all the terms that A and C may stand for, A is a grandparent of C if
there is a term B such that A is a parent of B and B is a parent of C.
In mathematical notation:
∀A∀C : grandparent(A,C)← ∃B : parent(A,B) ∧ parent(B,C)
The variables appearing in the head are universally quantified over the
entire clause, while variables appearing only in the body are existentially
quantified over the body.
NOTE
∀ is “forall”, the universal quantifier, while ∃ is “there exists”, the
existential quantifier. The sign ∧ denotes the logical “and” operation,
while the sign ∨ denotes the logical “or” operation, and the sign ¬ denotes
the logical “not” operation.
COMP90048 Declarative Programming Lecture 3 – 3 / 21
Logic and Resolution
The meaning of predicate definitions
A predicate is defined by a finite number of clauses, each of which is in the
form of an implication. A fact such as parent(queen_elizabeth,
prince_charles) represents this implication:
∀A∀B : parent(A,B)←
(A = queen_elizabeth ∧ B = prince_charles)
To represent the meaning of the predicate, create a disjunction of the
bodies of all the clauses:
∀A∀B : parent(A,B)←
(A = queen_elizabeth ∧ B = prince_charles) ∨
(A = prince_philip ∧ B = prince_charles) ∨
(A = prince_charles ∧ B = prince_william) ∨
(A = prince_charles ∧ B = prince_harry) ∨
(A = princess_diana ∧ B = prince_william) ∨
(A = princess_diana ∧ B = prince_harry)
NOTE
Obviously, this definition of the parent relationship is the correct one only
if you restrict the universe of discourse to this small set of people.
COMP90048 Declarative Programming Lecture 3 – 4 / 21
Logic and Resolution
The closed world assumption
To implement the closed world assumption, we only need to make the
implication arrow go both ways (if and only if):
∀A∀B : parent(A,B)↔
(A = queen_elizabeth ∧ B = prince_charles) ∨
(A = prince_philip ∧ B = prince_charles) ∨
(A = prince_charles ∧ B = prince_william) ∨
(A = prince_charles ∧ B = prince_harry) ∨
(A = princess_diana ∧ B = prince_william) ∨
(A = princess_diana ∧ B = prince_harry)
This means that A is not a parent of B unless they are one of the listed
cases.
Adding the reverse implication this way creates the Clark completion of
the program.
COMP90048 Declarative Programming Lecture 3 – 5 / 21
Logic and Resolution
Semantics of logic programs
A logic program P consists of a set of predicate definitions. The semantics
of this program (its meaning) is the set of its logical consequences as
ground atomic formulas.
A ground atomic formula a is a logical consequence of a program P if P
makes a true.
A negated ground atomic formula ¬a, written in Prolog as \+a, is a logical
consequence of P if a is not a logical consequence of P.
For most logic programs, the set of ground atomic formulas it entails is
infinite (as is the set it does not entail). As logicians, we do not worry
about this any more than a mathematician worries that there are an
infinite number of solutions to a + b = c.
COMP90048 Declarative Programming Lecture 3 – 6 / 21
Logic and Resolution
Finding the semantics
You can find the semantics of a logic program by working backwards.
Instead of reasoning from a query to find a satisfying substitution, you
reason from the program to find what ground queries will succeed.
The immediate consequence operator TP takes a set of ground unit
clauses C and produces the set of ground unit clauses implied by C
together with the program P.
This always includes all ground instances of all unit clauses in P. Also, for
each clause H : −G1, . . . ,Ga. in P, if C contains instances of G1, . . .Gn,
then the corresponding instance of H is also in the result.
Eg, if P = {q(X ,Z ) : −p(X ,Y ), p(Y ,Z )} and
C = {p(a, b).p(b, c).p(c, d).}, then TP(C) = {q(a, c).q(b, d).}.
The semantics of program P is always TP(TP(TP(· · · (∅) · · · )))
(TP applied infinitely many times to the empty set).
COMP90048 Declarative Programming Lecture 3 – 7 / 21
Logic and Resolution
Procedural Interpretation
The logical reading of the clause
grandparent(X, Z) :-
parent(X, Y), parent(Y, Z).
says “for all X, Y, Z, if X is parent of Y and Y is parent of Z, then X is
grandparent of Z”.
The procedural reading says “to show that X is a grandparent of Z, it is
sufficient to show that X is a parent of Y and Y is a parent of Z”.
SLD resolution, used by Prolog, implements this strategy.
COMP90048 Declarative Programming Lecture 3 – 9 / 21
Logic and Resolution
SLD Resolution
The consequences of a logic program are determined through a simple but
powerful deduction strategy called resolution.
SLD resolution is an efficient version of resolution. The basic idea is: given
this program, to show this goal is true
q :- b1a, b1b.
q :- b2a, b2b.
...
?- p, q, r.
it is sufficient to show any of
?- p, b1a, b1b, r.
?- p, b2a, b2b, r.
...
COMP90048 Declarative Programming Lecture 3 – 10 / 21
Logic and Resolution
SLD resolution in action
E.g., to determine if Queen Elizabeth is Prince Harry’s grandparent:
?- grandparent(queen_elizabeth, prince_harry).
with this program
grandparent(X, Z) :-
parent(X, Y), parent(Y, Z).
we unify query goal grandparent(queen_elizabeth, prince_harry)
with clause head grandparent(X, Z), apply the resulting substitution to
the clause, yielding the resolvent. Since the goal is identical to the
resolvent head, we can replace it with the resolvent body, leaving:
?- parent(queen_elizabeth, Y), parent(Y, prince_harry).
COMP90048 Declarative Programming Lecture 3 – 11 / 21
Logic and Resolution
SLD resolution can fail
Now we must pick one of these goals to resolve; we select the second.
The program has several clauses for parent, but only two can successfully
resolve with parent(Y, prince_harry):
parent(prince_charles, prince_harry).
parent(princess_diana, prince_harry).
We choose the second. After resolution, we are left with the query (note
the unifying substitution is applied to both the selected clause and the
query):
?- parent(queen_elizabeth, princess_diana).
No clause unifies with this query, so resolution fails. Sometimes, it may
take many resolution steps to fail.
COMP90048 Declarative Programming Lecture 3 – 12 / 21
Logic and Resolution
SLD resolution can succeed
Selecting the second of these matching clauses led to failure:
parent(prince_charles, prince_harry).
parent(princess_diana, prince_harry).
This does not mean we are through: we must backtrack and try the first
matching clause. This leaves
?- parent(queen_elizabeth, prince_charles).
There is one matching program clause, leaving nothing more to prove.
The query succeeds.
COMP90048 Declarative Programming Lecture 3 – 13 / 21
Logic and Resolution
Resolution
This derivation can be shown as an SLD tree:
failure
grandparent(queen_elizabeth, prince_harry).
parent(queen_elizabeth, Y), parent(Y, prince_harry).
parent(queen_elizabeth,
prince_charles).
parent(queen_elizabeth,
princess_diana).
success
COMP90048 Declarative Programming Lecture 3 – 14 / 21
Logic and Resolution
Order of execution
The order in which goals are resolved and the order in which clauses are
tried does not matter for correctness (in pure Prolog), but it does matter
for efficiency. In this example, resolving parent(queen_elizabeth, Y)
before parent(Y, prince_harry) is more efficient, because there is only
one clause matching the former, and two matching the latter.
parent(queen_elizabeth, Y), parent(Y, prince_harry).
success
grandparent(queen_elizabeth, prince_harry).
parent(prince_charles, prince_harry).
COMP90048 Declarative Programming Lecture 3 – 15 / 21
Logic and Resolution
SLD resolution in Prolog
At each resolution step we must make two decisions:
1 which goal to resolve
2 which clauses matching the selected goal to pursue
(though there may only be one choice for either or both).
Our procedure was somewhat haphazard when decisions needed to be
made. For pure logic programming, this does not matter for correctness.
All goals will need to be resolved eventually; which order they are resolved
in does not change the answers. All matching clauses may need to be
tried; the order in which we try them determines the order solutions are
found, but not which solutions are found.
Prolog always selects the first goal to resolve, and always selects the first
matching clause to pursue first. This gives the programmer more certainty,
and control, over execution.
COMP90048 Declarative Programming Lecture 3 – 17 / 21
Logic and Resolution
Backtracking
When there are multiple clauses matching a goal, Prolog must remember
which one to go back to if necessary. It must be able to return the
computation to the state it was in when the first matching clause was
selected, so that it can return to that state and try the next matching
clause. This is all done with a choicepoint.
When a goal fails, Prolog backtracks to the most recent choicepoint,
removing all variable bindings made since the choicepoint was created,
returning those variables to their unbound state. Then Prolog begins
resolution with the next matching clause, repeating the process until
Prolog detects that there are no more matching clauses, at which point it
removes that choicepoint. Subsequent failures will then backtrack to the
next most recent choicepoint.
COMP90048 Declarative Programming Lecture 3 – 18 / 21
Logic and Resolution
Indexing
Indexing can greatly improve Prolog efficiency
Most Prolog systems will automatically create an index for a predicate such
as parent/2 (Prolog uses name/arity to refer to predicates) with multiple
clauses the heads of which have distinct constants or functors. This means
that, for a call with the first argument bound, Prolog will immediately
jump to the first clause that matches. If backtracking occurs, the index
allows Prolog to jump straight to the next clause that matches, and so on.
If the first argument is unbound, then all clauses will have to be tried.
COMP90048 Declarative Programming Lecture 3 – 19 / 21
Logic and Resolution
Indexing
If some clauses have variables in the first argument of the head, those
clauses will be tried at the appropriate time regardless of the call. Indexing
changes performance, not behaviour. Consider:
p(a, z).
p(b, y).
p(X, X).
p(a, x).
For the call p(I, J), all clauses will be tried, in order. For p(a, J), the
first clause will be tried, then the third, then fourth. For p(b, J), the
second, then third, clause will be tried. For p(c, J), only the third clause
will be tried.
COMP90048 Declarative Programming Lecture 3 – 20 / 21
Logic and Resolution
Indexing
Some Prolog systems, such as SWI Prolog, will construct indices for
arguments other than the first. For parent/2, SWI Prolog will index on
both arguments, so finding the children of a parent or parents of a child
both benefit from indexing.
Just as important as jumping directly to the first matching clause,
indexing tells Prolog when no further clauses could possibly match the
goal, allowing it to remove the choicepoint, or even to avoid creating the
choicepoint in the first place. Even with only two clauses, such as for
append/3, indexing can substantially improve performance.
COMP90048 Declarative Programming Lecture 3 – 21 / 21
Understanding and Debugging Prolog code
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 4
Understanding and Debugging
Prolog code
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Understanding and Debugging Prolog code
List Reverse
To reverse a list, put the first element of the list at the end of the reverse
of the tail of the list.
rev1([], []).
rev1([A|BC], CBA) :-
rev1(BC, CB),
append(CB, [A], CBA).
reverse/2 is an SWI Prolog built-in, so we use a different name to avoid
conflict.
NOTE
This version of reverse/2 is called naive reverse, because it has quadratic
complexity. We’ll see a more efficient version a little later.
COMP90048 Declarative Programming Lecture 4 – 1 / 22
Understanding and Debugging Prolog code
List Reverse
The mode of a Prolog goal says which arguments are bound (inputs) and
which are unbound (outputs) when the predicate is called.
rev1/2 works as intended when the first argument is ground and the
second is free, but not for the opposite mode.
?- rev1([a,b,c], Y).
Y = [c, b, a].
?- rev1(X, [c,b,a]).
X = [a, b, c] ;
Prolog hangs at this point. We will use the Prolog debugger to understand
why. For now, hit control-C and then ’a’ to abort.
COMP90048 Declarative Programming Lecture 4 – 2 / 22
Understanding and Debugging Prolog code
The Prolog Debugger
To understand the debugger, you will need to understand the Byrd box
model. Think of goal execution as a box with a port for each way to enter
and exit.
A conventional language has only one way to enter and one way to exit;
Prolog has two of each.
The four debugger ports are:
initial entry
final failure
call
redo
exit
fail
successful completion
backtrack into goal
Turn on debugger with trace, and off with nodebug, at the Prolog
prompt.
COMP90048 Declarative Programming Lecture 4 – 3 / 22
Understanding and Debugging Prolog code
Using the Debugger
The debugger prints the current port, execution depth, and goal (with the
current variable bindings) at each step.
?- trace, rev1([a,b], Y).
Call: (7) rev1([a, b], _12717) ? creep
Call: (8) rev1([b], _12834) ? creep
Call: (9) rev1([], _12834) ? creep
Exit: (9) rev1([], []) ? creep
Call: (9) lists:append([], [b], _12838) ? creep
Exit: (9) lists:append([], [b], [b]) ? creep
Exit: (8) rev1([b], [b]) ? creep
Call: (8) lists:append([b], [a], _12717) ? creep
Exit: (8) lists:append([b], [a], [b, a]) ? creep
Exit: (7) rev1([a, b], [b, a]) ? creep
Y = [b, a].
“lists:” in front of append is a module name.
COMP90048 Declarative Programming Lecture 4 – 4 / 22
Understanding and Debugging Prolog code
Reverse backward
Now try the “backwards” mode of rev1/2. We shall use a smaller test
case to keep it manageable.
?- trace, rev1(X, [a]).
Call: (7) rev1(_11553, [a]) ? creep
Call: (8) rev1(_11661, _11671) ? creep
Exit: (8) rev1([], []) ? creep
Call: (8) lists:append([], [_11660], [a]) ? creep
Exit: (8) lists:append([], [a], [a]) ? creep
Exit: (7) rev1([a], [a]) ? creep
X = [a] ;
COMP90048 Declarative Programming Lecture 4 – 5 / 22
Understanding and Debugging Prolog code
Reverse backward, continued
after showing the first solution, Prolog goes on forever like this:
Redo: (8) rev1(_11661, _11671) ? creep
Call: (9) rev1(_11664, _11674) ? creep
Exit: (9) rev1([], []) ? creep
Call: (9) lists:append([], [_11663], _11678) ? creep
Exit: (9) lists:append([], [_11663], [_11663]) ? creep
Exit: (8) rev1([_11663], [_11663]) ? creep
Call: (8) lists:append([_11663], [_11660], [a]) ? creep
Fail: (8) lists:append([_11663], [_11660], [a]) ? creep
Redo: (9) rev1(_11664, _11674) ? creep
Call: (10) rev1(_11667, _11677) ? creep
Exit: (10) rev1([], []) ? creep
Call: (10) lists:append([], [_11666], _11681) ? creep
Exit: (10) lists:append([], [_11666], [_11666]) ? creep
Exit: (9) rev1([_11666], [_11666]) ? creep
Call: (9) lists:append([_11666], [_11663], _11684) ? creep
Exit: (9) lists:append([_11666], [_11663], [_11666, _11663]) ? creep
Exit: (8) rev1([_11663, _11666], [_11666, _11663]) ? creep
Call: (8) lists:append([_11666, _11663], [_11660], [a])
Fail: (8) lists:append([_11666, _11663], [_11660], [a])
...
COMP90048 Declarative Programming Lecture 4 – 6 / 22
Understanding and Debugging Prolog code
Infinite backtracking loop
rev1([], []).
rev1([A|BC], CBA) :-
rev1(BC, CB),
append(CB, [A], CBA).
The problem is that the goal rev1(X,[a]), resolves to the goal rev1(BC,
CB), append(CB, [A], [a]). The call rev1(BC, CB) produces an
infinite backtracking sequence of solutions {BC 7→ [], CB 7→ []},
{BC 7→ [Z], CB 7→ [Z]}, {BC 7→ [Y,Z], CB 7→ [Z,Y]}, . . .. For each of
these solutions, we call append(CB, [A], [a]).
append([], [A], [a]) succeeds, with {A 7→ [a]}. However,
append([Z], [A], [a]) fails, as does this goal for all following solutions
for CB. This is an infinite backtracking loop.
COMP90048 Declarative Programming Lecture 4 – 7 / 22
Understanding and Debugging Prolog code
Infinite backtracking loop
We could fix this problem by executing the body goals in the other order:
rev2([], []).
rev2([A|BC], CBA) :-
append(CB, [A], CBA),
rev2(BC, CB).
But this definition does not work in the forward direction:
?- rev2(X, [a,b]).
X = [b, a] ;
false.
?- rev2([a,b], Y).
Y = [b, a] ;
^CAction (h for help) ? abort
% Execution Aborted
NOTE
Prolog uses a fixed left-to-right execution order, so neither order works for both
directions.
COMP90048 Declarative Programming Lecture 4 – 8 / 22
Understanding and Debugging Prolog code
Working in both directions
The solution is to ensure that when rev1 is called, the first argument is
always bound to a list. We do this by observing that the length of a list
must always be the same as that of its reverse. When same_length/2
succeeds, both arguments are bound to lists of the same fixed length.
rev3(ABC, CBA) :-
same_length(ABC, CBA),
rev1(ABC, CBA).
same_length([], []).
same_length([_|Xs], [_|Ys]) :-
same_length(Xs, Ys).
?- rev3(X, [a,b]).
X = [b, a].
?- rev3([a,b], Y).
Y = [b, a].
COMP90048 Declarative Programming Lecture 4 – 9 / 22
Understanding and Debugging Prolog code
More on the Debugger
Some useful debugger commands:
h display debugger help
c creep to the next port (also space, enter)
s skip over goal; go straight to exit or fail port
r back to initial call port of goal, undoing all bindings done
since starting it;
a abort whole debugging session
+ set spypoint (like breakpoint) on this pred
- remove spypoint from this predicate
l leap to the next spypoint
b pause this debugging session and enter a “break level,”
giving a new Prolog prompt; end of file reenters debugger
COMP90048 Declarative Programming Lecture 4 – 11 / 22
Understanding and Debugging Prolog code
More on the Debugger
Built-in predicates for controlling the debugger:
spy(Predspec) Place a spypoint on Predspec, which can be a
name/arity pair, or just a predicate name.
nospy(Predspec) Remove the spypoint from Predspec.
trace Turn on the debugger
debug Turn on the debugger and leap to first spypoint
nodebug Turn off the debugger
A “Predspec” is a predicate name or name/arity
COMP90048 Declarative Programming Lecture 4 – 12 / 22
Understanding and Debugging Prolog code
Using the debugger
Note the r (retry) debugger command restarts a goal from the beginning,
“time travelling” back to the time when starting to execute that goal.
The s (skip) command skips forward in time, over the whole execution of
a goal, to its exit or fail port.
This leads to a quick way of tracking down most bugs:
1 When you arrive at a call or redo port: skip.
2 If you come to an exit port with the correct results (or a correct fail
port): creep.
3 If you come to an incorrect exit or fail port: retry, then creep.
Eventually you will find a clause that has the right input and wrong output
(or wrong failure); this is the bug. This will not help find infinite recursion,
though.
COMP90048 Declarative Programming Lecture 4 – 13 / 22
Understanding and Debugging Prolog code
Spypoints
For larger computations, it may take some time to get to the part of the
computation where the bug lies. Usually, you will have a good idea, or at
least a few good guesses, which predicates you suspect of being buggy
(usually the predicates you have edited most recently). In cases of infinite
recursion you may suspect certain predicates of being involved in the loop.
In these cases, spypoints will be helpful. Like a breakpoint in most
debuggers, when Prolog reaches any port of a predicate with a spypoint
set, Prolog stops and shows the port. The l (leap) command tells Prolog
to run quietly until it reaches a spypoint. Use the spy(pred) goal at the
Prolog prompt to set a spypoint on the named predicate, nospy(pred) to
remove one. You can also add a spypoint on the predicate of the current
debugger port with the + command, and remove it with -.
COMP90048 Declarative Programming Lecture 4 – 14 / 22
Understanding and Debugging Prolog code
Documenting Prolog Code
Your code files should have two levels of documentation – file level
documentation and predicate level comments. Each file should start with
comments that outline: the purpose of the file; its author; the date at
which the code was written; and a brief summary of what the code does
and any underlying rationale.
Comments should be provided above all significant and non-trivial
predicates in a consistent format. These comments should identify: the
meaning of each argument; what the predicate does; and the modes in
which the predicate is designed to operate.
An excellent resource on coding standards in Prolog is the paper “Coding
guidelines for Prolog” by Covington et al. (2011).
COMP90048 Declarative Programming Lecture 4 – 15 / 22
Understanding and Debugging Prolog code
Predicate level documentation
The following is an example of predicate level documentation from
Covington et al. (2011). This predicate removes duplicates from a list.
%% remove_duplicates(+List, -Result)
%
% Removes the duplicates in List, giving Result
% Elements are considered to match if they can be
% unified with each other; thus, a partly
% uninstantiated element may become further
% instantiated during testing. If several elements
% match, the last of them is preserved.
Predicate arguments are prefaced with a: + to indicate that the argument
is an input and must be instantiated to a term that is not an unbound
variable; - if the argument is an output and may be an unbound variable;
or a ? to indicate that the argument can be either an input or an output.
COMP90048 Declarative Programming Lecture 4 – 16 / 22
Understanding and Debugging Prolog code
Managing nondeterminism
This is a common mistake in defining factorial:
fact(0, 1).
fact(N, F) :-
N1 is N - 1,
fact(N1, F1),
F is N * F1.
?- fact(5, F).
F = 120 ;
ERROR: Out of local stack
fact(5,F) has only one solution, why was Prolog looking for another?
COMP90048 Declarative Programming Lecture 4 – 17 / 22
Understanding and Debugging Prolog code
Managing nondeterminism
This is a common mistake in defining factorial:
fact(0, 1).
fact(N, F) :-
N1 is N - 1,
fact(N1, F1),
F is N * F1.
?- fact(5, F).
F = 120 ;
ERROR: Out of local stack
fact(5,F) has only one solution, why was Prolog looking for another?
COMP90048 Declarative Programming Lecture 4 – 17 / 22
Understanding and Debugging Prolog code
Correctness
The second clause promises that for all n, n! = n× (n− 1)!. This is wrong
for n < 1.
Even if one clause applies, later clauses are still tried. After finding 0! = 1,
Prolog thinks 0! = 0×−1!; tries to compute −1!,−2!, . . .
The simple solution is to ensure each clause is a correct (part of the)
definition.
fact(0, 1).
fact(N, F) :-
N > 0,
N1 is N - 1,
fact(N1, F1),
F is F1 * N.
COMP90048 Declarative Programming Lecture 4 – 18 / 22
Understanding and Debugging Prolog code
Choicepoints
This definition is correct, but it could be more efficient.
When a clause succeeds but there are later clauses that could possibly
succeed, Prolog will leave a choicepoint so it can later backtrack and try
the later clause.
In this case, backtracking to the second clause will fail unless N > 0. This
test is quick. However, as long as the choicepoint exists, it inhibits the
very important last call optimisation (discussed later). Therefore, where
efficiency matters, it is important to make your recursive predicates not
leave choicepoints when they should be deterministic.
In this case, N = 0 and N > 0 are mutually exclusive, so at most one
clause can apply, so fact/2 should not leave a choicepoint.
NOTE
Actually, you will probably never take the factorial of a very large number,
so the loss in efficiency from this definition is unlikely to make a detectable
difference. For deeper arithmetic recursions or other recursions that are
not structural inductions, however, this technique is useful.
COMP90048 Declarative Programming Lecture 4 – 19 / 22
Understanding and Debugging Prolog code
If-then-else
We can avoid the choicepoint with Prolog’s if-then-else construct:
fact(N, F) :-
( N =:= 0 ->
F = 1
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is F1 * N
).
The -> is treated like a conjunction (,), except that when it is crossed,
any alternative solutions of the goal before the ->, as well as any
alternatives following the ; are forgotten. Conversely, if the goal before
the -> fails, then the goal after the ; is tried. So this is deterministic
whenever both the code between -> and ;, and the code after the ;, are.
COMP90048 Declarative Programming Lecture 4 – 20 / 22
Understanding and Debugging Prolog code
If-then-else caveats
However, you should prefer indexing (discussed next time) and avoid
if-then-else, when you have a choice. If-then-else usually leads to code
that will not work smoothly in multiple modes. For example, append could
be written with if-then-else:
ap(X, Y, Z) :-
( X = [] ->
Z = Y
; X = [U|V],
ap(V, Y, W),
Z = [U|W]
).
This may appear correct, and may follow the logic you would use to code
it in another language, but it is not appropriate for Prolog.
COMP90048 Declarative Programming Lecture 4 – 21 / 22
Understanding and Debugging Prolog code
If-then-else caveats
With that definition of ap:
?- ap([a,b,c], [d,e], L).
L = [a, b, c, d, e].
?- ap(L, [d,e], [a,b,c,d,e]).
false.
?- ap(L, M, [a,b,c,d,e]).
L = [],
M = [a, b, c, d, e].
Because the if-then-else commits to binding the first argument to [] when
it can, this version of append will not work correctly unless the first
argument is bound when append is called.
COMP90048 Declarative Programming Lecture 4 – 22 / 22
Tail Recursion
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 5
Tail Recursion
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Tail Recursion
Tail recursion
A predicate (or function, or procedure, or method, or. . . ) is tail recursive if
the only recursive call on any execution of that predicate is the last code
executed before returning to the caller. For example, the usual definition
of append/3 is tail recursive, but rev1/2 is not:
append([], C, C).
append([A|B], C, [A|BC]) :-
append(B, C, BC).
rev1([], []).
rev1([A|BC], CBA) :-
rev1(BC, CB),
append(CB, [A], CBA).
COMP90048 Declarative Programming Lecture 5 – 1 / 21
Tail Recursion
Tail recursion optimisation
Like most declarative languages, Prolog performs tail recursion
optimisation (TRO). This is important for declarative languages, since
they use recursion more than non-declarative languages. TRO makes
recursive predicates behave as if they were loops.
Note that TRO is more often directly applicable in Prolog than in other
languages because more Prolog code is tail recursive. For example, while
append/3 in Prolog is tail recursive, an equivalent in most other languages
would not be. In most languages, after recursively calling append, you
would have to construct a new list cell whose head is the head of the first
input list and whose tail is the result of the recursive call.
In Prolog, that list cell is constructed in the head of the clause, making
use of the fact that in Prolog we can use an unbound variable in
constructing a term, and only bind it later.
COMP90048 Declarative Programming Lecture 5 – 2 / 21
Tail Recursion
The stack
To understand TRO, it is important to understand how
programming languages (not just Prolog) implement call
and return using a stack. While a is executing, it stores
its local variables, and where to return to when finished,
in a stack frame or activation record. When a calls b,
it creates a fresh stack frame for b’s local variables and
return address, preserving a’s frame, and similarly when
b calls c, as shown to the right.
But if all b will do after calling c is return to a, then
there is no need to preserve its local variables. Prolog
can release b’s frame before calling c, as shown
c
Growth
a
b
c
Growth
a
COMP90048 Declarative Programming Lecture 5 – 3 / 21
Tail Recursion
TRO and choicepoints
Tail recursion optimisation is a special case of last call
optimisation where the last call is recursive. This is espe-
cially beneficial, since recursion is used to replace looping.
Without TRO, this would require a stack frame for each
iteration, and would quickly exhaust the stack. With
TRO, tail recursive predicates execute in constant stack
space, just like a loop.
However, if b leaves a choicepoint, it sits on the stack
above b’s frame, “freezing” that and all earlier frames so
that they are not reclaimed. This is necessary because
when Prolog backtracks to that choicepoint, b’s argu-
ments must be ready to try the next matching clause for
b. The same is true if c or any predicate called later
leaves a choicepoint, but choicepoints before the call to
b do not interfere.
Choicepoint
Growth
a
b
c
COMP90048 Declarative Programming Lecture 5 – 4 / 21
Tail Recursion
Making code tail recursive
Our factorial predicate was not tail recursive, as the last thing it does is
perform arithmetic.
fact(N, F) :-
( N =:= 0 ->
F = 1
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is F1 * N
).
Note that Prolog’s if-then-else construct does not leave a choicepoint. A
choicepoint is created, but is removed as soon as the condition succeeds or
fails. So fact would be subject to TRO, if only it were tail recursive.
COMP90048 Declarative Programming Lecture 5 – 5 / 21
Tail Recursion
Adding an accumulator
We make factorial tail recursive by introducing an accumulating
parameter, or just an accumulator. This is an extra parameter to the
predicate that holds a partially computed result.
Usually the base case for the recursion will specify that the partially
computed result is actually the result. The recursive clause usually
computes more of the partially computed result, and passes this in the
recursive goal.
The key to getting the implementation correct is specifying what the
accumulator means and how it relates to the final result. To see how to
add an accumulator, determine what is done after the recursive call, and
then respecify the predicate so it performs this task, too.
COMP90048 Declarative Programming Lecture 5 – 6 / 21
Tail Recursion
Adding an accumulator
For factorial, we compute fact(N1, F1), F is F1 * N last, so the tail
recursive version will need to perform the multiplication too. We must
define a predicate fact1(N, A, F) so that F is A times the factorial of N.
In most cases, it is not difficult to see how to transform the original
definition to the tail recursive one.
fact(N, F) :-
( N =:= 0 ->
F = 1
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is F1 * N
).
fact(N, F) :- fact1(N, 1, F).
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
A1 is A * N,
fact1(N1, A1, F)
).
COMP90048 Declarative Programming Lecture 5 – 7 / 21
Tail Recursion
Adding an accumulator
Finally, define the original predicate in terms of the new one. Again, it is
usually easy to see how to do that.
Another way to think about writing a tail recursive implementation of a
predicate is to realise that it will essentially be a loop, so think of how you
would write it as a while loop, and then write that loop in Prolog.
A = 1;
while (N > 0) {
A *= N;
N--;
}
if (N == 0) return A;
else FAIL;
fact(N, F) :- fact1(N, 1, F).
fact1(N, A, F) :-
( N > 0 ->
A1 is A * N,
N1 is N - 1,
fact1(N1, A1, F)
; N =:= 0 ->
F = A
).
COMP90048 Declarative Programming Lecture 5 – 8 / 21
Tail Recursion
Transformation
Another approach is to systematically transform the non-tail recursive
version into an equivalent tail recursive predicate. Start by defining a
predicate to do the work of the recursive call to fact/2 and everything
following it. Then replace the call to fact(N, F2) by the definition of
fact/2. This is called unfolding.
fact1(N, A, F) :-
fact(N, F2),
F is F2 * A
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1
; N > 0,
N1 is N - 1,
fact(N1, F1),
F2 is F1 * N
),
F is F2 * A.
COMP90048 Declarative Programming Lecture 5 – 11 / 21
Tail Recursion
Transformation
Next we move the final goal into both the then and else branches.
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1
; N > 0,
N1 is N - 1,
fact(N1, F1),
F2 is F1 * N
),
F is F2 * A.
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1,
F is F2 * A
; N > 0,
N1 is N - 1,
fact(N1, F1),
F2 is F1 * N,
F is F2 * A
).
COMP90048 Declarative Programming Lecture 5 – 12 / 21
Tail Recursion
Transformation
The next step is to simplify the arithmetic goals.
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1,
F is F2 * A
; N > 0,
N1 is N - 1,
fact(N1, F1),
F2 is F1 * N,
F is F2 * A
).
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is (F1 * N) * A
).
COMP90048 Declarative Programming Lecture 5 – 13 / 21
Tail Recursion
Transformation
Now we utilise the associativity of multiplication. This is the insightful
step that is necessary to be able to make the next step.
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is (F1 * N) * A
).
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is F1 * (N * A)
).
COMP90048 Declarative Programming Lecture 5 – 14 / 21
Tail Recursion
Transformation
Now part of the computation can be moved before the call to fact/2.
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
fact(N1, F1),
F is F1 * (N * A)
).
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
A1 is N * A,
fact(N1, F1),
F is F1 * A1
).
COMP90048 Declarative Programming Lecture 5 – 15 / 21
Tail Recursion
Transformation
The final step is to recognise that the last two goals look very much like
the body of the original definition of fact1/3, with the substitution
{N 7→ N1, F2 7→ F1, A 7→ A1}. So we replace those two goals with the
clause head with that substitution applied. This is called folding.
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
A1 is N * A,
fact(N1, F1),
F is F1 * A1
).
fact1(N, A, F) :-
( N =:= 0 ->
F = A
; N > 0,
N1 is N - 1,
A1 is N * A,
fact1(N1, A1, F)
).
COMP90048 Declarative Programming Lecture 5 – 16 / 21
Tail Recursion
Accumulating Lists
The tail recursive version of fact is a constant factor more efficient,
because it behaves like a loop. Sometimes accumulators can make an
order difference, if it can replace an operation with a computation of lower
asymptotic complexity, for example replacing append/3 (linear time) with
list construction (constant time).
rev1([], []).
rev1([A|BC], CBA) :-
rev1(BC, CB),
append(CB, [A], CBA).
This definition of rev1/2 is of quadratic complexity, because for the nth
element from the end of the first argument, we append a list of length
n − 1 to a singleton list. Doing this for each of the n elements gives time
proportional to n(n−1)2 .
COMP90048 Declarative Programming Lecture 5 – 17 / 21
Tail Recursion
Tail recursive rev1/2
The first step in making a tail recursive version of rev1/2 is to specify the
new predicate. It must combine the work of rev1/2 with that of
append/3. The specification is:
% rev(BCD, A, DCBA)
% DCBA is BCD reversed, with A appended
We could develop this by transformation as we did for fact1/3, but we
implement it directly here. We begin with the base case, for BCD = []:
rev([], A, A).
COMP90048 Declarative Programming Lecture 5 – 18 / 21
Tail Recursion
Tail recursive rev1/2
The first step in making a tail recursive version of rev1/2 is to specify the
new predicate. It must combine the work of rev1/2 with that of
append/3. The specification is:
% rev(BCD, A, DCBA)
% DCBA is BCD reversed, with A appended
We could develop this by transformation as we did for fact1/3, but we
implement it directly here. We begin with the base case, for BCD = []:
rev([], A, A).
COMP90048 Declarative Programming Lecture 5 – 18 / 21
Tail Recursion
Tail recursive rev1/2
For the recursive case, take BCD = [B|CD]:
rev([B|CD], A, DCBA) :-
the result, DCBA, must be the reverse of CD, with [B] appended to the
end, and A appended after that. In a functional notation where + is list
concatenation, this is:
(rev(cd) + [b]) + a.
Because append is associative, this is the same as:
rev(cd) + ([b] + a) ≡ rev(cd) + [b | a].
We can use our rev/3 predicate to compute that:
rev([B|CD], A, DCBA) :-
rev(CD, [B|A], DCBA).
COMP90048 Declarative Programming Lecture 5 – 19 / 21
Tail Recursion
Tail recursive rev1/2
rev([], A, A).
rev([B|CD], A, DCBA) :-
rev(CD, [B|A], DCBA).
At each recursive step, this code removes an element from the head of the
input list and adds it to the head of the accumulator. The cost of each
step is therefore a constant, so the overall cost is linear in the length of
the list.
Accumulator lists work like a stack: the last element of the accumulator is
the first element that was added to it, and so on. Thus at the end of the
input list, the accumulator is the reverse of the original input.
COMP90048 Declarative Programming Lecture 5 – 20 / 21
Tail Recursion
Difference pairs
The trick used for a tail recursive reverse predicate is often used in
Prolog: a predicate that generates a list takes an extra argument specifying
what should come after the list. This avoids the need to append to the list.
In Prolog, if you do not know what will come after at the time you call the
predicate, you can pass an unbound variable, and bind that variable when
you do know what should come after. Thus many predicates intended to
produce a list have two arguments, the first is the list produced, and the
second is what comes after. This is called a difference pair, because the
predicate generates the difference between the first and second list.
flatten(empty, List, List).
flatten(node(L,E,R), List, List0) :-
flatten(L, List, List1),
List1 = [E|List2],
flatten(R, List2, List0).
COMP90048 Declarative Programming Lecture 5 – 21 / 21
Higher Order Programming , and Impurity
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 6
Higher Order Programming
and Impurity
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Higher Order Programming , and Impurity
Homoiconicity
Prolog is a Homoiconic language. This means that Prolog programs can
manipulate Prolog programs as data.
The built-in predicate clause(+Head,-Body) allows a running program
to access the clauses of the program.
?- clause(append(X,Y,Z), Body).
false.
Many SWI Prolog “built-ins”, such as append/3, are not actually built-in,
but are auto-loaded. The first time you use them, Prolog detects that they
are undefined, discovers that they can be auto-loaded, quietly loads the,
and continues the computation.
Because append/3 hasn’t been auto-loaded yet, clause/2 can’t find its
code.
COMP90048 Declarative Programming Lecture 6 – 1 / 18
Higher Order Programming , and Impurity
clause/2
If we call append/3 ourselves, Prolog will load it, so we can access its
definition.
?- append([a],[b],L).
L = [a, b].
?- clause(append(X,Y,Z), Body).
X = [],
Y = Z,
Body = true ;
X = [_7184|_7186],
Z = [_7184|_7192],
Body = append(_7186, Y, _7192).
COMP90048 Declarative Programming Lecture 6 – 2 / 18
Higher Order Programming , and Impurity
A Prolog interpreter
This makes it very easy to write a Prolog interpreter:
interp(Goal) :-
( Goal = true
-> true
; Goal = (G1,G2)
-> interp(G1),
interp(G2)
; clause(Goal,Body),
interp(Body)
).
There is a more complete definition, supporting disjunction and negation,
in interp.pl in the examples directory.
COMP90048 Declarative Programming Lecture 6 – 3 / 18
Higher Order Programming , and Impurity
Higher Order Programming
The call/1 built-in predicate executes a term as a goal, capitalising on
Prolog’s homoiconicity.
?- X=append(A, B, [1]), call(X).
X = append([], [1], [1]),
A = [],
B = [1] ;
X = append([1], [], [1]),
A = [1],
B = [] ;
false.
This allows you to write a predicate that takes a goal as an argument and call
that goal.
This is called higher order programming. We will discuss it in some depth when
we cover Haskell.
COMP90048 Declarative Programming Lecture 6 – 4 / 18
Higher Order Programming , and Impurity
Currying
It is often useful to provide a goal omitting some arguments, which are
supplied when the goal is called. This allows the same goal to be used
many times with different arguments.
To support this, many Prologs, including SWI, support versions of call of
higher arity. All arguments to call/n after the goal (first) argument are
added as extra arguments at the end of the goal.
?- X=append([1,2],[3]), call(X, L).
X = append([1, 2], [3]),
L = [1, 2, 3].
When some arguments are supplied with the goal, as we have done here,
they are said to be “curried”. We will cover this in greater depth later.
COMP90048 Declarative Programming Lecture 6 – 5 / 18
Higher Order Programming , and Impurity
Writing higher-order code
It is fairly straightforward to write higher order predicates using call/n.
For example, this predicate will apply a predicate to corresponding
elements of two lists.
maplist(_, [], []).
maplist(P, [X|Xs], [Y|Ys]) :-
call(P, X, Y),
maplist(P, Xs, Ys).
?- maplist(length, [[a,b],[a],[a,b,c]], Lens).
Lens = [2, 1, 3].
?- length(List,2), maplist(same_length(List), List).
List = [[_9378, _9384], [_9390, _9396]].
This is defined in the SWI Prolog library. There are versions with arities 2–5,
allowing 1–4 extra arguments to be passed to the goal argument.
COMP90048 Declarative Programming Lecture 6 – 6 / 18
Higher Order Programming , and Impurity
All solutions
Sometimes one would like to bring together all solutions to a goal.
Prolog’s all solutions predicates do exactly this.
setof(Template, Goal, List) binds List to sorted list of all distinct
instances of Template satisfying Goal
Template can be any term, usually containing some of the variables
appearing in Goal. On completion, setof/3 binds its List argument, but
does not further bind any variables in the Template.
?- setof(P-C, parent(P, C), List).
List = [duchess_kate-prince_george, prince_charles-prince_harry,
prince_charles-prince_william, prince_philip-prince_charles,
prince_william-prince_george, princess_diana-prince_harry,
princess_diana-prince_william, queen_elizabeth-prince_charles].
COMP90048 Declarative Programming Lecture 6 – 8 / 18
Higher Order Programming , and Impurity
All solutions
If Goal contains variables not appearing in Template, setof/3 will
backtrack over each distinct binding of these variables, for each of them
binding List to the list of instances of Template for that binding.
?- setof(C, parent(P, C), List).
P = duchess_kate,
List = [prince_george] ;
P = prince_charles,
List = [prince_harry, prince_william] ;
P = prince_philip,
List = [prince_charles] ;
P = prince_william,
List = [prince_george] ;
P = princess_diana,
List = [prince_harry, prince_william] ;
P = queen_elizabeth,
List = [prince_charles].
COMP90048 Declarative Programming Lecture 6 – 9 / 18
Higher Order Programming , and Impurity
Existential quantification
Use existential quantification, written with infix caret (ˆ), to collect
solutions for a template regardless of the bindings of some of the variables
not in the Template.
E.g., to find all the people in the database who are parents of any child:
?- setof(P, C^parent(P, C), Parents).
Parents = [duchess_kate, prince_charles, prince_philip,
prince_william, princess_diana, queen_elizabeth].
COMP90048 Declarative Programming Lecture 6 – 10 / 18
Higher Order Programming , and Impurity
Unsorted solutions
The bagof/3 predicate is just like setof/3, except that it does not sort
the result or remove duplicates.
?- bagof(P, C^parent(P, C), Parents).
Parents = [queen_elizabeth, prince_philip, prince_charles,
prince_charles, princess_diana, princess_diana, prince_william,
duchess_kate].
Solutions are collected in the order they are produced. This is not purely
logical, because the order of solutions should not matter, nor should the
number of times a solution is produced.
COMP90048 Declarative Programming Lecture 6 – 11 / 18
Higher Order Programming , and Impurity
Input/Output
Prolog’s Input/Output facility does not even try to be pure. I/O
operations are executed when they are reached according to Prolog’s
simple execution order. I/O is not “undone” on backtracking.
Prolog has builtin predicates to read and write arbitrary Prolog terms.
Prolog also allows users to define their own operators. This makes Prolog
very convenient for applications involving structured I/O.
?- op(800, xfy, wibble).
true.
?- read(X).
|: p(x,[1,2],X>Y wibble z).
X = p(x, [1, 2], _1274>_1276 wibble z).
?- write(p(x,[1,2],X>Y wibble z)).
p(x,[1,2],_1464>_1466 wibble z)
true.
COMP90048 Declarative Programming Lecture 6 – 12 / 18
Higher Order Programming , and Impurity
Input/Output
write/1 is handy for printing messages:
?- write(’hello ’), write(’world!’).
hello world!
true.
?- write(’world!’), write(’hello ’).
world!hello
true.
This demonstrates that Prolog’s input/output predicates are non-logical. These
should be equivalent, because conjunction should be commutative.
Code that performs I/O must be handled carefully — you must be aware of the
modes. It is recommended to isolate I/O in a small part of the code, and keep the
bulk of your code I/O-free. (This is a good idea in any language.)
COMP90048 Declarative Programming Lecture 6 – 13 / 18
Higher Order Programming , and Impurity
Comparing terms
All Prolog terms can be compared for ordering using the built-in predicates
@<, @=<, @>, and @>=. Prolog, somewhat arbitrarily, uses the ordering
Variables < Numbers < Atoms < CompoundTerms
but most usefully, within these classes, terms are ordered as one would
expect: numbers by value and atoms are sorted alphabetically. Compound
terms are ordered first by arity, then alphabetically by functor, and finally
by arguments, left-to-right. It is best to use these only for ground terms.
?- hello @< hi.
true.
?- X @< 7, X = foo.
X = foo.
?- X = foo, X @< 7.
false.
COMP90048 Declarative Programming Lecture 6 – 14 / 18
Higher Order Programming , and Impurity
Sorting
There are three SWI Prolog builtins for sorting ground lists according to
the @< ordering: sort/2 sorts a list, removing duplicates, msort/2 sorts a
list, without removing duplicates, and keysort/2 stably sorts list of X-Y
terms, only comparing X parts:
?- sort([h,e,l,l,o], L).
L = [e, h, l, o].
?- msort([h,e,l,l,o], L).
L = [e, h, l, l, o].
?- keysort([7-a, 3-b, 3-c, 8-d, 3-a], L).
L = [3-b, 3-c, 3-a, 7-a, 8-d].
COMP90048 Declarative Programming Lecture 6 – 15 / 18
Higher Order Programming , and Impurity
Determining term types
integer/1 holds for integers and fails for anything else. It also fails for
variables.
?- integer(3).
true.
?- integer(a).
false.
?- integer(X).
false.
Similarly, float/1 recognises floats, number recognises either kind of
number, atom/1 recognises atoms, and compound/1 recognises compound
terms. All of these fail for variables, so must be used with care.
COMP90048 Declarative Programming Lecture 6 – 16 / 18
Higher Order Programming , and Impurity
Recognising variables
var/1 holds for unbound variables, nonvar/1 holds for any term other
than an unbound variable, and ground/1 holds for ground terms (this
requires traversing the whole term). Using these or the predicates on the
previous slide can make your code behave differently in different modes.
But they can also be used to write code that works in multiple modes.
Here is a tail-recursive version of len/2 that works whenever the length is
known:
len2(L, N) :-
( N =:= 0
-> L = []
; N1 is N - 1,
L = [_|L1],
len2(L1, N1)
).
COMP90048 Declarative Programming Lecture 6 – 17 / 18
Higher Order Programming , and Impurity
Recognising variables
This version works when the length is unknown:
len1([], N, N).
len1([_|L], N0, N) :-
N1 is N0 + 1,
len1(L, N1, N).
This code chooses between the two:
len(L, N) :-
( integer(N)
-> len2(L, N)
; nonvar(N)
-> throw(error(type_error(integer, N),
context(len/2, ’’)))
; len1(L, 0, N)
).
COMP90048 Declarative Programming Lecture 6 – 18 / 18
Constraint (Logic) Programming
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 7
Constraint (Logic) Programming
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Constraint (Logic) Programming
Constraint (Logic) Programming
An imperative program specifies the exact sequence of actions to be
executed by the computer.
A functional program specifies how the result of the program is to be
computed at a more abstract level. One can read function definitions as
suggesting an order of actions, but the language implementation can and
sometimes will deviate from that order, due to laziness, parallel execution,
and various optimizations.
A logic program is in some ways more declarative, as it specifies a set of
equality constraints that the terms of the solution must satisfy, and then
searches for a solution.
A constraint program is more declarative still, as it allows more general
constraints than just equality constraints. The search for a solution will
typically follow an algorithm whose relationship to the specification can be
recognized only by experts.
COMP90048 Declarative Programming Lecture 7 – 1 / 23
Constraint (Logic) Programming
Problem specification
The specification of a constraint problem consists of
a set of variables, each variable having a known domain,
a set of constraints, with each constraint involving one or more
variables, and
an optional objective function.
The job of the constraint programming system is to find a solution, a set
of assignments of values to variables (with the value of each variable being
drawn from its domain) that satisfies all the constraints.
The objective function, if there is one, maps each solution to a number.
If this number represents a cost, you want to pick the cheapest solution;
if this number represents a profit, you want to pick the most profitable
solution.
NOTE
The objective function is optional because sometimes you care only about
the existence of a solution, and sometimes all solutions are equally good.
The set of constraints may be given in advance, or it may be discovered
piecemeal, as you go along. The latter occurs reasonably often in planning
problems.
COMP90048 Declarative Programming Lecture 7 – 2 / 23
Constraint (Logic) Programming
Kinds of constraint problems
There are several kinds of constraints, of which the following four are the
most common. Most CP systems handle only one or two kinds.
In Herbrand constraint systems, the variables represent terms, and the
basic constraints are unifications, i.e. they have the form term1 = term2.
This is the constraint domain implemented by Prolog.
In finite domain or FD constraint systems, each variable’s domain has a
finite number of elements.
In boolean satisfiability or SAT systems, the variables represent booleans,
and each constraint asserts the truth of an expression constructed using
logical operations such as AND, OR, NOT and implication.
In linear inequality constraint systems, the variables represent real numbers
(or sometimes integers), and the constraints are of the form ax + by ≤ c
(where x and y are variables, and a, b and c are constants).
NOTE
You can view boolean satisfiability (SAT) problems as a subclass of finite
domain problems, but there are specialized algorithms that work only on
booleans and not on finite domains with more than two values, Research
into solving SAT problems has made great progress over the last 20 years,
and modern SAT solvers are surprisingly efficient at solving NP complete
problems. So in practice, the two classes of problems should be handled
differently.
COMP90048 Declarative Programming Lecture 7 – 3 / 23
Constraint (Logic) Programming
Herbrand Constraints
Herbrand constraints are just equality constraints over Herbrand terms —
exactly what we have been using since we started with Prolog.
In Prolog, we can constrain variables to be equal, and Prolog will succeed
if that is possible, and fail if not.
?- length(Word, 5), reverse(Word, Word).
Word = [_2056, _2062, _2068, _2062, _2056].
?- length(Word, 5), reverse(Word, Word), Word=[r,a,d,a,r].
Word = [r, a, d, a, r].
?- length(Word, 5), reverse(Word, Word), Word=[l,a,s,e,r].
false.
COMP90048 Declarative Programming Lecture 7 – 4 / 23
Constraint (Logic) Programming
Search
Prolog normally employs a strategy known as “generate and test” to
search for variable bindings that satisfy constraints. Nondeterministic goals
generate potential solutions; later goals test those solutions, imposing
further constraints and rejecting some candidate solutions.
For example, in
?- between(1,9,X), 0 =:= X mod 2, X =:= X * X mod 10.
The first goal generates single-digit numbers, the second tests that it is
even, and the third that its square ends in the same digit.
Constraint logic programming uses the more efficient “constrain and
generate” strategy. In this approach, constraints on variables can be more
sophisticated than simply binding to a Herbrand term. This is generally
accomplished in Prolog systems with attributed variables, which allow
constraint domains to control unification of constrained variables.
COMP90048 Declarative Programming Lecture 7 – 5 / 23
Constraint (Logic) Programming
Propagation
The usual algorithm for solving a set of FD constraints consists of two
steps: propagation and labelling.
In the propagation step, we try to reduce the domain of each variable as
much as possible.
For each constraint, we check whether the constraint rules out any values
in the current domains of any of the variables in that constraint. If it does,
then we remove that value from the domain of that variable, and schedule
the constraints involving that variable to be looked at again.
The propagation step ends
if every variable has a domain of size one, which represents a fixed
value, since this represents a solution;
if some variable has an empty domain, since this represents failure; or
if there are no more constraints to look at, in which case propagation
can do no more.
COMP90048 Declarative Programming Lecture 7 – 6 / 23
Constraint (Logic) Programming
Labelling
If propagation cannot do any more, we go on to the labelling step, which
picks a not-yet-fixed variable,
partitions its current domain (of size n) into k parts d1, ..., dk , where
usually k = 2 but may be any value satisfying 2 ≤ k ≤ n, and
recursively invokes the whole constraint solving algorithm k times,
with invocation i restricting the domain of the chosen variable to di .
Each recursive invocation also consists of a propagation step and
(if needed) a labelling step. The whole computation therefore consists of
alternating propagation and labelling steps.
The labelling steps generate a search tree. The size of the tree depends on
the effectiveness of propagation: the more effective propagation is at
removing values from domains, the smaller the tree will be, and the less
time searching it will take.
NOTE
The root node of the tree represents the computation for solving the whole
constraint problem. Every other node represents the choice within the
computation represented by its parent node. If it is the ith child of its
parent node, then it represents the choice in the labelling step to set the
domain of the selected variable to di .
If the current domain of the chosen variable is e.g. [1..5], then the selected
partition maybe contain two to five parts. The only partition that divides
[1..5] into five parts is of course [1], [2], [3], [4], [5], but there are many
partitions that divide [1..5] into two parts, with [1..2], [3..5] and
[1, 3, 5], [2, 4] being two of them.
Note that one can view the labelling step as imposing an extra constraint
on each branch of the search tree, with the extra constraint for branch i
requiring the chosen variable to have a value in di of its current domain.
COMP90048 Declarative Programming Lecture 7 – 7 / 23
Constraint (Logic) Programming
Prolog Arithmetic Revisted
In earlier lectures we introduced a number of built-in arithmetic predicates,
including (is)/2, (=:=)/2, (=\=)/2, and (=<)/2. Recall that these
predicates could only be used in certain modes. The predicate (is)/2, for
example, only works when its second argument is ground.
?- 25 is X * X.
ERROR: Arguments are not sufficiently instantiated
?- X is 5 * 5.
X = 25.
The CLP(FD) library (constraint logic programming over finite domains)
provides replacements for these lower-level arithmetic predicates. These
new predicates are called arithmetic constraints and can be used in both
directions (i.e., in,out and out,in).
COMP90048 Declarative Programming Lecture 7 – 8 / 23
Constraint (Logic) Programming
CLP(FD) Arithmetic Constraints
CLP(FD) provides the following arithmetic constraints:
Expr1 #= Expr2 Expr1 equals Expr2
Expr1 #\= Expr2 Expr1 is not equal to Expr2
Expr1 #> Expr2 Expr1 is greater than Expr2
Expr1 #< Expr2 Expr1 is less than Expr2
Expr1 #>= Expr2 Expr1 is greater than or equal to Expr2
Expr1 #=< Expr2 Expr1 is less than or equal to Expr2
Var in Low..High Low ≤ Var ≤ High
List ins Low..High every Var in List is between Low and High
?- 25 #= X * X.
X in -5\/5.
?- 25 #= X * 5.
X = 5.
COMP90048 Declarative Programming Lecture 7 – 9 / 23
Constraint (Logic) Programming
Propagation and Labelling with CLP(FD)
Recall that the domain of a CLP(FD) variable is the set of all integers. We
reduce or restrict the domain of these variables with the use of CLP(FD)
constraints. When a constraint is posted, the library automatically revises
the domains of relevant variables if necessary. This is called propagation.
Sometimes propagation alone is enough to reduce the domains of each
variable to a single element. In other cases, we need to tell Prolog to
perform the labelling step.
label/1 is an enumeration predicate that searches for an assignment to
each variable in a list that satisfies all posted constraints.
?- 25 #= X * X, label([X]).
X = -5;
X = 5.
COMP90048 Declarative Programming Lecture 7 – 10 / 23
Constraint (Logic) Programming
Propagation and Labelling with CLP(FD)
What we expressed in Prolog using generate-and-test
?- between(1,9,X), 0 =:= X mod 2, X =:= X * X mod 10.
would be expressed with constrain and generate as:
?- X in 1..9, 0 #= X mod 2, X #= X * X mod 10.
X in 2..8,
_12562 mod 10#=X,
X^2#=_12562,
X mod 2#=0,
_12562 in 4..64.
And with labelling:
?- X in 1..9, 0 #= X mod 2, X #= X * X mod 10, label([X]).
X = 6.
COMP90048 Declarative Programming Lecture 7 – 11 / 23
Constraint (Logic) Programming
Sudoku
Sudoku is a class of puzzles played
on a 9x9 grid. Each grid position
should hold a number between 1
and 9. The same integer may not
appear twice
in a single row,
in a single column, or
in one of the nine 3x3 boxes.
The puzzle creator provides some
of the numbers; the challenge is to
fill in the rest.
This is a classic finite domain con-
straint satisfaction problem.
r1 5 3 7
r2 6 1 9 5
r3 9 8 6
r4 8 6 3
r5 4 8 3 1
r6 7 2 6
r7 6 2 8
r8 4 1 9 5
r9 8 7 9
c1 c2 c3 c4 c5 c6 c7 c8 c9
COMP90048 Declarative Programming Lecture 7 – 12 / 23
Constraint (Logic) Programming
Sudoku as finite domain constraints
You can represent the rules of sudoku as a set of 81 constraint variables
(r1c1, r1c2 etc) each with the domain 1..9, and 27 all-different
constraints: one for each row, one for each column, and one for each box.
For example, the constraint for the top left box would be
all_different([r1c1, r1c2, r1c3, r2c1, r2c2, r2c3, r3c1, r3c2, r3c3]).
Initially, the domain of each variable is [1..9]. If you fix the value of a
variable e.g. by setting r1c1 = 5, this means that the other variables that
share a row, column or box with r1c1 (and that therefore appear in an
all-different constraint with it) cannot be 5, so their domain can be
reduced to [1..4, 6..9].
This is how the variables fixed by our example puzzle reduce the domain of
r3c1 to only [1..2], and the domain of r5c5 to only [5].
Fixing r5c5 to be 5 gives us a chance to further reduce the domains of the
other variables linked to r5c5 by constraints, e.g. r7c5.
COMP90048 Declarative Programming Lecture 7 – 13 / 23
Constraint (Logic) Programming
Sudoku in SWI Prolog
Using SWI’s library(clpfd), Sudoku problems can be solved:
sudoku(Rows) :-
length(Rows, 9), maplist(same_length(Rows), Rows),
append(Rows, Vs), Vs ins 1..9,
maplist(all_distinct, Rows),
transpose(Rows, Columns),
maplist(all_distinct, Columns),
Rows = [A,B,C,D,E,F,G,H,I],
blocks(A, B, C), blocks(D, E, F), blocks(G, H, I).
blocks([], [], []).
blocks([A,B,C|Bs1], [D,E,F|Bs2], [G,H,I|Bs3]) :-
all_distinct([A,B,C,D,E,F,G,H,I]),
blocks(Bs1, Bs2, Bs3).
COMP90048 Declarative Programming Lecture 7 – 14 / 23
Constraint (Logic) Programming
Sudoku in SWI Prolog
?- Puzzle=[[5,3,_, _,7,_, _,_,_],
| [6,_,_, 1,9,5, _,_,_],
| [_,9,8, _,_,_, _,6,_],
|
| [8,_,_, _,6,_, _,_,3],
| [4,_,_, 8,_,3, _,_,1],
| [7,_,_, _,2,_, _,_,6],
|
| [_,6,_, _,_,_, 2,8,_],
| [_,_,_, 4,1,9, _,_,5],
| [_,_,_, _,8,_, _,7,9]],
| sudoku(Puzzle),
| write(Puzzle).
COMP90048 Declarative Programming Lecture 7 – 15 / 23
Constraint (Logic) Programming
Sudoku solution
In less than 120 of a second, this produces the solution:
Puzzle=[[5,3,4, 6,7,8, 9,1,2],
[6,7,2, 1,9,5, 3,4,8],
[1,9,8, 3,4,2, 5,6,7],
[8,5,9, 7,6,1, 4,2,3],
[4,2,6, 8,5,3, 7,9,1],
[7,1,3, 9,2,4, 8,5,6],
[9,6,1, 5,3,7, 2,8,4],
[2,8,7, 4,1,9, 6,3,5],
[3,4,5, 2,8,6, 1,7,9]]
COMP90048 Declarative Programming Lecture 7 – 16 / 23
Constraint (Logic) Programming
Linear inequality constraints
Suppose you want to make banana and/or chocolate cakes for a bake sale,
and you have 10 kg of flour, 30 bananas, 1.2 kg of sugar, 1.5 kg of butter,
and 700 grams of cocoa on hand. You can charge $4.00 for a banana cake
and $6.50 for a chocolate one. Each kind of cake requires a certain
quantity of each ingredient. How do you determine how many of each
cake to make so as to maximise your profit?
To solve such a problem, you need to set up a system of constraints
saying, for example, that the number of each kind of cake times the
amount of flour needed for that kind of cake must add to no more than
the amount of flour you have, and so on.
You also need to specify that the number of each kind of cake must be
non-negative. Finally, you need to define your revenue as the sum of the
number of each kind of cake times its price, and specify that you would
like to maximise revenue.
COMP90048 Declarative Programming Lecture 7 – 21 / 23
Constraint (Logic) Programming
Linear inequality constraints
We can use SWI Prolog’s library(clpr) to solve such problems. This
library requires constraints to be enclosed in curly braces.
?- {250*B + 200*C =< 10000},
| {2*B =< 30},
| {75*B + 150*C =< 1200},
| {100*B + 150*C =< 1500},
| {75*C =< 700},
| {B>=0}, {C>=0},
| {Revenue = 4*B + 6.5*C}, maximize(Revenue).
B = 12.0,
C = 2.0,
Revenue = 61.0
So we can make $61.00 by making 12 Banana and 2 Chocolate cakes.
COMP90048 Declarative Programming Lecture 7 – 22 / 23
Constraint (Logic) Programming
Mercury
The Mercury language was developed at The University of Melbourne as a
purely declarative successor to Prolog. Mercury is strongly typed, with a
type system very similar to Haskell’s. It is also strongly moded, which
means that the binding state of all variables is determined at compile-time.
Mercury has a strong determinism system, as well, which allows the
compiler to determine, for each predicate mode, whether it will always be
deterministic.
All these properties, along with aggressive compiler optimisations, make
Mercury among the highest-performing declarative languages. Although
Mercury’s learning curve is somewhat steeper than Prolog’s, it is a more
realistic choice for serious development.
COMP90048 Declarative Programming Lecture 7 – 23 / 23
Introduction to Functional Programming
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 8
Introduction to
Functional Programming
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Introduction to Functional Programming
Functional programming
The basis of functional programming is equational reasoning. This is a
grand name for a simple idea:
if two expressions have equal values, then one can be replaced by the
other.
You can use equational reasoning to rewrite a complex expression to be
simpler and simpler, until it is as simple as possible. Suppose x = 2 and
y = 4, and you start with the expression x + (3 * y):
step 0: x + (3 * y)
step 1: 2 + (3 * y)
step 2: 2 + (3 * 4)
step 3: 2 + 12
step 4: 14
COMP90048 Declarative Programming Lecture 8 – 1 / 16
Introduction to Functional Programming
Lists
Of course, programs want to manipulate more complex data than just
simple numbers.
Like most functional programming languages, Haskell allows programmers
to define their own types, using a much more expressive type system than
the type system of e.g. C.
Nevertheless, the most frequently used type in Haskell programs is
probably the builtin list type.
The notation [] means the empty list, while x:xs means a nonempty list
whose head (first element) is represented by the variable x, and whose tail
(all the remaining elements) is represented by the variable xs.
The notation ["a", "b"] is syntactic sugar for "a":"b":[]. As in most
languages, "a" represents the string that consists of a single character, the
first character of the alphabet.
COMP90048 Declarative Programming Lecture 8 – 2 / 16
Introduction to Functional Programming
Functions
A function definition consists of equations, each of which establishes an
equality between the left and right hand sides of the equal sign.
len [] = 0
len (x:xs) = 1 + len xs
Each equation typically expects the input arguments to conform to a given
pattern; [] and (x:xs) are two patterns.
The set of patterns should be exhaustive: at least one pattern should
apply for any possible call.
It is good programming style to ensure that the set of patterns is also
exclusive, which means that at most one pattern should apply for any
possible call.
If the set of patterns is both exhaustive and exclusive, then exactly one
pattern will apply for any possible call.
COMP90048 Declarative Programming Lecture 8 – 3 / 16
Introduction to Functional Programming
Aside: syntax
In most languages, a function call looks like f(fa1, fa2, fa3).
In Haskell, it looks like f fa1 fa2 fa3.
If the second argument is not fa2 but the function g applied to the single
argument ga1, then in Haskell you would need to write
f fa1 (g ga1) fa3
since Haskell would interpret f fa1 g ga1 fa3 as a call to f with four
arguments.
In Haskell, there are no parentheses around the whole argument list of a
function call, but parentheses may be needed around individual arguments.
This applies on the left as well the right hand sides of equations.
This is why the recursive call is len xs and not len(xs), and why the
left hand side of the second equation is len (x:xs) instead of len x:xs.
COMP90048 Declarative Programming Lecture 8 – 4 / 16
Introduction to Functional Programming
More syntax issues
Comments start with two minus signs and continue to the end of the line.
The names of functions and variables are sequences of letters, numbers
and/or underscores that must start with a lower case letter.
Suppose line1 starts in column n, and the following nonblank line, line2,
starts in column m. The offside rule says that
if m > n, then line2 is a continuation of the construct on line1;
if m = n, then line2 is the start of a new construct at the same level
as line1;
if m < n, then line2 is either the continuation of something else that
line1 is part of, or a new item at the same level as something else that
line1 is part of.
This means that the structure of the code as shown by indentation must
match the structure of the code.
NOTE
Actually, it is also ok for function names to consist wholly of graphic
characters like +, but only builtin functions should have such names.
If your source code includes tabs as well as spaces, two lines can look like
they have the same level of indentation even if they do not. It is therefore
best to ensure that Haskell source files do not contain tabs.
If you use vim as your editor, you can ensure this by putting
-- vim: ts=4 sw=4 expandtab syntax=haskell
at the top of the source file. The expandtab tells vim to expand all tabs
into several spaces, while the ts=4 and sw=4 tell vim to set up tab stops
every four columns. The syntax=haskell specifies the set of rules vim
should use for syntax highlighting.
The following function definition is wrong: since the second line is
indented further than the first, Haskell considers it to be a continuation of
the first equation, rather than a separate equation.
len [] = 0
len (x:xs) = 1 + len xs
The following definition is acceptable to the offside rule, though it is not
an example of good style:
len [] =
0
len (x:xs) =
1 +
len xs
COMP90048 Declarative Programming Lecture 8 – 5 / 16
Introduction to Functional Programming
Recursion
The definition of a function to compute the length of a list, like many
Haskell functions, reflect the structure of the data: a list is either empty,
or has a head and a tail.
The first equation for len handles the empty list case.
len [] = 0
This is called the base case.
The second equation handles nonempty lists.
This is called the recursive case, since it contains a recursive call.
len (x:xs) = 1 + len xs
If you want to be a good programmer in a declarative language, you have
to get comfortable with recursion, because most of the things you need to
do involve recursion.
COMP90048 Declarative Programming Lecture 8 – 6 / 16
Introduction to Functional Programming
Recursion
The definition of a function to compute the length of a list, like many
Haskell functions, reflect the structure of the data: a list is either empty,
or has a head and a tail.
The first equation for len handles the empty list case.
len [] = 0
This is called the base case. The second equation handles nonempty lists.
This is called the recursive case, since it contains a recursive call.
len (x:xs) = 1 + len xs
If you want to be a good programmer in a declarative language, you have
to get comfortable with recursion, because most of the things you need to
do involve recursion.
COMP90048 Declarative Programming Lecture 8 – 6 / 16
Introduction to Functional Programming
Recursion
The definition of a function to compute the length of a list, like many
Haskell functions, reflect the structure of the data: a list is either empty,
or has a head and a tail.
The first equation for len handles the empty list case.
len [] = 0
This is called the base case. The second equation handles nonempty lists.
This is called the recursive case, since it contains a recursive call.
len (x:xs) = 1 + len xs
If you want to be a good programmer in a declarative language, you have
to get comfortable with recursion, because most of the things you need to
do involve recursion.
COMP90048 Declarative Programming Lecture 8 – 6 / 16
Introduction to Functional Programming
Using a function
len [] = 0
len (x:xs) = 1 + len xs
Given a function definition like this, the Haskell implementation can use it
to replace calls to the function with the right hand side of an applicable
equation.
step 0: len ["a", "b"] -- ("a":("b":[]))
step 1: 1 + len ["b"] -- ("b":[])
step 2: 1 + 1 + len []
step 3: 1 + 1 + 0
step 4: 1 + 1
step 5: 2
NOTE
In general, if there is more than one applicable equation, the Haskell
implementation picks the first one.
You can think of builtin functions as being implicitly defined by a very long
list of equations; on a 32 bit machine, you would need 232 ∗ 232 = 264
equations. In practice, such functions are of course implemented using the
arithmetic instructions of the platform.
COMP90048 Declarative Programming Lecture 8 – 7 / 16
Introduction to Functional Programming
Expression evaluation
To evaluate an expression, the Haskell runtime system conceptually
executes a loop, each iteration of which consists of these steps:
looks for a function call in the current expression,
searches the list of equations defining the function from the top
downwards, looking for a matching equation,
sets the values of the variables in the matching pattern to the
corresponding parts of the actual arguments, and
replaces the left hand side of the equation with the right hand side.
The loop stops when the current expression contains no function calls, not
even calls to such builtin “functions” as addition.
The actual Haskell implementation is more sophisticated than this loop,
but the effect it achieves is the same.
COMP90048 Declarative Programming Lecture 8 – 8 / 16
Introduction to Functional Programming
Order of evaluation
The first step in each iteration of the loop, “look for a function call in the
current expression”, can find more than one function call. Which one
should we select?
len [] = 0
len (x:xs) = 1 + len xs
evaluation order A:
step 0: len ["a"] + len []
step 1: 1 + len [] + len []
step 2: 1 + 0 + len []
step 3: 1 + len []
step 4: 1 + 0
step 5: 1
evaluation order B:
step 0: len ["a"] + len []
step 1: len ["a"] + 0
step 2: 1 + len [] + 0
step 3: 1 + 0 + 0
step 4: 1 + 0
step 5: 1
NOTE
In this example, evaluation order A chooses the leftmost call, while
evaluation order B chooses the rightmost call.
COMP90048 Declarative Programming Lecture 8 – 9 / 16
Introduction to Functional Programming
Church-Rosser theorem
In 1936, Alonzo Church and J. Barkley Rosser proved a famous theorem,
which says that for the rewriting system known as the lambda calculus,
regardless of the order in which the original term’s subterms are rewritten,
the final result is always the same.
This theorem also holds for Haskell and for several other functional
programming languages (though not for all).
This is not that surprising, since most modern functional languages are
based on one variant or another of the lambda calculus.
We will ignore the order of evaluation of Haskell expressions for now, since
in most cases it does not matter. We will come back to the topic later.
The Church-Rosser theorem is not applicable to imperative languages.
NOTE
Each rewriting step replaces the left hand side of an equation with the
right hand side.
COMP90048 Declarative Programming Lecture 8 – 10 / 16
Introduction to Functional Programming
Order of evaluation: efficiency
all_pos [] = True
all_pos (x:xs) = x > 0 && all_pos xs
evaluation order A:
0: all_pos [-1, 2]
1: -1 > 0 & all_pos [2]
2: False & all_pos [2]
3: False
evaluation order B:
0: all_pos [-1, 2]
1: -1 > 0 & all_pos [2]
2: -1 > 0 & 2 > 0 & all_pos []
3: -1 > 0 & 2 > 0 & True
4: -1 > 0 & True & True
5: -1 > 0 & True
6: False & True
7: False
NOTE
The definition of conjunction for Booleans does not need to know the
value of second operand if the value of the first operand is False:
False & _ = False
True & b = b
Note that all_pos, like len, has one equation for empty lists and one
equation for nonempty lists. Functions that operate on data with the
similar structures often have similar structures themselves.
COMP90048 Declarative Programming Lecture 8 – 11 / 16
Introduction to Functional Programming
Order of evaluation: efficiency
all_pos [] = True
all_pos (x:xs) = x > 0 && all_pos xs
evaluation order A:
0: all_pos [-1, 2]
1: -1 > 0 & all_pos [2]
2: False & all_pos [2]
3: False
evaluation order B:
0: all_pos [-1, 2]
1: -1 > 0 & all_pos [2]
2: -1 > 0 & 2 > 0 & all_pos []
3: -1 > 0 & 2 > 0 & True
4: -1 > 0 & True & True
5: -1 > 0 & True
6: False & True
7: False
NOTE
The definition of conjunction for Booleans does not need to know the
value of second operand if the value of the first operand is False:
False & _ = False
True & b = b
Note that all_pos, like len, has one equation for empty lists and one
equation for nonempty lists. Functions that operate on data with the
similar structures often have similar structures themselves.
COMP90048 Declarative Programming Lecture 8 – 11 / 16
Introduction to Functional Programming
Order of evaluation: efficiency
all_pos [] = True
all_pos (x:xs) = x > 0 && all_pos xs
evaluation order A:
0: all_pos [-1, 2]
1: -1 > 0 & all_pos [2]
2: False & all_pos [2]
3: False
evaluation order B:
0: all_pos [-1, 2]
1: -1 > 0 & all_pos [2]
2: -1 > 0 & 2 > 0 & all_pos []
3: -1 > 0 & 2 > 0 & True
4: -1 > 0 & True & True
5: -1 > 0 & True
6: False & True
7: False
NOTE
The definition of conjunction for Booleans does not need to know the
value of second operand if the value of the first operand is False:
False & _ = False
True & b = b
Note that all_pos, like len, has one equation for empty lists and one
equation for nonempty lists. Functions that operate on data with the
similar structures often have similar structures themselves.
COMP90048 Declarative Programming Lecture 8 – 11 / 16
Introduction to Functional Programming
Imperative vs declarative languages
In the presence of side effects, a program’s behavior depends on history;
that is, the order of evaluation matters.
Because understanding an effectful program requires thinking about all
possible histories, side effects often make a program harder to understand.
When developing larger programs or working in teams, managing
side-effects is critical and difficult; Haskell guarantees the absence of
side-effects.
What really distinguishes pure declarative languages from imperative
languages is that they do not allow side effects.
There is only one benign exception to that: they do allow programs to
generate exceptions.
We will ignore exceptions from now on, since in the programs we deal
with, they have only one effect: they abort the program.
NOTE
This is OK. The only thing that depends on the order of evaluation is
which of several exceptions that a program can raise will actually be
raised. Unless you are writing the exception handler, you still don’t have to
understand all possible histories.
COMP90048 Declarative Programming Lecture 8 – 13 / 16
Introduction to Functional Programming
Referential transparency
The absence of side effects allows pure functional languages to achieve
referential transparency, which means that an expression can be replaced
with its value. This requires that the expression has no side effects and is
pure, i.e. always returns the same results on the same input.
By contrast, in imperative languages such as C, functions in general are
not pure and are thus not functions in a mathematical sense: two identical
calls may return different results.
Impure functional languages such as Lisp are called impure precisely
because they do permit side effects like assignments, and thus their
programs are not referentially transparent.
NOTE
Impure functional languages share characteristics both with imperative
languages and with pure functional languages, so they are effectively
somewhere between them on the spectrum of programming languages.
In the rest of the subject, we will use “functional languages” as a
shorthand to mean “pure functional languages”, except in contexts where
we are specifically talking about impure functional languages.
NOTE
Even in imperative languages, some functions are pure. This is usually true
e.g. of implementations of actual mathematical functions, such as
logarithm, sine, cosine, etc, but also of many others.
COMP90048 Declarative Programming Lecture 8 – 14 / 16
Introduction to Functional Programming
Single assignment
One consequence of the absence of side effects is that assignment means
something different in a functional language than in an imperative
language.
In conventional, imperative languages, even object-oriented ones
(including C, Java, and Python), each variable has a current value
(a garbage value if not yet initialized), and assignment statements
can change the current value of a variable.
In functional languages, variables are single assignment, and there are
no assignment statements. You can define a variable’s value, but you
cannot redefine it. Once a variable has a value, it has that value until
the end of its lifetime.
COMP90048 Declarative Programming Lecture 8 – 15 / 16
Introduction to Functional Programming
Giving variables values
Haskell programs can give a variable a value in one of two ways.
The explicit way is to use a let clause:
let pi = 3.14159 in ...
This defines pi to be the given value in the expression represented by the
dots. It does not define pi anywhere else.
The implicit way is to put the variable in a pattern on the left hand side of
an equation:
len (x:xs) = 1 + len xs
If len is called with a nonempty list, Haskell will bind x to its head and xs
to its tail.
NOTE
Actually, there are other ways, both explicit and implicit, but these are
enough for now.
COMP90048 Declarative Programming Lecture 8 – 16 / 16
Builtin Haskell types
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 9
Builtin Haskell types
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Builtin Haskell types
The Haskell type system
Haskell has a strong, safe and static type system.
The strong part means that the system has no loopholes; one cannot tell
Haskell to e.g. consider an integer to be a pointer, as one can in C with
(char *) 42.
The safe part means that a running program is guaranteed never to crash
due to a type error. (A C program that dereferenced the above pointer
would almost certainly crash.)
The static part means that types are checked when the program is
compiled, not when the program is run.
This is partly what makes the safe part possible; Haskell will not even start
to run a program with a type error.
NOTE
There is an “out” from the static nature of the type system, for use in
cases where this is warranted Haskell also supports a dynamic type, and
operations on values of this type are checked for type correctness only at
runtime.
Note also that different people mean different things when they talk about
e.g. the strength or safety of a type system, so these are not the only
definitions you will see in the computer science literature. However, we will
use these definitions in this subject.
COMP90048 Declarative Programming Lecture 9 – 1 / 17
Builtin Haskell types
Basic Haskell types
Haskell has the usual basic types. These include:
The Boolean type is called Bool. It has two values: True and False.
The native integer type is called Int. Values of this type are 32 or 64
bits in size, depending on the platform. Haskell also has a type for
integers of unbounded size: Integer.
The usual floating-point type is Double. (Float is also available, but
its use is discouraged.)
The character type is called Char.
There are also others, e.g. integer types with 8, 16, 32 and 64 bits
regardless of platform.
There are more complex types as well.
COMP90048 Declarative Programming Lecture 9 – 2 / 17
Builtin Haskell types
The types of lists
In Haskell, list is not a type; it is a type constructor.
Given any type t, it constructs a type for lists whose elements are all of
type t. This type is written as [t], and it is pronounced as “list of t”.
You can have lists of any type. For example,
[Bool] is the type of lists of Booleans,
[Int] is the type of lists of native integers,
[[Int]] is the type of lists of lists of native integers.
These are similar to LinkedList, LinkedList, and
LinkedListHaskell considers strings to be lists of characters, whose type is [Char];
String is a synonym for [Char].
The names of types and type constructors should be identifiers starting
with an upper case letter; the list type constructor is an exception.
NOTE
In fact, you can have lists of lists of lists of lists [[[[Int]]]], lists of lists
of lists of lists of lists [[[[[Int]]]]], and so on. The only effective limit
is the programmer’s ability to do something useful with the values of the
type.
COMP90048 Declarative Programming Lecture 9 – 3 / 17
Builtin Haskell types
ghci
The usual implementation of Haskell is ghc, the Glasgow Haskell
Compiler. It also comes with an interpreter, ghci.
$ ghci
...
Prelude> let x = 2
Prelude> let y = 4
Prelude> x + (3 * y)
14
The prelude is Haskell’s standard library.
ghci uses its name as the prompt to remind users that they can call its
functions.
NOTE
Once you invoke ghci,
you type an expression on a line;
it typechecks the expression;
it evaluates the expression (if it is type correct);
it prints the resulting value.
You can also load Haskell code into ghci with :load filename.hs. The
suffix .hs, the standard suffix for Haskell source files, can be omitted.
COMP90048 Declarative Programming Lecture 9 – 4 / 17
Builtin Haskell types
Types and ghci
You can ask ghci to tell you the type of an expression by prefacing that
expression with :t. The command :set +t tells ghci to print the type as
well as the value of every expression it evaluates.
Prelude> :t "abc"
"abc" :: [Char]
Prelude> :set +t
Prelude> "abc"
"abc"
...
it :: [Char]
The notation x::y says that expression x is of type y. In this case, it says
"abc" is of type [Char].
it is ghci’s name for the value of the expression just evaluated.
COMP90048 Declarative Programming Lecture 9 – 6 / 17
Builtin Haskell types
Function types
You can also ask ghci about the types of functions. Consider this
function, which checks whether a list is empty:
isEmpty [] = True
isEmpty (_:_) = False
(Just as in Prolog, _ is a special pattern that matches anything.)
If you ask ghci about its type, you get
> :t isEmpty
isEmpty :: [a] -> Bool
A function type lists the types of all the arguments and the result, all
separated by arrows. We’ll see what the a means a bit later.
NOTE
This function is already defined in standard Haskell; it is called null. The
len function is also already defined, under the name length.
COMP90048 Declarative Programming Lecture 9 – 7 / 17
Builtin Haskell types
Function types
Programmers should declare the type of each function. The syntax for this
is similar to the notation printed by ghci: the function name, a double
colon, and the type.
module Emptiness where
isEmpty :: [t] -> Bool
isEmpty [] = True
isEmpty _ = False
Declaring the type of functions is required only by good programming
style. The Haskell implementation will infer the types of functions if not
declared.
Haskell also infers the types of all the local variables.
Later in the subject, we will briefly introduce the algorithm Haskell uses for
type inference.
NOTE
A Haskell source file should contain one Haskell module. We will cover the
Haskell module system later.
COMP90048 Declarative Programming Lecture 9 – 8 / 17
Builtin Haskell types
Function type declarations
With type declarations, Haskell will report an error and refuse to compile
the file if the declared type of a function is incompatible with its definition.
It’s also an error if a call to the function is incompatible with its declared
type.
Without declarations, Haskell will report an error if the types in any call to
any function are incompatible with its definition. Haskell will never allow
code to be run with a type error.
Type declarations improve Haskell’s error messages, and make function
definitions much easier to understand.
COMP90048 Declarative Programming Lecture 9 – 9 / 17
Builtin Haskell types
Number types
Haskell has several numeric types, including Int, Integer, Float, and
Double. A plain integer constant belongs to all of them. So what does
Haskell say when asked what the type of e.g. 3 is?
Prelude> :t 3
3 :: Num p => p
Prelude> :t [1, 2]
[1, 2] :: Num a => [a]
In these messages, a and p are type variables; they are variables that stand
for types, not values.
The notation Num p means “the type p is a member of type class Num”.
Num is the class of numeric types, including the four types above.
The notation 3 :: Num p => p means that “if p is a numeric type, then
3 is a value of that type”.
NOTE
Similarly, the notation [1, 2] :: Num a => [a] means that “if a is a
numeric type, then [1, 2] is a list of values of that type”.
COMP90048 Declarative Programming Lecture 9 – 11 / 17
Builtin Haskell types
Number type flexibility
The usual arithmetic operations, such as addition, work for any numeric
type:
Prelude> :t (+)
(+) :: (Num a) => a -> a -> a
The notation a -> a -> a denotes a function that takes two arguments
and returns a result, all of which have to be of the same type (since they
are denoted by the same type variable, a), which in this case must be a
member of the Num type class.
This flexibility is nice, but it does result in confusing error messages:
Prelude> [1, True]
No instance for (Num Bool) arising from the literal ‘1’
...
NOTE
The error message is trying to say that Bool, the type of True, is not a
member or instance of Num type class. If it were a numeric type, then the
list could a list of elements of that type, since the integer constant is a
member of any numeric type, and thus the types of the two elements
would be the same (which must hold for all Haskell lists).
The fix suggested by the error message is to have the programmer add a
declaration to the effect that Bool is a numeric type. Since Booleans are
not numbers, this fix would be worse than useless, but since ghc has no
idea about what each type actually means, it does not know that.
Programmers new to Haskell should probably handle type errors by simply
going to the location specified in the error message (which has been
removed from this example to make it fit on the slide), and looking around
for the error, ignoring the rest of the error message.
Normally, + is a binary infix operator, but you can tell Haskell you want to
use it as an ordinary function name by wrapping it in parentheses. (You
can do the same with any other operator.) This means that (+) 1 2
means the same as 1 + 2.
COMP90048 Declarative Programming Lecture 9 – 12 / 17
Builtin Haskell types
if-then-else
-- Definition A
iota n = if n == 0 then [] else iota (n-1) ++ [n]
Definition A uses an if-then-else. If-then-else in Haskell differs from
if-then-elses in imperative languages in that
the else arm is not optional, and
the then and else arms are expressions, not statements.
NOTE
The expressions representing the then and else arms must be of the same
type, since the result of the if-then-else will be one of them. It will be the
expression in the then arm if the condition is true, and it will be the
expression in the else arm if the condition is false.
In Haskell, = separates the left and right hand sides of equations, while ==
represents a test for equality.
A language called APL in the 1960s had a whole bunch of builtin functions
working with numbers and vectors and matrices, and many of these were
named after letters of the greek alphabet. This function is named after the
iota function of APL, which also returned the first n integers when invoked
with n as its argument.
COMP90048 Declarative Programming Lecture 9 – 14 / 17
Builtin Haskell types
Guards
-- Definition B
iota n
| n == 0 = []
| n > 0 = iota (n-1) ++ [n]
Definition B uses guards to specify cases. Note the first line does not end
with an “=”; each guard line specifies a case and the value for that case,
much as in definition A.
Note that the second guard specifies n > 0. What should happen if you
do iota (-3)? What do you expect to happen? What about for
definition A?
COMP90048 Declarative Programming Lecture 9 – 15 / 17
Builtin Haskell types
Structured definitions
Some Haskell equations do not fit on one line, and even the ones that do
fit are often better split across several. Guards are only one example of
this.
-- Definition C
iota n =
if n == 0
then
[]
else
iota (n-1) ++ [n]
The offside rule says that
the keywords then and else, if they start a line, must be at the same
level of indentation as the corresponding if, and
if the then and else expressions are on their own lines, these must
be more indented than those keywords.
COMP90048 Declarative Programming Lecture 9 – 16 / 17
Builtin Haskell types
Parametric polymorphism
Here is a version of the code of len complete with type declaration:
len :: [t] -> Int
len [] = 0
len (_:xs) = 1 + len xs
This function, like many others in Haskell, is polymorphic. The phrase
“poly morph” means “many shapes” or “many forms”. In this context,
it means that len can process lists of type t regardless of what type t is,
i.e. regardless of what the form of the elements is.
The reason why len works regardless of the type of the list elements is
that it does not do anything with the list elements.
This version of len shows this in the second pattern: the underscore is a
placeholder for a value you want to ignore.
NOTE
This is called parametric polymorphism because the type variable t is
effectively a type parameter.
Since the underscore matches all values in its position, it is often called a
wild card.
COMP90048 Declarative Programming Lecture 9 – 17 / 17
Defining Haskell types
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 10
Defining Haskell types
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Defining Haskell types
Type definitions
Lihese are similar to Li programmers to define their own types. The
simplest type definitions define types that are similar to enumerated types
in C:
data Gender = Female | Male
data Role = Staff | Student
This defines two new types. The type called Gender has the two values
Female and Male, while the type called Role has the two values Staff
and Student.
Both types are also considered arity-0 type constructors; given zero
argument types, they each construct a type.
The four values are also called data constructors. Given zero arguments,
they each construct a value (a piece of data).
The names of type constructors and data constructors must be identifiers
starting with upper-case letters.
NOTE
"Arity" means the number of arguments. A function of arity 0 takes 0
arguments, a function of arity 1 takes 1 argument, a function of arity 2
takes 2 arguments, and so on. Similarly for type constructors. A type
constructor of arity 0 constructs a type from 0 other types, a type
constructor of arity 1 constructs a type from 1 other type, a type
constructor of arity 2 constructs a type from 2 other types, and so on.
COMP90048 Declarative Programming Lecture 10 – 1 / 14
Defining Haskell types
Using Booleans
You do not have to use such types. If you wish, you can use the standard
Boolean type instead, like this:
show1 :: Bool -> Bool -> String
-- intended usage: show1 isFemale isStaff
show1 True True = "female staff"
show1 True False = "female student"
show1 False True = "male staff"
show1 False False = "male student"
You can use such a function like this:
> let isFemale = True
> let isStaff = False
> show1 isFemale isStaff
COMP90048 Declarative Programming Lecture 10 – 2 / 14
Defining Haskell types
Using defined types vs using Booleans
> show1 isFemale isStaff
> show1 isStaff isFemale
The problem with using Booleans is that of these two calls to show1, only
one matches the programmer’s intention, but since both are type correct
(both supply two Boolean arguments), Haskell cannot catch errors that
switch the arguments.
show2 :: Gender -> Role -> String
With show2, Haskell can and will detect and report any accidental switch.
This makes the program safer and the programmer more productive.
In general, you should use separate types for separate semantic
distinctions. You can use this technique in any language that supports
enumerated types.
COMP90048 Declarative Programming Lecture 10 – 3 / 14
Defining Haskell types
Creating structures
data Card = Card Suit Rank
In this definition, Card is not just the name of the type (from its first
appearance), but (from its second appearance) also the name of the data
constructor which constructs the “structure” from its arguments.
In languages like C, creating a structure and filling it in requires a call to
malloc or its equivalent, a check of its return value, and an assignment to
each field of the structure. This typically takes several lines of code.
In Haskell, you can construct a structure just by writing down the name of
the data constructor, followed by its arguments, like this: Card Club Ace.
This typically takes only part of one line of code.
In practice, this seemingly small difference has a significant impact,
because it removes much clutter (details irrelevant to the main objective).
NOTE
This is the reason for the name “data constructor”.
COMP90048 Declarative Programming Lecture 10 – 5 / 14
Defining Haskell types
Printing values
Many programs have code whose job it is to print out the values of a given
type in a way that is meaningful to the programmer. Such functions are
particularly useful during various forms of debugging.
The Haskell approach is use a function that returns a string. However,
writing such functions by hand can be tedious, because each data
constructor requires its own case:
showrank :: Rank -> String
showrank R2 = "R2"
showrank R3 = "R3"
...
COMP90048 Declarative Programming Lecture 10 – 6 / 14
Defining Haskell types
Show
The Haskell prelude has a standard string conversion function called show.
Just as the arithmetic functions are applicable to all types that are
members of the type class Num, this function is applicable to all types that
are members of the type class Show.
You can tell Haskell that the show function for values of the type Rank is
showrank:
instance Show Rank where show = showrank
This of course requires defining showrank. If you don’t want to do that,
you can get Haskell to define the show function for a type by adding
deriving Show to the type’s definition, like this:
data Rank =
R2 | R3 | R4 | R5 | R6 | R7 | R8 |
R9 | R10 | Jack | Queen | King | Ace
deriving Show
NOTE
The offside rule applies to type definitions as well as function definitions,
so e.g.
data Rank =
R2 | R3 | R4 | R5 | R6 | R7 | R8 |
R9 | R10 | Jack | Queen | King | Ace
deriving Show
would not be valid Haskell: since deriving is part of the definition of the
type Rank, it must be indented more than the line that starts the type
definition.
COMP90048 Declarative Programming Lecture 10 – 7 / 14
Defining Haskell types
Eq and Ord
Another operation even more important than string conversion is
comparison for equality.
To be able to use Haskell’s == comparison operation for a type, it must be
in the Eq type class. This can also be done automatically by putting
deriving Eq at the end of a type definition.
To compare values of a type for order (using <, <=, etc.), the type must be
in the Ord type class, which can also be done by putting deriving Ord at
the end of a type definition. To be in Ord, the type must also be in Eq.
To derive multiple type classes, parenthesise them:
data Suit = Club | Diamond | Heart | Spade
deriving (Show, Eq, Ord)
COMP90048 Declarative Programming Lecture 10 – 8 / 14
Defining Haskell types
Disjunction and conjunction
data Suit = Club | Diamond | Heart | Spade
data Card = Card Suit Rank
A value of type Suit is either a Club or a Diamond or a Heart or a
Spade. This disjunction of values corresponds to an enumerated type.
A value of type Card contains a value of type Suit and a value of type
Rank. This conjunction of values corresponds to a structure type.
In most imperative languages, a type can represent either a disjunction or
a conjunction, but not both at once.
Haskell and related languages do not have this limitation.
COMP90048 Declarative Programming Lecture 10 – 9 / 14
Defining Haskell types
Discriminated union types
Haskell has discriminated union types, which can include both disjunction
and conjunction at once.
Since disjunction and conjunction are operations in Boolean algebra, type
systems that allows them to be combined in this way are often called
algebraic type systems, and their types algebraic types.
data JokerColor = Red | Black
data JCard = NormalCard Suit Rank | JokerCard JokerColor
A value of type JCard is constructed
either using the NormalCard constructor, in which case it contains a
value of type Suit and a value of type Rank,
or using the JokerCard constructor, in which case it contains a value
of type JokerColor.
COMP90048 Declarative Programming Lecture 10 – 10 / 14
Defining Haskell types
Discriminated vs undiscriminated unions
In C, you could try to represent JCard like this:
struct normalcard_struct { ... };
struct jokercard_struct { ... };
union card_union {
struct normalcard_struct normal;
struct jokercard_struct joker;
};
but you wouldn’t know which field of the union is applicable in any given
case. In Haskell, you do (the data constructor tells you), which is why
Haskell’s unions are said to be discriminated.
Note that unlike C’s union types, C’s enumeration types and structure
types are special cases of Haskell’s discriminated union types.
Discriminated union types allow programmers to define types that describe
exactly what they mean.
NOTE
C’s unions are undiscriminated.
A Haskell discriminated union type in which none of the data constructors
have arguments corresponds to a C enumeration type. A Haskell
discriminated union type with only one data constructor corresponds to a
C structure type.
COMP90048 Declarative Programming Lecture 10 – 11 / 14
Defining Haskell types
Maybe
In languages like C, if you have a value of type *T for some type T, or in
languages like Java, if you have a value of some non-primitive type, can
this value be null?
If not, the value represents a value of type T. If yes, the value may
represent a value of type T, or it may represent nothing. The problem is,
often the reader of the code has no idea whether it can be null.
And even if the value must not be null, there’s no guarantee it won’t be.
This can lead to segfaults or NullPointerExceptions.
In Haskell, if a value is optional, you indicate this by using the maybe type
defined in the prelude:
data Maybe t = Nothing | Just t
For any type t, a value of type Maybe t is either Nothing, or Just x,
where x is a value of type t. This is a polymorphic type, like [t].
COMP90048 Declarative Programming Lecture 10 – 14 / 14
Using Haskell Types
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 11
Using Haskell Types
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Using Haskell Types
Representing expressions in C
typedef enum {
EXPR_NUM, EXPR_VAR, EXPR_BINOP, EXPR_UNOP
} ExprKind;
typedef struct expr_struct *Expr;
struct expr_struct
{
ExprKind kind;
int value; /* if EXPR_NUM */
char *name; /* if EXPR_VAR */
Binop binop; /* if EXPR_BINOP */
Unop unop; /* if EXPR_UNOP */
Expr subexpr1; /* if EXPR_BINOP or EXPR_UNOP */
Expr subexpr2; /* if EXPR_BINOP */
};
COMP90048 Declarative Programming Lecture 11 – 1 / 17
Using Haskell Types
Representing expressions in Java
public abstract class Expr {
... abstract methods ...
}
public class NumExpr extends Expr {
int value;
... implementation of abstract methods ...
}
public class VarExpr extends Expr {
String name;
... implementation of abstract methods ...
}
COMP90048 Declarative Programming Lecture 11 – 2 / 17
Using Haskell Types
Representing expressions in Java (2)
public class BinExpr extends Expr {
Binop binop;
Expr arg1;
Expr arg2;
... implementation of abstract methods ...
}
public class UnExpr extends Expr {
Unop unop;
Expr arg;
... implementation of abstract methods ...
}
COMP90048 Declarative Programming Lecture 11 – 3 / 17
Using Haskell Types
Representing expressions in Haskell
data Expr
= Number Int
| Variable String
| Binop Binopr Expr Expr
| Unop Unopr Expr
data Binopr = Plus | Minus | Times | Divide
data Unopr = Negate
As you can see, this is a much more direct definition of the set of values
that the programmer wants to represent.
It is also much shorter, and entirely free of notes that are meaningless to
the compiler and understood only by humans.
COMP90048 Declarative Programming Lecture 11 – 4 / 17
Using Haskell Types
Comparing representions: errors
By far the most important difference is that the C representation is quite
error-prone.
You can access a field when that field is not meaningful, e.g. you can
access the subexpr2 field instead of the subexpr1 field when kind is
EXPR_UNOP.
You can forget to initialize some of the fields, e.g. you can forget to
assign to the name field when setting kind to EXPR_VAR.
You can forget to process some of the alternatives, e.g. when
switching on the kind field, you may handle only three of the four
enum values.
The first mistake is literally impossible to make with Haskell, and would be
caught by the Java compiler. The second is guaranteed to be caught by
Haskell, but not Java. The third will be caught by Java, and by Haskell if
you ask ghc to be on the lookout for it.
NOTE
You can do this by invoking ghc with the option
-fwarn-incomplete-patterns, or by setting the same option inside
ghci with
COMP90048 Declarative Programming Lecture 11 – 5 / 17
Using Haskell Types
Comparing representions: memory
The C representation requires more memory: seven words for every
expression, whereas the Java and Haskell representations needs a
maximum of four (one for the kind, and three for arguments/members).
Using unions can make the C representation more compact, but only at
the expense of more complexity, and therefore a higher probability of
programmer error.
Even with unions, the C representation needs four words for all kinds of
expressions. The Java and Haskell representations need only two for
numbers and variables, and three for expressions built with unary operators.
This is an example where a Java or Haskell program can actually be more
efficient than a C program.
COMP90048 Declarative Programming Lecture 11 – 6 / 17
Using Haskell Types
Comparing representions: maintenance
Adding a new kind of expression requires:
Java: Adding a new class and implementing all the methods for it
C: Adding a new alternative to the enum and adding the needed
members to the type, and adding code for it to all functions
handling that type
Haskell Adding a new alternative, with arguments, to the type, and
adding code for it to all functions handling that type
Adding a new operation for expressions requires:
Java: Adding a new method to the abstract Expr class, and
implementing it for each class
C: Writing one new function
Haskell Writing one new function
COMP90048 Declarative Programming Lecture 11 – 7 / 17
Using Haskell Types
Switching on alternatives
You do not have to have separate equations for each possible shape of the
arguments. You can test the value of a variable (which may or may not be
the value of an argument) in the body of an equation, like this:
is_static :: Expr -> Bool
is_static expr =
case expr of
Number _ -> True
Variable _ -> False
Unop _ expr1 -> is_static expr1
Binop _ expr1 expr2 ->
is_static expr1 && is_static expr2
This function figures out whether the value of an expression can be known
statically, i.e. without having to know the values of variables.
NOTE
As with equations, Haskell matches the value being switched on against
the given patterns in order, from the top down. If you want, you can use
an underscore as a wildcard pattern that matches any value, like in this
example:
COMP90048 Declarative Programming Lecture 11 – 8 / 17
Using Haskell Types
Missing alternatives
If you specify the option -fwarn-incomplete-patterns, ghc and ghci
will warn about any missing alternatives, both in case expressions and in
sets of equations.
This option is particularly useful during program maintenance. When you
add a new alternative to an existing type, all the switches on values of
that type instantly become incorrect. To fix them, a programmer must add
a case to each such switch to handle the new alternative.
If you always compile the program with this option, the compiler will tell
you all the switches in the program that must be modified.
Without such help, programmers must look for such switches themselves,
and they may not find them all.
COMP90048 Declarative Programming Lecture 11 – 9 / 17
Using Haskell Types
The consequences of missing alternatives
If a Haskell program finds a missing alternative at runtime, it will throw an
exception, which (unless caught and handled) will abort the program.
Without a default case, a C program would simply go on and silently
compute an incorrect result. If a default case is provided, it is likely to just
print an error message and abort the program. C programmers thus have
to do more work than Haskell programmers just to get up to the level of
safety offered by Haskell.
If an abstract method is used in Java, this gives the same safety as Haskell.
However, if overriding is used alone, forgetting to write a method for a
subclass will just inherit the (probably wrong) behaviour of the superclass.
NOTE
Switches in C usually need default clauses to compensate for the absence
of type safety. Consider our expression representation example. In theory,
a switch on expr->kind should need only four cases: EXPR_NUM,
EXPR_VAR, EXPR_BINOP, and EXPR_UNOP. However, some other part of the
program could have violated type safety by assigning e.g. (ExprKind) 42
to expr->kind. You need a default case to catch and report such errors.
In Haskell, such bugs cannot happen, since the compiler will not let the
programmer violate type safety.
COMP90048 Declarative Programming Lecture 11 – 10 / 17
Using Haskell Types
Binary search trees
Here is one possible representation of binary search trees in C:
typedef struct bst_struct *BST;
struct bst_struct {
char *key;
int value;
BST left;
BST right;
};
Here it is in Haskell:
data Tree = Leaf | Node String Int Tree Tree
The Haskell version has two alternatives, one of which has no associated
data. The C version uses a null pointer to represent this alternative.
COMP90048 Declarative Programming Lecture 11 – 11 / 17
Using Haskell Types
Counting nodes in a BST
countnodes :: Tree -> Int
countnodes Leaf = 0
countnodes (Node _ _ l r) =
1 + (countnodes l) + (countnodes r)
int countnodes(BST tree)
{
if (tree == NULL) {
return 0;
} else {
return 1 +
countnodes(tree->left) +
countnodes(tree->right);
}
}
NOTE
Since all we are doing is counting nodes, the values of the keys and values
in nodes do not matter.
In most cases, using one-character variable names is not a good idea, since
such names are usually not readable. However, for anyone who knows what
binary search trees are, it should be trivial to figure out that l and r refer
to the left and right subtrees. Similarly for k and v for keys and values.
COMP90048 Declarative Programming Lecture 11 – 13 / 17
Using Haskell Types
Pattern matching vs pointer dereferencing
The left-hand-side of the second equation in the Haskell definition
naturally gives names to each of the fields of the node that actually need
names (because they are used in the right hand side).
These variables do not have to be declared, and Haskell infers their types.
The C version refers to these fields using syntax that dereferences the
pointer tree and accesses one of the fields of the structure it points to.
The C code is longer, and using Haskell-like names for the fields would
make it longer still:
BST l = tree->left;
BST r = tree->right;
...
COMP90048 Declarative Programming Lecture 11 – 14 / 17
Using Haskell Types
Searching a BST in C (iteration)
int search_bst(BST tree, char *key, int *value_ptr)
{
while (tree != NULL) {
int cmp_result;
cmp_result = strcmp(key, tree->key);
if (cmp_result == 0) {
*value_ptr = tree->value;
return TRUE;
} else if (cmp_result < 0) {
tree = tree->left;
} else {
tree = tree->right;
}
}
return FALSE;
}
NOTE
C allows functions to assign to their formal parameters (tree in this case),
since it considers these to be ordinary local variables that just happen to
be initialized by the caller.
COMP90048 Declarative Programming Lecture 11 – 15 / 17
Using Haskell Types
Searching a BST in Haskell
search_bst :: Tree -> String -> Maybe Int
search_bst Leaf _ = Nothing
search_bst (Node k v l r) sk
| sk == k = Just v
| sk < k = search_bst l sk
| otherwise = search_bst r sk
If the search succeeds, this function returns Just v, where v is the
searched-for value.
If the search fails, it returns Nothing.
We could have used Haskell’s if-then-else for this, but guards make
the code look much nicer and easier to read.
NOTE
When the tree being searched is the empty tree, the value of the key being
searched for doesn’t matter.
COMP90048 Declarative Programming Lecture 11 – 16 / 17
Using Haskell Types
Data structure and code structure
The Haskell definitions of countnodes and search_bst have similar
structures:
an equation handling the case where the tree is empty (a Leaf), and
an equation handling the case where the tree is nonempty (a Node).
The type we are processing has two alternatives, so these two functions
have two equations: one for each alternative.
This is quite a common occurrence:
a function whose input is a data structure will need to process all or a
selected part of that data structure, and
what the function needs to do often depends on the shape of the data,
so the structure of the code often mirrors the structure of the data.
NOTE
If a function doesn’t need to process any part of a data structure, it
shouldn’t be passed that data structure as an argument.
COMP90048 Declarative Programming Lecture 11 – 17 / 17
Adapting to Declarative Programming
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 12
Adapting to Declarative
Programming
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Adapting to Declarative Programming
Writing code
Consider a C function with two loops, and some other code around and
between the loops:
... somefunc(...)
{
straight line code A
loop 1
straight line code B
loop 2
straight line code C
}
How can you get the same effect in Haskell?
COMP90048 Declarative Programming Lecture 12 – 1 / 18
Adapting to Declarative Programming
The functional equivalent
loop1func base case
loop1func recursive case
loop2func base case
loop2func recursive case
somefunc =
let ... = ... in
let r1 = loop1func ... in
let ... = ... in
let r2 = loop2func ... in
...
The only effect of the absence of iteration constructs is that instead of
writing a loop inside somefunc, the Haskell programmer needs to write an
auxiliary recursive function, usually outside somefunc.
NOTE
In fact, Haskell allows one function definition to contain another, and if he
or she wished, the programmer could put the definition of e.g. loop1func
inside the definition of somefunc.
COMP90048 Declarative Programming Lecture 12 – 2 / 18
Adapting to Declarative Programming
Example: C
int f(int *a, int size)
{
int i;
int target;
int first_gt_target;
i = 0;
while (i < size && a[i] <= 0) {
i++;
}
target = 2 * a[i];
i++;
while (i < size && a[i] <= target) {
i++;
}
first_gt_target = a[i];
return 3 * first_gt_target;
}
COMP90048 Declarative Programming Lecture 12 – 3 / 18
Adapting to Declarative Programming
Example: Haskell version
f :: [Int] -> Int
f list =
let after_skip = skip_init_le_zero list in
case after_skip of
(x:xs) ->
let target = 2 * x in
let first_gt_target = find_gt xs target in
3 * first_gt_target
skip_init_le_zero :: [Int] -> [Int]
skip_init_le_zero [] = []
skip_init_le_zero (x:xs) =
if x <= 0 then skip_init_le_zero xs else (x:xs)
find_gt :: [Int] -> Int -> Int
find_gt (x:xs) target =
if x <= target then find_gt xs target else x
NOTE
This version has the steps of f directly one after another.
COMP90048 Declarative Programming Lecture 12 – 4 / 18
Adapting to Declarative Programming
Recursion vs iteration
Functional languages do not have language constructs for iteration. What
imperative language programs do with iteration, functional language
programs do with recursion.
For a programmer who has known nothing but imperative languages, the
absence of iteration can seem like a crippling limitation.
In fact, it is not a limitation at all. Any loop can be implemented with
recursion, but some recursions are difficult to implement with iteration.
There are several viewpoints to consider:
How does this affect the process of writing code?
How does this affect the reliability of the resulting code?
How does this affect the productivity of the programmers?
How does this affect the efficiency of the resulting code?
NOTE
There are a few languages called functional languages, such as Lisp, that
do have constructs for iteration. However, these languages are hybrids,
with some functional programming features and some imperative
programming features, and their constructs for iteration belong to the
second category.
COMP90048 Declarative Programming Lecture 12 – 6 / 18
Adapting to Declarative Programming
C version vs Haskell versions
The Haskell versions use lists instead of arrays, since in Haskell, lists are
the natural representation.
With -fwarn-incomplete-patterns, Haskell will warn you that
there may not be a strictly positive number in the list;
there may not be a number greater than the target in the list,
and that these situations need to be handled.
The C compiler cannot generate such warnings. If the Haskell code
operated on an array, the Haskell compiler couldn’t either.
The Haskell versions give meaningful names to the jobs done by the loops.
NOTE
Haskell supports arrays, but their use requires concepts we have not
covered yet.
COMP90048 Declarative Programming Lecture 12 – 7 / 18
Adapting to Declarative Programming
Reliability
The names of the auxiliary functions should remind readers of their tasks.
These functions should be documented like other functions. The
documentation should give the meaning of the arguments, and describe
the relationship between the arguments and the return value.
This description should allow readers to construct a correctness argument
for the function.
The imperative language equivalent of these function descriptions are loop
invariants, but they are as rare as hen’s teeth in real-world programs.
The act of writing down the information needed for a correctness
argument gives programmers a chance to notice situations where the
(implicit or explicit) correctness argument doesn’t hold water.
The fact that such writing down occurs much more often with functional
programs is one factor that tends to make them more reliable.
NOTE
As we saw on the previous slide, the Haskell compiler can help spot bugs
as well.
COMP90048 Declarative Programming Lecture 12 – 8 / 18
Adapting to Declarative Programming
Productivity
Picking a meaningful name for each auxiliary function and writing down its
documentation takes time.
This cost imposed on the original author of the code is repaid manyfold
when
other members of the team read the code, and find it easier to read
and understand,
the original author reads the code much later, and finds it easier to
read and understand.
Properly documented functions, whether created as auxiliary functions or
not, can be reused. Separating the code of a loop out into a function
allows the code of that function to be reused, requiring less code to be
written overall.
In fact, modern functional languages come with large libraries of
prewritten useful functions.
NOTE
Programmers who do not document their code often find later themselves
in the position of reading a part of the program they are working on,
finding that they do not understand it, ask “who wrote this unreadable
mess?”, and then finding out they they wrote it.
Just because you understand your code today does not mean that you will
understand it a few months or few years from now. Using meaningful
names, documenting the meaning of each data structure, the purpose of
each function, the reasons for each design decision, and in general
following the rules of good programming style will help your future self as
well as your teammates in both the present and the future.
COMP90048 Declarative Programming Lecture 12 – 9 / 18
Adapting to Declarative Programming
Efficiency
The recursive version of e.g. search_bst will allocate one stack frame for
each node of the tree it traverses, while the iterative version will just
allocate one stack frame period.
The recursive version will therefore be less efficient, since it needs to
allocate, fill in and then later deallocate more stack frames.
The recursive version will also need more stack space. This should not be
a problem for search_bst, but the recursive versions of some other
functions can run out of stack space.
However, compilers for declarative languages put huge emphasis on the
optimization of recursive code. In many cases, they can take a recursive
algorithm in their source language (e.g. Haskell), and generate iterative
code in their target language.
COMP90048 Declarative Programming Lecture 12 – 10 / 18
Adapting to Declarative Programming
Efficiency in general
Overall, programs in declarative languages are typically slower than they
would be if written in C. Depending on which declarative language and
which language implementation you are talking about, and on what the
program does, the slowdown can range from a few percent to huge integer
factors, such as 10% to a factor of a 100.
However, popular languages like Python and Javascript typically also yield
significantly slower programs than C. In fact, their programs will typically
be significantly slower than corresponding Haskell programs.
In general, the higher the level of a programming language (the more it
does for the programmer), the slower its programs will be on average.
The price of C’s speed is the need to handle all the details yourself.
The right point on the productivity vs efficiency tradeoff continuum
depends on the project (and component of the project).
COMP90048 Declarative Programming Lecture 12 – 11 / 18
Adapting to Declarative Programming
Sublists
Suppose we want to write a Haskell function
sublists :: [a] -> [[a]]
that returns a list of all the “sublists” of a list. A list a is a sublist of a list
b iff every element of a appears in b in the same order, though some
elements of b may be omitted from a. It does not matter in what order
the sublists appear in the resulting list.
For example:
sublists "ABC" = ["ABC","AB","AC","A","BC","B","C",""]
sublists "BC" = ["BC","B","C",""]
How would you implement this?
COMP90048 Declarative Programming Lecture 12 – 12 / 18
Adapting to Declarative Programming
Declarative thinking
Some problems are difficult to approach imperatively, and are much easier
to think about declaratively.
The imperative approach is procedural: we devise a way to solve the
problem step by step. As an afterthought we may think about grouping
the steps into chunks (methods, procedures, functions, etc.)
The declarative approach breaks down the problem into chunks
(functions), assembling the results of the chunks to construct the result.
You must be careful in imperative languages, because the chunks may not
compose due to side-effects. The chunks always compose in purely
declarative languages.
COMP90048 Declarative Programming Lecture 12 – 13 / 18
Adapting to Declarative Programming
Recursive thinking
One especially useful approach is recursive thinking: use the function you
are defining as one of the chunks.
To take this approach:
1 Determine how to produce the result for the whole problem from the
results for the parts of the problem (recursive case);
2 Determine the solution for the smallest part of the input (base case).
Keep in mind the specification of the problem, but it also helps to think of
concrete examples.
For lists, (1) usually means generating the result for the whole list from
the list head and the result for the tail; (2) usually means the result for the
empty list.
This works perfectly well in most imperative languages, if you’re careful to
ensure your function composes. But it takes practice to think this way.
COMP90048 Declarative Programming Lecture 12 – 14 / 18
Adapting to Declarative Programming
Sublists again
Write a Haskell function:
sublists :: [a] -> [[a]]
sublists "ABC" = ["ABC","AB","AC","A","BC","B","C",""]
sublists "BC" = ["BC","B","C",""]
How can we produce ["ABC","AB","AC","A","BC","B","C",""] from
["BC","B","C",""] and A?
It is just ["BC","B","C",""] with A added to the front of each string,
followed by ["BC","B","C",""] itself.
For the base case, the only sublist of [] is [] itself, so the list of sublists
of [] is [[]].
COMP90048 Declarative Programming Lecture 12 – 15 / 18
Adapting to Declarative Programming
Sublists again
The problem becomes quite simple when we think about it declaratively
(recursively). The sublists of a list is the sublists of its tail both with and
without the head of the list added to the front of each sublist.
sublists :: [a] -> [[a]]
sublists [] = [[]]
sublists (e:es) = addToEach e restSeqs ++ restSeqs
where restSeqs = sublists es
addToEach :: a -> [[a]] -> [[a]]
addToEach h [] = []
addToEach h (t:ts) = (h:t):addToEach h ts
COMP90048 Declarative Programming Lecture 12 – 16 / 18
Adapting to Declarative Programming
Immutable data structures
In declarative languages, data structures are immutable: once created,
they cannot be changed. So what do you do if you do need to update a
data structure?
You create another version of the data structure, one which has the
change you want to make, and use that version from then on.
However, if you want to, you can hang onto the old version as well. You
will definitely want to do so if some part of the system still needs the old
version (in which case imperative code must also make a modified copy).
The old version can also be used
because both old and new are needed, as in sublists
to implement undo
to gather statistics, e.g. about how the size of a data structure
changes over time
COMP90048 Declarative Programming Lecture 12 – 17 / 18
Adapting to Declarative Programming
Updating a BST
insert_bst :: Tree -> String -> Int -> Tree
insert_bst Leaf ik iv = Node ik iv Leaf Leaf
insert_bst (Node k v l r) ik iv
| ik == k = Node ik iv l r
| ik < k = Node k v (insert_bst l ik iv) r
| otherwise = Node k v l (insert_bst r ik iv)
Note that all of the code of this function is concerned with the job at
hand; there is no code concerned with memory management.
In Haskell, as in Java, memory management is automatic. Any
unreachable cells of memory are recovered by the garbage collector.
COMP90048 Declarative Programming Lecture 12 – 18 / 18
Polymorphism
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 13
Polymorphism
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Polymorphism
Polymorphic types
Our definition of the tree type so far was this:
data Tree = Leaf | Node String Int Tree Tree
This type assumes that the keys are strings and the values are integers.
However, the functions we have written to handle trees (countnodes and
search_bst do not really care about the types of the keys and values.
We could also define trees like this:
data Tree k v = Leaf | Node k v (Tree k v) (Tree k v)
In this case, k and v are type variables, variables standing in for the types
of keys and values, and Tree is a type constructor, which constructs a new
type from two other types.
COMP90048 Declarative Programming Lecture 13 – 1 / 18
Polymorphism
Using polymorphic types: countnodes
With the old, monomorphic definition of Tree, the type declaration or
signature of countnodes was:
countnodes :: Tree -> Int
With the new, polymorphic definition of Tree, it will be
countnodes :: Tree k v -> Int
Regardless of the types of the keys and values in the tree, countnodes will
count the number of nodes in it.
The exact same code works in these cases.
NOTE
In C++, you can use templates to arrange to get the same piece of code
to work on values of different types, but the C++ compiler will generate
different object code for each type. This can make executables
significantly bigger than they need to be.
Generics in Java bear more similarity to Haskell Type Classes, however,
they have some limitations that Type Classes do not share, because of the
way they are implemented.
COMP90048 Declarative Programming Lecture 13 – 2 / 18
Polymorphism
Using polymorphic types: search_bst
countnodes does not touch keys or values, but search_bst does perform
some operations on keys. Replacing
search_bst :: Tree -> String -> Maybe Int
with
search_bst :: Tree k v -> k -> Maybe v
will not work; it will yield an error message.
The reason is that search_bst contains these two tests:
a comparison for equality: sk == k, and
a comparison for order: sk < k.
COMP90048 Declarative Programming Lecture 13 – 3 / 18
Polymorphism
Comparing values for equality and order
Some types cannot be compared for equality. For example, two functions
should be considered equal if for all sets of input argument values, they
compute the same result. Unfortunately, it has been proven that testing
whether two functions are equal is undecidable. This means that building
an algorithm that is guaranteed to decide in finite time whether two
functions are equal is impossible.
Some types that can be compared for equality cannot be compared for
order. Consider a set of integers. It is obvious that {1, 5} is not equal to
{2, 4}, but using the standard method of set comparison (set inclusion),
they are otherwise incomparable; neither can be said to be greater than
the other.
COMP90048 Declarative Programming Lecture 13 – 4 / 18
Polymorphism
Eq and Ord
In Haskell,
comparison for equality can only be done on values of types that
belong to the type class Eq, while
comparison for order can only be done on values of types that belong
to the type class Ord.
Membership of Ord implies membership of Eq, but not vice versa.
The declaration of search_bst should be this:
search_bst :: Ord k => Tree k v -> k -> Maybe v
The construct Ord k => is a type class constraint; it says search_bst
requires whatever type k stands for to be in Ord. This guarantees its
membership of Eq as well.
COMP90048 Declarative Programming Lecture 13 – 5 / 18
Polymorphism
Data.Map
The polymorphic Tree type described above is defined in the standard
library with the name Map, in the module Data.Map. You can import it
with the declaration:
import Data.Map as Map
The key functions defined for Maps include:
insert :: Ord k => k -> a -> Map k a -> Map k a
Map.lookup :: Ord k => k -> Map k a -> Maybe a
(!) :: Ord k => Map k a -> k -> a -- infix operator
size :: Map k a -> Int
. . . and many, many more functions; see the documentation.
NOTE
(!) is an infix operator, and m ! k throws an exception if key k is not
present in map m.
COMP90048 Declarative Programming Lecture 13 – 6 / 18
Polymorphism
Deriving membership automatically
data Suit = Club | Diamond | Heart | Spade
deriving (Show, Eq, Ord)
data Card = Card Suit Rank
deriving (Show, Eq, Ord)
The automatically created comparison function takes the order of data
constructors from the order in the declaration itself: a constructor listed
earlier is less than a constructor listed later (e.g. Club < Diamond).
If the two values being compared have the same top level data
constructor, the automatically created comparison function compares their
arguments in turn, from left to right. This means the argument types
must also be instances of Ord. If the corresponding arguments are not
equal, the comparison stops (e.g. Card Club Ace < Card Spade Jack);
if the corresponding argument are equal, it goes on to the next argument,
if there is one (e.g. Card Spade Ace > Card Spade Jack). This is
called lexicographic ordering.
COMP90048 Declarative Programming Lecture 13 – 8 / 18
Polymorphism
Recursive vs nonrecursive types
data Tree = Leaf | Node String Int Tree Tree
data Card = Card Suit Rank
Tree is a recursive type because some of its data constructors have
arguments of type Tree.
Card is a non-recursive type because none of its data constructors have an
arguments of type Card.
A recursive type needs a nonrecursive alternative, because without one, all
values of the type would have infinite size.
COMP90048 Declarative Programming Lecture 13 – 9 / 18
Polymorphism
Mutually recursive types
Some types are recursive but not directly recursive.
data BoolExpr
= BoolConst Bool
| BoolOp BoolOp BoolExpr BoolExpr
| CompOp CompOp IntExpr IntExpr
data IntExpr
= IntConst Int
| IntOp IntOp IntExpr IntExpr
| IntIfThenElse BoolExpr IntExpr IntExpr
In a mutually recursive set of types, it is enough for one of the types to
have a nonrecursive alternative.
These types represent Boolean- and integer-valued expressions in a
program. They must be mutually recursive because comparison of integers
returns a Boolean and integer-valued conditionals use a Boolean.
NOTE
Provided that at least one of the recursive alternatives is only mutually
recursive, not directly recursive. That way, a value built using that
alternative can be of finite size.
COMP90048 Declarative Programming Lecture 13 – 10 / 18
Polymorphism
Structural induction
Code that follows the shape of a nonrecursive type tends to be simple.
Code that follows the shape of a directly or mutually recursive type tends
to be more interesting.
Consider a recursive type with one nonrecursive data constructor (like
Leaf in Tree) and one recursive data constructor (like Node in Tree).
A function that follows the structure of this type will typically have
an equation for the nonrecursive data constructor, and
an equation for the recursive data constructor.
Typically, recursive calls will occur only in the second equation, and the
switched-on argument in the recursive call will be strictly smaller than the
corresponding argument in the left hand side of the equation.
COMP90048 Declarative Programming Lecture 13 – 11 / 18
Polymorphism
Proof by induction
You can view the function definition’s structure as the outline of a
correctness argument.
The argument is a proof by induction on n, the number of data
constructors of the switched-on type in the switched-on argument.
Base case: If n = 1, then the applicable equation is the base case.
If the first equation is correct, then the function correctly handles the
case where n = 1.
Induction step: Assume the induction hypothesis: the function
correctly handles all cases where n ≤ k. This hypothesis implies that
all the recursive calls are correct. If the second equation is correct,
then the function correctly handles all cases where n ≤ k + 1.
The base case and the induction step together imply that the function
correctly handles all inputs.
COMP90048 Declarative Programming Lecture 13 – 12 / 18
Polymorphism
Formality
If you want, you can use these kinds of arguments to formally prove the
correctness of functions, and of entire functional programs.
This typically requires a formal specification of the expected relationship
between each function’s arguments and its result.
Typical software development projects do not do formal proofs of
correctness, regardless of what kind of language their code is written in.
However, projects using functional languages do tend to use informal
correctness arguments slightly more often.
The support for this provided by the original programmer usually consists
of nothing more than a natural language description of the criterion of
correctness of each function. Readers who want a correctness argument
can then construct it for themselves from this and the structure of the
code.
COMP90048 Declarative Programming Lecture 13 – 13 / 18
Polymorphism
Structural induction for more complex types
If a type has nr nonrecursive data constructors and r recursive data
constructors, what happens when nr > 1 or r > 1, like BoolExpr and
IntExpr?
You can do structural induction on such types as well.
Such functions will typically have nr nonrecursive equations and r
recursive equations, but not always. Sometimes you need more than one
equation to handle a constructor, and sometimes one equation can handle
more than one constructor. For example, sometimes all base cases need
the same treatment.
Picking the right representation of the data is important in every program,
but when the structure of the code follows the structure of the data, it is
particularly important.
NOTE
If all the base cases can be handled the same way, then the function can
start with r equations for the recursive constructors, followed by a single
equation using a wildcard pattern that handles all the nonrecursive
constructors.
COMP90048 Declarative Programming Lecture 13 – 14 / 18
Polymorphism
Let clauses and where clauses
assoc_list_to_bst ((hk, hv):kvs) =
let t0 = assoc_list_to_bst kvs
in insert_bst t0 hk hv
A let clause let name = expr in mainexpr introduces a name for a
value to be used in the main expression.
assoc_list_to_bst ((hk, hv):kvs) = insert_bst t0 hk hv
where t0 = assoc_list_to_bst kvs
A where clause mainexpr where name = expr has the same meaning,
but has the definition of the name after the main expression.
Which one you want to use depends on where you want to put the
emphasis.
But you can only use where clauses at the top level of a function, while
you can use a let for any expression.
NOTE
In theory, you can also mix the two, like this:
let name1 = expr1
in mainexpr
where name2 = expr2
However, you should not do this, since it is definitely bad programming
style.
COMP90048 Declarative Programming Lecture 13 – 17 / 18
Polymorphism
Defining multiple names
You can define multiple names with a single let or where clause:
let name1 = expr1
name2 = expr2
in mainexpr
or
mainexpr
where
name1 = expr1
name2 = expr2
The scope of each name includes the right hand sides of the definitions of
the following names, as well as the main expression, unless one of the later
definitions defines the same name, in which case the original definition is
shadowed and not visible from then on.
COMP90048 Declarative Programming Lecture 13 – 18 / 18
Higher order functions
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 14
Higher order functions
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Higher order functions
First vs higher order
First order values are data.
Second order values are functions whose arguments and results are first
order values.
Third order values are functions whose arguments and results are first or
second order values.
In general, nth order values are functions whose arguments and results are
values of any order from first up to n − 1.
Values that belong to an order higher than first are higher order values.
Java 8, released mid-2014, supports higher order programming. C also
supports it, if you work at it. Higher order programming is a central aspect
of Haskell, often allowing Haskell programmers to avoid writing recursive
functions.
COMP90048 Declarative Programming Lecture 14 – 1 / 19
Higher order functions
A higher order function in C
IntList filter(Bool (*f)(int), IntList list)
{
IntList filtered_tail, new_list;
if (list == NULL) {
return NULL;
} else {
filtered_tail = filter(f, list->tail);
if ((*f)(list->head)) {
new_list = checked_malloc(sizeof(*new_list));
new_list->head = list->head;
new_list->tail = filtered_tail;
return new_list;
} else {
return filtered_tail;
}
}
}
NOTE
This code assumes type definitions like these:
typedef struct intlist_struct *IntList;
struct intlist_struct {
int head;
IntList tail;
};
typedef int Bool;
Unfortunately, although the last typedef allows programmers to write Bool
instead of int to tell readers of the code that something (in this case, the
return value of the function) is meant to be used to represent only a
TRUE/FALSE distinction, the C compiler will not report as errors any
arithmetic operations on booleans, any mixing of booleans and integers, or
in general any unintended use of a boolean as an integer or vice versa.
Since booleans and integers are distinct builtin types in Haskell, any such
errors in Haskell programs will be caught by the Haskell implementation.
COMP90048 Declarative Programming Lecture 14 – 2 / 19
Higher order functions
A higher order function in Haskell
Haskell’s syntax for passing a function as an argument is much simpler
than C’s syntax. All you need to do is wrap the type of the higher order
argument in parentheses to tell Haskell it is one argument.
filter :: (a -> Bool) -> [a] -> [a]
filter _ [] = []
filter f (x:xs) =
if f x then x:fxs else fxs
where fxs = filter f xs
Even though it is significantly shorter, this function is actually more
general than the C version, since it is polymorphic, and thus works for lists
with any type of element.
filter is defined in the Haskell prelude.
COMP90048 Declarative Programming Lecture 14 – 3 / 19
Higher order functions
Using higher order functions
You can call filter like this:
... filter is_even [1, 2, 3, 4] ...
... filter is_pos [0, -1, 1, -2, 2] ...
... filter is_long ["a", "abc", "abcde"] ...
given definitions like this:
is_even :: Int -> Bool
is_even x = if (mod x 2) == 0 then True else False
is_pos :: Int -> Bool
is_pos x = if x > 0 then True else False
is_long :: String -> Bool
is_long x = if length x > 3 then True else False
NOTE
length is a function defined in the Haskell prelude. As its name implies, it
returns the length of a list. Remember that in Haskell, a string is a list of
characters.
The mod function is similar to the % operator in C: in this case, it returns
the remainder after dividing x by 2.
COMP90048 Declarative Programming Lecture 14 – 4 / 19
Higher order functions
Backquote
Modulo is a built-in infix operator in many languages. For example, in C or
Java, 5 modulo 2 would be written 5 % 2.
Haskell uses mod for the modulo operation, but Haskell allows you to make
any function an infix operator by surrounding the function name with
backquotes (backticks, written ‘).
So a friendlier way to write the is_even function would be:
is_even :: Int -> Bool
is_even x = x ‘mod‘ 2 == 0
Operators written with backquotes have high precedence and associate to
the left.
It’s also possible to explicitly declare non-alphanumeric operators, and
specify their associativity and fixity, but this feature should be used
sparingly.
COMP90048 Declarative Programming Lecture 14 – 6 / 19
Higher order functions
Anonymous functions
In some cases, the only thing you need a function for is to pass as an
argument to a higher order function like filter. In such cases, readers
may find it more convenient if the call contained the definition of the
function, not its name.
In Haskell, anonymous functions are defined by lambda expressions, and
you use them like this.
... filter (\x -> x ‘mod‘ 2 == 0) [1, 2, 3, 4] ...
... filter (\s -> length s > 3) ["a", "abc", "abcde"] ...
This notation is based on the lambda calculus, the basis of functional
programming.
In the lambda calculus, each argument is preceded by a lambda, and the
argument list is followed by a dot and the expression that is the function
body. For example, the function that adds together its two arguments is
written as λa.λb.a + b.
NOTE
These calls are equivalent to the calls
... filter is_even [1, 2, 3, 4] ...
... filter is_long ["a", "abc", "abcde"] ...
given our earlier definitions of is_even and is_long.
COMP90048 Declarative Programming Lecture 14 – 7 / 19
Higher order functions
Map
(Not to be confused with Data.Map.)
map is one of the most frequently used Haskell functions. (It is defined in
the Haskell prelude.) Given a function and a list, map applies the function
to every member of the list.
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = (f x):(map f xs)
Many things that an imperative programmer would do with a loop,
a functional programmer would do with a call to map. An example:
get_names :: [Customer] -> [String]
get_names customers = map customer_name customers
This assumes that customer_name is a function whose type is
Customer -> String.
COMP90048 Declarative Programming Lecture 14 – 8 / 19
Higher order functions
Partial application
Given a function with n arguments, partially applying that function means
giving it its first k arguments, where k < n.
The result of the partial application is a closure that records the identity of
the function and the values of those k arguments.
This closure behaves as a function with n − k arguments. A call of the
closure leads to a call of the original function with both sets of arguments.
is_longer :: Int -> String -> Bool
is_longer limit x = length x > limit
... filter (is_longer 4) ["ab", "abcd", "abcdef"] ...
In this case, the function is_longer takes two arguments. The expression
is_longer 4 partially applies this function, and creates a closure which
records 4 as the value of the first argument.
NOTE
There is no way to partially apply a function (as opposed to an operator,
see below) by supplying it with k arguments if those not are not the first k
arguments.
Sometimes, programmers define small, sometimes anonymous helper
functions which simply call another function with a different order of
arguments, the point being to bring the k arguments you want to supply
to that function to the start of the argument list of the helper function.
COMP90048 Declarative Programming Lecture 14 – 9 / 19
Higher order functions
Calling a closure: an example
filter f (x:xs) =
if f x then x:fxs else fxs
where fxs = filter f xs
... filter (is_longer 4) ["ab", "abcd", "abcdef"] ...
In this case, the code of filter will call is_longer three times:
is_longer 4 "ab"
is_longer 4 "abcd"
is_longer 4 "abcdef"
Each of these calls comes from the higher order call f x in filter. In this
case f represents the closure is_longer 4. In each case, the first
argument comes from the closure, with the second being the value of x.
COMP90048 Declarative Programming Lecture 14 – 10 / 19
Higher order functions
Operators and sections
If you enclose an infix operator in parentheses, you can partially apply it by
enclosing its left or right operand with it; this is called a section.
Prelude> map (*3) [1, 2, 3]
[3,6,9]
You can use section notation to partially apply either of its arguments.
Prelude> map (5 ‘mod‘) [3, 4, 5, 6, 7]
[2,1,0,5,5]
Prelude> map (‘mod‘ 3) [3, 4, 5, 6, 7]
[0,1,2,0,1]
COMP90048 Declarative Programming Lecture 14 – 12 / 19
Higher order functions
Types for partial application
In most languages, the type of a function with n arguments would be
something like:
f :: (at1, at2, ... atn) -> rt
where at1, at2 etc are the argument types, (at1, at2, ... atn) is
the type of a tuple containing all the arguments, and rt is the result type.
To allow the function to be partially applied by supplying the first
argument, you need a function with a different type:
f :: at1 -> ((at2, ... atn) -> rt)
This function takes a single value of type at1, and returns as its result
another function, which is of type (at2, ... atn) -> rt.
COMP90048 Declarative Programming Lecture 14 – 14 / 19
Higher order functions
Currying
You can keep transforming the function type until every single argument is
supplied separately:
f :: at1 -> (at2 -> (at3 -> ... (atn -> rt)))
The transformation from a function type in which all arguments are
supplied together to a function type in which the arguments are supplied
one by one is called currying.
In Haskell, all function types are curried. This is why the syntax for
function types is what it is. The arrow that makes function types is right
associative, so the second declaration below just shows explicitly the
parenthesization implicit in the first:
is_longer :: Int -> String -> Bool
is_longer :: Int -> (String -> Bool)
NOTE
Currying and the Haskell programming language are both named after the
same person, the English mathematician Haskell Brooks Curry. He did a
lot to develop the lambda calculus, the mathematical foundation of
functional programming, and a result, he is a popular guy in functional
programming circles. In fact, there are two functional programming
languages named for him: besides Haskell, there is another one named
Curry.
COMP90048 Declarative Programming Lecture 14 – 15 / 19
Higher order functions
Functions with all their arguments
Given a function with curried argument types, you can supply the function
its first argument, then its second, then its third, and so on. What
happens when you have supplied them all?
is_longer 3 "abcd"
There are two things you can get:
a closure that contains all the function’s arguments, or
the result of the evaluation of the function.
In C and in most other languages, these would be very different, but in
Haskell, as we will see later, they are equivalent.
COMP90048 Declarative Programming Lecture 14 – 16 / 19
Higher order functions
Composing functions
Any function that makes a higher order function call or creates a closure
(e.g. by partially applying another function) is a second order function.
This means that both filter and its callers are second order functions.
filter has a piece of data as an argument (the list to filter) as well as a
function (the filtering function). Some functions do not take any piece of
data as arguments; all their arguments are functions.
The builtin operator ‘.’ composes two functions. The expression f . g
represents a function which first calls g, and then invokes f on the result:
(f . g) x = f (g x)
If the type of x is represented by the type variable a, then the type of g
must be a -> b for some b, and the type of f must be b -> c for some c.
The type of . itself is therefore (b -> c) -> (a -> b) -> (a -> c).
NOTE
There is nothing preventing some or all of those type variables actually
standing for the same type.
COMP90048 Declarative Programming Lecture 14 – 17 / 19
Higher order functions
Composing functions: some examples
Suppose you already have a function that sorts a list and a function that
returns the head of a list, if it has one. You can then compute the
minimum of the list like this:
minimum = head . sort
If you also have a function that reverses a list, you can also compute the
maximum with very little extra code:
maximum = head . reverse . sort
This shows that functions created by composition, such as reverse .
sort, can themselves be part of further compositions.
This style of programming is sometimes called point-free style, though
value-free style would be a better description, since its distinguishing
characteristic is the absence of variables representing (first order) values.
NOTE
The . operator is right associative, so head . reverse . sort
parenthesizes as head . (reverse . sort), not as (head . reverse)
. sort, even though the two parenthesizations in fact yield functions that
compute the same answers for all possible argument values.
Given the above definition, maximum xs is equivalent to (head .
reverse . sort) xs, which in turn is equivalent to head (reverse
(sort xs)).
Code written in point-free style can contain variables representing
functions; the definition of . is an example.
Code written in point free style is usually very short, and it can be very
elegant. However, while elegance is nice, it is not the most important
characteristic that programmers should strive for. If most readers cannot
understand a piece of code, its elegance to the few readers that do
understand it is of little concern, and unfortunately, a large fraction of
programmers find code written in point-free style hard to understand.
COMP90048 Declarative Programming Lecture 14 – 18 / 19
Higher order functions
Composition as sequence
Function composition is one way to express a sequence of operations.
Consider the function composition step3f . step2f . step1f.
1 You start with the input, x.
2 You compute step1f x.
3 You compute step2f (step1f x).
4 You compute step3f (step2f (step1f x)).
This idea is the basis of monads, which is the mechanism Haskell uses to
do input/output.
COMP90048 Declarative Programming Lecture 14 – 19 / 19
Functional design patterns
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 15
Functional design patterns
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Functional design patterns
Higher order programming
Higher order programming is widely used by functional programmers.
Its advantages include
code reuse,
a higher level of abstraction, and
a set of canned solutions to frequently encountered problems.
In programs written by programmers who do not use higher order
programming, you frequently find pieces of code that have the same
structure but slot different pieces of code into that structure.
Such code typically qualifies as an instance of the copy-and-paste
programming antipattern, a pattern that programmers should strive to
avoid.
NOTE
This is an antipattern because the many different copies of the code
structure violate a variant of the principle of single point of control. If you
find a bug in one of the copies, that bug may be present in other copies as
well, but you don’t get help in finding out where those copies are. With
higher order code, fixing the code of the higher order function itself fixes
that bug in all calls to it.
COMP90048 Declarative Programming Lecture 15 – 1 / 21
Functional design patterns
Folds
We have already seen the functions map and filter, which operate on
and transform lists.
The other class of popular higher order functions on lists are the reduction
operations, which reduce a list to a single value.
The usual reduction operations are folds. There are three main folds: left,
right and balanced.
left ((((I X1) X2) . . .) Xn)
right (X1 (X2 (. . . (Xn I))))
balanced ((X1 X2) (X3 X4)) . . .
Here denotes a binary function, the folding operation, and I denotes the
identity element of that operation. (The balanced fold also needs the
identity element in case the list is empty.)
COMP90048 Declarative Programming Lecture 15 – 2 / 21
Functional design patterns
Foldl
foldl :: (v -> e -> v) -> v -> [e] -> v
foldl _ base [] = base
foldl f base (x:xs) =
let newbase = f base x in
foldl f newbase xs
suml :: Num a => [a] -> a
suml = foldl (+) 0
productl :: Num a => [a] -> a
productl = foldl (*) 1
concatl :: [[a]] -> [a]
concatl = foldl (++) []
COMP90048 Declarative Programming Lecture 15 – 3 / 21
Functional design patterns
Foldr
foldr :: (e -> v -> v) -> v -> [e] -> v
foldr _ base [] = base
foldr f base (x:xs) =
let fxs = foldr f base xs in
f x fxs
sumr = foldr (+) 0
productr = foldr (*) 1
concatr = foldr (++) []
You can define sum, product and concatenation in terms of both foldl
and foldr because addition and multiplication on integers, and list
append, are all associative operations.
NOTE
The declarations of sumr, productr and concatr differ from the
declarations of suml, productl and concatl only in the function name.
Addition and multiplication are not associative on floating point numbers,
because of their limited precision. For the sake of simplicity in the
discussion, suppose the limit is four decimal digits in the fraction, and
suppose you want to sum up the list of numbers [0.25, 0.25,0.25, 0.25,
1000]. If you do the additions from left to right, the first addition adds
0.25 and 0.25 giving 0.5, the second adds 0.5 and 0.25 giving 0.75, the
third adds 0.75 and 0.25 giving 1.0, and the fourth adds 1 and 1000,
yielding 1001. The final addition of the identity element 0 does not change
this result. However, if you do the additions right to left, the result is
different. This is because adding 0.25 and 1000 cannot yield 1000.25,
since that has too many digits. Instead, 1000.25 must be rounded to the
nearest number with four decimal digits, which will be 1000. The next
three additions of 0.25 and the final addition of 0 will still leave the overall
result at 1000.
While concatl and concatr are guaranteed to generate the same result,
concatr is much more efficient than concatl. We will discuss this later.
COMP90048 Declarative Programming Lecture 15 – 4 / 21
Functional design patterns
Balanced fold
balanced_fold :: (e -> e -> e) -> e -> [e] -> e
balanced_fold _ b [] = b
balanced_fold _ _ (x:[]) = x
balanced_fold f b l@(_:_:_) =
let
len = length l
(half1, half2) = splitAt (len ‘div‘ 2) l
value1 = balanced_fold f b half1
value2 = balanced_fold f b half2
in
f value1 value2
splitAt n l returns a pair of the first n elements of l and the rest of l.
It is defined in the standard Prelude.
NOTE
This code does some wasted work. Each call to balanced_fold computes
the length of its list, but for recursive calls, the caller already knows the
length. (It is div len 2 for half1 and len - (div len 2) for half2.)
Also, balanced_fold does not make recursive calls unless the length of
the list is at least two. This means that the length of both half1 and
half2 will be at least one, which means that the test for zero length lists
can succeed only for the top level call; for all the recursive calls, that test
is wasted work. It is of course possible to write a balanced fold that does
not have either of these performance problems.
COMP90048 Declarative Programming Lecture 15 – 5 / 21
Functional design patterns
More folds
The Haskell prelude defines sum, product, and concat.
For maximum and minimum, there is no identity element, and it is an error
if the list is empty. For such cases, the Haskell Prelude defines:
foldl1 :: (a -> a -> a) -> [a] -> a
foldr1 :: (a -> a -> a) -> [a] -> a
that compute
foldl1 o [X1,X2, . . . ,Xn] = ((X1 X2) . . . Xn)
foldr1 o [X1,X2, . . . ,Xn] = (X1 (X2 . . .Xn))
maximum = foldr1 max
minimum = foldr1 min
You could equally well use foldl1 for these.
COMP90048 Declarative Programming Lecture 15 – 7 / 21
Functional design patterns
Folds are really powerful
You can compute the length of a list by summing 1 for each element,
instead of the element itself. So if we can define a function that takes
anything and returns 1, together with (+) we can use fold to define
length.
const :: a -> b -> a
const a b = a
length = foldr ((+) . const 1) 0
You can map over a list with foldr:
map f = foldr ((:) . f) []
COMP90048 Declarative Programming Lecture 15 – 8 / 21
Functional design patterns
Fold can reverse a list
If we had a “backwards” (:) operation, call it snoc, then foldl could
reverse a list:
reverse [X1,X2, . . .Xn] = [] 8snoc8 X1 8snoc8 X2 . . . 8snoc8 Xn
snoc :: [a] -> a -> [a]
snoc tl hd = hd:tl
reverse = foldl snoc []
But the Haskell Prelude defines a function to flip the arguments of a
binary function:
flip :: (a -> b -> c) -> b -> a -> c
flip f x y = f y x
reverse = foldl (flip (:)) []
COMP90048 Declarative Programming Lecture 15 – 9 / 21
Functional design patterns
Foldable
But what about types other than lists? Can we fold over them?
Actually, we can:
Prelude> sum (Just 7)
7
Prelude> sum Nothing
0
In fact we can fold over any type in the type class Foldable
Prelude> :t foldr
foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b
We can declare our own types to be instances of Foldable by defining
foldr for our type; then many standard functions, such length, sum, etc.
will work on that type, too.
COMP90048 Declarative Programming Lecture 15 – 10 / 21
Functional design patterns
List comprehensions
Haskell has special syntax for one class of higher order operations. These
two implementations of quicksort do the same thing, with the first using
conventional higher order code, and the second using list comprehensions:
qs1 [] = []
qs1 (x:xs) = qs1 littles ++ [x] ++ qs1 bigs
where
littles = filter (bigs = filter (>=x) xs
qs2 [] = []
qs2 (x:xs) = qs2 littles ++ [x] ++ qs2 bigs
where
littles = [l | l <- xs, l < x]
bigs = [b | b <- xs, b >= x]
COMP90048 Declarative Programming Lecture 15 – 11 / 21
Functional design patterns
List comprehensions
List comprehensions can be used for things other than filtering a single list.
In general, a list comprehension consists of
a template (an expression, which is often just a variable)
one or more generators (each of the form var <- list),
zero or more tests (boolean expressions),
zero or more let expressions defining local variables.
Some more examples:
columns = "abcdefgh"
rows = "12345678"
chess_squares = [[c, r] | c <- columns, r <- rows]
chess_squares = [[c, r]
| c <- columns, r <- rows]
pairs = [(a, b) | a <- [1, 2, 3], b <- [1, 2, 3]]
nums = [10*a+b | a <- [1, 2, 3], b <- [1, 2, 3]]COMP90048 Declarative Programming Lecture 15 – 12 / 21
Functional design patterns
Traversing HTML documents
Types to represent HTML documents:
type HTML = [HTML_element]
data HTML_element
= HTML_text String
| HTML_font Font_tag HTML
| HTML_p HTML
data Font_tag = Font_tag (Maybe Int)
(Maybe String)
(Maybe Font_color)
data Font_color
= Colour_name String
| Hex Int
| RGB Int Int Int
COMP90048 Declarative Programming Lecture 15 – 13 / 21
Functional design patterns
Collecting font sizes
font_sizes_in_html :: HTML -> Set Int -> Set Int
font_sizes_in_html elements sizes =
foldr font_sizes_in_elt sizes elements
font_sizes_in_elt :: HTML_element -> Set Int -> Set Int
font_sizes_in_elt (HTML_text _) sizes = sizes
font_sizes_in_elt (HTML_font font_tag html) sizes =
let
Font_tag maybe_size _ _ = font_tag
newsizes = case maybe_size of
Nothing -> sizes
Just fontsize -> Data.Set.insert fontsize sizes
in
font_sizes_in_html html newsizes
font_sizes_in_elt (HTML_p html) sizes =
font_sizes_in_html html sizes
NOTE
Normally, font_sizes_in_html would be invoked with the empty set as
the second argument, which would mean that the returned set is the set of
font sizes appearing in the given HTML page description. The second
argument of font_sizes_in_html thus plays the role of an accumulator.
COMP90048 Declarative Programming Lecture 15 – 14 / 21
Functional design patterns
Collecting font names
font_names_in_html :: HTML -> Set String -> Set String
font_names_in_html elements names =
foldr font_names_in_elt names elements
font_names_in_elt :: HTML_element -> Set String -> Set String
font_names_in_elt (HTML_text _) names = names
font_names_in_elt (HTML_font font_tag html) names =
let
Font_tag _ maybe_name _ = font_tag
newnames = case maybe_name of
Nothing -> names
Just fontname -> Data.Set.insert fontname names
in
font_names_in_html html newnames
font_names_in_elt (HTML_p html) names =
font_names_in_html html names
COMP90048 Declarative Programming Lecture 15 – 15 / 21
Functional design patterns
Collecting any font information
font_stuff_in_html :: (Font_tag -> a -> a) -> HTML -> a -> a
font_stuff_in_html f elements stuff =
foldr (font_stuff_in_elt f) stuff elements
font_stuff_in_elt :: (Font_tag -> a -> a) ->
HTML_element -> a -> a
font_stuff_in_elt f (HTML_text _) stuff = stuff
font_stuff_in_elt f (HTML_font font_tag html) stuff =
let newstuff = f font_tag stuff in
font_stuff_in_html f html newstuff
font_stuff_in_elt f (HTML_p html) stuff =
font_stuff_in_html f html stuff
COMP90048 Declarative Programming Lecture 15 – 16 / 21
Functional design patterns
Collecting font sizes again
font_sizes_in_html’ :: HTML -> Set Int -> Set Int
font_sizes_in_html’ html sizes =
font_stuff_in_html accumulate_font_sizes html sizes
accumulate_font_sizes font_tag sizes =
let Font_tag maybe_size _ _ = font_tag in
case maybe_size of
Nothing ->
sizes
Just fontsize ->
Data.Set.insert fontsize sizes
Using the higher order version avoids duplicating the code that traverses
the data structure. The benefit you get from this scales linearly with the
complexity of the data structure being traversed.
COMP90048 Declarative Programming Lecture 15 – 17 / 21
Functional design patterns
Comparison to the visitor pattern
The function font_stuff_in_html does a job that is very similar to the
job that the visitor design pattern would do in an object-oriented language
like Java: they both traverse a data structure, invoking a function at one
or more selected points in the code. However, there are also differences.
In the Haskell version, the type of the higher order function makes it
clear whether the code executed at the selected points just gathers
information, or whether it modifies the traversed data structure. In
Java, the invoked code is imperative, so it can do either.
The Java version needs an accept method in every one of the classes
that correspond to Haskell types in the data structure (in this case,
HTML and HTML_element).
In the Haskell version, the functions that implement the traversal can
be (and typically are) next to each other. In Java, the corresponding
methods have to be dispersed to the classes to which they belong.
COMP90048 Declarative Programming Lecture 15 – 19 / 21
Functional design patterns
Libraries vs frameworks
The way typical libraries work in any language (including C and Java as
well as Haskell) is that code written by the programmer calls functions in
the library.
In some cases, the library function is a higher order function, and thus it
can call back a function supplied to it by the programmer.
Application frameworks are libraries but they are not typical libraries,
because they are intended to be the top layer of a program.
When a program uses a framework, the framework is in control, and it
calls functions written by the programmer when circumstances call for it.
For example, a framework for web servers would handle all communication
with remote clients. It would itself implement the event loop that waits for
the next query to arrive, and would invoke user code only to generate the
response to each query.
COMP90048 Declarative Programming Lecture 15 – 20 / 21
Functional design patterns
Frameworks: libraries vs application generators
Frameworks in Haskell can be done like this, with framework simply being
a library function:
main = framework plugin1 plugin2 plugin3
plugin1 = ...
plugin2 = ...
plugin3 = ...
This approach could also be used in other languages, since even C and
Java support callbacks, though sometimes clumsily.
Unfortunately, many frameworks instead just generate code (in C, C#,
Java, ...) that the programmer is then expected to modify. This approach
throws abstraction out the window, and is much more error-prone.
NOTE
Application generators expose to the programmer all the details of their
implementation. This allows the programmer to modify those details if
needed, but the next invocation of the application generator will destroy
those modifications, which means the application generator can be used
only once. If a new version of the application generator comes out that
fixes some problems with the old version, the programmer has no good
options: there is no easy way to integrate the new version’s bug fixes with
his or her own earlier modifications.
However, bugs in which programmers modify the wrong part of the
generated code or forget to modify a part they should have modified can
be expected to be made considerably more frequently.
COMP90048 Declarative Programming Lecture 15 – 21 / 21
Exploiting the type system
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 16
Exploiting the type system
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Exploiting the type system
Representation of C programs in gcc
The gcc compiler has one main data type to represent the code being
compiled. The node type is a giant union which has different fields for
different kinds of entities. A node can represent, amongst other things,
a data type,
a variable,
an expression or
a statement.
Every link to another part of a program (such as the operands of an
operator) is a pointer to a tree node of this can-represent-everything type.
When Stallman chose this design in the 1980s, he was a Lisp programmer.
Lisp does not have a static type system, so the Blub paradox applies here
in reverse: even C has a better static type system than Lisp. It’s up to the
programmer to design types to exploit the type system.
NOTE
The giant union is definitely very big: in gcc 4.4.1, it has 40 alternatives,
many more than the four listed above.
Like Python, Lisp does have a type system, but it operates only at
runtime.
COMP90048 Declarative Programming Lecture 16 – 1 / 18
Exploiting the type system
Representation of if-then-elses
To represent if-then-else expressions such as C’s ternary operator
(x > y) ? x : y
the relevant union field is a structure that has an array of operands, which
should have exactly three elements (the condition, the then part and the
else part). All three should be expressions.
This representation is subject to two main kinds of error.
The array of operands could have the wrong number of operands.
Any operand in the array could point to the wrong kind of tree node.
gcc has extensive infrastructure designed to detect these kinds of errors,
but this infrastructure itself has three problems:
it makes the source code harder to read and write;
if enabled, it slows down gcc by about 5 to 15%; and
it detects violations only at runtime.
NOTE
When trying to compiler some unusual C programs, usually those
generated by a program rather than a programmer, you may gcc to abort
with a message talking about an "internal compiler error". The usual
reason for such aborts is that the compiler expected to find a node of a
particular kind in a given position in the tree, but found a node of a
different kind. In other words, the problem is the failure of a runtime type
check. A better representation that took advantage of the type system
would allow such bugs to be caught at compile time.
COMP90048 Declarative Programming Lecture 16 – 2 / 18
Exploiting the type system
Exploiting the type system
A well designed representation using algebraic types is not vulnerable to
either kind of error, and is not subject any of those three kinds of problems.
data Expr
= Const Const
| Var String
| Binop Binop Expr Expr
| Unop Unop Expr
| Call String [Expr]
| ITE Expr Expr Expr
data Binop = Add | Sub | ...
data Unop = Uminus | ...
data Const = IntConst Int | FloatConst Double | ...
NOTE
You can do much better than gcc’s design even in C. A native C
programmer would have chosen to have represent these four very different
kinds of entities using four different main types, together with more types
representing their components. This representation could start from
something like the following.
typedef enum {
EXPR_CONST, EXPR_VAR, EXPR_BINOP, EXPR_UNOP, EXPR_CALL, EXPR_ITE
} ExprKind;
typedef struct expr_struct Expr;
typedef union expr_union ExprUnion;
typedef struct exprs_struct Exprs;
typedef struct const_struct Const;
typedef struct var_struct Var;
struct expr_struct {
ExprKind expr_kind;
ExprUnion *expr_union;
};
union expr_union {
Const *expr_const;
Var *expr_var;
struct unop_expr_struct *expr_unop;
struct binop_expr_struct *expr_binop;
struct call_expr_struct *expr_call;
struct ite_expr_struct *expr_ite;
};
typedef enum {
ADD, SUB, ...
} Binop;
typedef enum {
UMINUS, ...
} Unop;
struct binop_expr_struct {
Binop *binop;
Expr binop_arg_1;
Stmt binop_arg_2;
};
struct unop_expr_struct {
Unop *unop;
Expr unop_arg_1;
};
struct call_expr_struct {
char *funcname;
Exprs func_args;
};
struct ite_expr_struct {
Expr *cond;
Stmt then_part;
Stmt else_part;
};
struct exprs_struct {
Expr *head;
Exprs *tail;
};
There is a paper titled "Statically typed trees in GCC" by Sidwell and
Weinberg that describes the drawbacks of gcc’s current program
representation and proposes a roadmap for switching to a more type-safe
representation, something like the representation shown above.
Unfortunately, that roadmap has not been implemented, and probably
never will be, partly because its implementation interferes considerably
with the usual development of gcc.
COMP90048 Declarative Programming Lecture 16 – 3 / 18
Exploiting the type system
Generic lists in C
typedef struct generic_list List;
struct generic_list {
void *head;
List *tail;
};
...
List *int_list;
List *p;
int item;
for (p = int_list; p != NULL; p = p->tail) {
item = (int) p->head;
... do something with item ...
}
NOTE
Despite the name of the variable holding the list being int_list, C’s type
system cannot guarantee that the list elements are in fact integers.
COMP90048 Declarative Programming Lecture 16 – 4 / 18
Exploiting the type system
Type system expressiveness
Programmers who choose to use generic lists in a C program need only
one list type and therefore one set of functions operating on lists.
The downside is that every loop over lists needs to cast the list element to
the right type, and this cast is a frequent source of bugs.
The other alternative in a C program is to define and use a separate list
type for every different type of item that the program wants to put into a
list. This is type safe, but requires repeated duplication of the functions
that operate on lists. Any bugs in those those functions must be fixed in
each copy.
Haskell has a very expressive type system that is increasingly being copied
by other languages. Some OO/procedural languages now support generics.
A few such languages (Rust, Swift, Java 8) support option types, like
Haskell’s Maybe. No well-known such languages support full algebraic
types.
NOTE
However, even in Haskell, some programming practices help the compiler
to catch problems early (at compile time), and some do not.
COMP90048 Declarative Programming Lecture 16 – 5 / 18
Exploiting the type system
Units
One typical bug type in programs that manipulate physical measurements
is unit confusion, such as adding 2 meters and 3 feet, and thinking the
result is 5 meters. Mars Climate Orbiter was lost because of such a bug.
Such bugs can be prevented by wrapping the number representing the
length in a data constructor giving its unit.
data Length = Meters Double
meters_to_length :: Double -> Length
meters_to_length m = Meters m
feet_to_length :: Double -> Length
feet_to_length f = Meters (f * 0.3048)
add_lengths :: Length -> Length -> Length
add_lengths (Meters a) (Meters b) = Meters (a+b)
NOTE
Making Length an abstract type, and exporting only type-safe operations
to the rest of the program, improves safety even further.
Wrapping a data constructor around a number looks like adding overhead,
since normally, each data constructor requires a cell of its own on the heap
for itself and its arguments. However, types that have exactly one data
constructor with exactly one argument can be implemented as if the data
constructor were not there, eliminating the overhead. The Mercury
language uses this representation scheme for types like this, and in some
circumstances GHC can avoid the indirection.
For Mars Climate Orbiter, the Jet Propulsion Laboratory expected a
contractor to provide thruster control data expressed in newtons, the SI
unit of force, but the contractor provided the data in pounds-force, the
usual unit of force in the old English system of measurement. Since one
pound of force is 4.45 newtons, the thruster calculations were off by a
factor of 4.45. The error was not caught because the data files sent from
the contractor to JPL contained only numbers and not units, and because
other kinds of tests that could have caught the error were eliminated in an
effort to save money. The result was that Mars Climate Orbiter dived too
steeply into the Martian atmosphere and burned up.
COMP90048 Declarative Programming Lecture 16 – 6 / 18
Exploiting the type system
Different uses of one unit
Sometimes, you want to prevent confusion even between two kinds of
quantities measured in the same units.
For example, many operating systems represent time as the number of
seconds elapsed since a fixed epoch. For Unix, the epoch is 0:00am on
1 Jan 1970.
data Duration = Seconds Int
data Time = SecondsSinceEpoch Int
add_durations :: Duration -> Duration -> Duration
add_durations (Seconds a) (Seconds b) = Seconds (a+b)
add_duration_to_time :: Time -> Duration -> Time
add_duration_to_time (SecondsSinceEpoch sse) (Seconds t) =
SecondsSinceEpoch (sse + t)
NOTE
It makes sense to add together two durations or a time and a duration, but
it does not make sense to add two times.
COMP90048 Declarative Programming Lecture 16 – 8 / 18
Exploiting the type system
Different units in one type
Sometimes, you cannot apply a fixed conversion rate between different
units. In such applications, each operation may need to do conversion on
demand at whatever rate is applicable at the time of its execution.
data Money
= USD_dollars Double
| AUD_dollars Double
| GBP_pounds Double
For financial applications, using Doubles would not be a good idea, since
accounting rules that precede the use of computers specify rounding
methods (e.g. for interest calculations) that binary floating point numbers
do not satisfy.
One workaround is to use fixed-point numbers, such as integers in which 1
represents not one dollar, but one one-thousandth of one cent.
NOTE
Those accounting rules specified rounding algorithms for human
accountants working with decimal numbers, not for automatic computers
working with binary numbers. The difference is significant, since
floating-point numbers cannot even represent exactly such simple but
important fractions as one tenth and one percent.
COMP90048 Declarative Programming Lecture 16 – 9 / 18
Exploiting the type system
Mapping over a Maybe
Suppose we have a type
type Marks = Map String [Int]
giving a list of all the marks for each student.
(A type declaration like this declares that Marks is an alias for
Map String [Int], just the way String is an alias for [Char].)
We want to write a function
studentTotal :: Marks -> String -> Maybe Int
that returns Just the total mark for the specified student, or Nothing if
the specified student is unknown. Nothing means something different
from Just 0.
COMP90048 Declarative Programming Lecture 16 – 10 / 18
Exploiting the type system
Mapping over a Maybe
This definition will work, but it’s a bit verbose:
studentTotal marks student =
case Map.lookup student marks of
Nothing -> Nothing
Just ms -> Just (sum ms)
We’d like to do something like this:
studentTotal marks student =
sum (Map.lookup student marks)
but it’s a type error:
Couldn’t match expected type ‘Maybe Int’
with actual type ‘Int’
COMP90048 Declarative Programming Lecture 16 – 11 / 18
Exploiting the type system
The Functor class
If we think of a Maybe a as a “box” that holds an a (or maybe not), we
want to apply a function inside the box, leaving the box in place but
replacing its content with the result of the function.
That’s actually what the map function does: it applies a function to the
contents of a list, returning a list of the results.
What we want is to apply a function a -> b to the contents of a Maybe
a, returning a Maybe b. We want to map over a Maybe.
The Functor type class is the class of all types that can be “mapped”
over. This includes [a], but also Maybe a.
COMP90048 Declarative Programming Lecture 16 – 12 / 18
Exploiting the type system
The Functor class
You can map over a Functor type with the fmap function:
fmap :: Functor f => (a -> b) -> f a -> f b
This gives us a much more succinct definition:
studentTotal marks student =
fmap sum (Map.lookup student marks)
Functor is defined in the standard prelude, so you can use fmap over
Maybes (and lists) without importing anything.
The standard prelude also defines a function <$> as an infix operator alias
for fmap, so the following code will also work:
sum <$> Map.lookup student marks
COMP90048 Declarative Programming Lecture 16 – 13 / 18
Exploiting the type system
Beyond Functors
Suppose our students work in pairs, with either teammate submitting the
pair’s work. We want to write a function
pairTotal :: Marks -> String -> String -> Maybe Int
to return the total of the assessments of two students, or Nothing if either
or both of the students are not enrolled.
This code works, but is disappointingly verbose:
pairTotal marks student1 student2 =
case studentTotal marks student1 of
Nothing -> Nothing
Just t1 -> case studentTotal marks student2 of
Nothing -> Nothing
Just t2 -> Just (t1 + t2)
COMP90048 Declarative Programming Lecture 16 – 14 / 18
Exploiting the type system
Putting functions in Functors
Functor works nicely for unary functions, but not for greater arities. If we
try to use fmap on a binary function and a Maybe, we wind up with a
function in a Maybe.
Remembering the type of fmap,
fmap :: Functor f => (a -> b) -> f a -> f b
if we take f to be Maybe, a to be Int and b to be Int -> Int, then we
see that
fmap (+) (studentTotal marks student1)
returns a value of type Maybe (Int -> Int).
So all we need is a way to extract a function from inside a functor and
fmap that over another functor. We want to apply one functor to another.
COMP90048 Declarative Programming Lecture 16 – 15 / 18
Exploiting the type system
Applicative functors
Enter applicative functors. These are functors that can contain functions,
which can be applied to other functors, defined by the Applicative class.
The most important function of the Applicative class is
(<*>) :: f (a -> b) -> f a -> f b
which does exactly what we need.
Happily, Maybe is in the Applicative class, so the following definition
works:
pairTotal marks student1 student2 =
let mark = studentTotal marks
in (+) <$> mark student1 <*> mark student2
COMP90048 Declarative Programming Lecture 16 – 16 / 18
Exploiting the type system
Applicative
The second function defined for every Applicative type is
pure :: a -> f a
which just inserts a value into the applicative functor. For the Maybe class,
pure = Just. For lists, pure = (:[]) (creating a singleton list).
For example, if we wanted to subtract a student’s score from 100, we
could do:
(-) <$> pure 100 <*> studentTotal marks student
but a simpler way to do the same thing is:
(100-) <$> studentTotal marks student
In fact, every Applicative must also be a Functor, just as every Ord
type must be Eq.
COMP90048 Declarative Programming Lecture 16 – 17 / 18
Exploiting the type system
Lists are Applicative
<*> gets even more interesting for lists:
...> (++) <$> ["do","something","good"] <*> pure "!"
["do!","something!","good!"]
...> (++) <$> ["a","b","c"] <*> ["1","2"]
["a1","a2","b1","b2","c1","c2"]
You can think of <*> as being like a Cartesian product, hence the “*”.
COMP90048 Declarative Programming Lecture 16 – 18 / 18
Monads
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 17
Monads
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Monads
any and all
Other useful higher-order functions are
any :: (a -> Bool) -> [a] -> Bool
all :: (a -> Bool) -> [a] -> Bool
For example, to see if every word in a list contains the letter ’e’:
Prelude> all (elem ’e’) ["eclectic", "elephant", "legion"]
True
To check if a word contains any vowels:
Prelude> any (\x -> elem x "aeiou") "sky"
False
COMP90048 Declarative Programming Lecture 17 – 1 / 20
Monads
flip
If the order of arguments of elem were reversed, we could have used
currying rather than the bulky \x -> elem x "aeiou". The flip
function takes a function and returns a function with the order of
arguments flipped.
flip :: (a -> b -> c) -> (b -> a -> c)
flip f x y = f y x
Prelude> any (flip elem "aeiou") "hmmmm"
False
The ability to write functions to construct other functions is one of
Haskell’s strengths.
COMP90048 Declarative Programming Lecture 17 – 2 / 20
Monads
Monads
Monads build on this strength. A monad is a type constructor that
represents a computation. These computations can then be composed to
create other computations, and so on. The power of monads lies in the
programmer’s ability to determine how the computations are composed.
Phil Wadler, who introduced monads to Haskell, describes them as
“programmable semicolons”.
A monad M is a type constructor that supports two operations:
A sequencing operation, denoted »=, whose type is
M a -> (a -> M b) -> M b.
An identity operation, denoted return, whose type is a -> M a.
COMP90048 Declarative Programming Lecture 17 – 3 / 20
Monads
Monads
Think of the type M a as denoting a computation that produces an a, and
possibly carries something extra. For example, if M is the Maybe type
constructor, that something extra is an indication of whether an error has
occurred so far.
You can take a value of type a and use the (misnamed) identity
operation to wrap it in the monad’s type constructor.
Once you have such a wrapped value, you can use the sequencing
operation to perform an operation on it. The »= operation will
unwrap its first argument, and then typically it will invoke the
function given to it as its second argument, which will return a
wrapped up result.
NOTE
You can apply the sequencing operation to any value wrapped up in the
monad’s type constructor; it does not have to have been generated by the
monad’s identity function.
COMP90048 Declarative Programming Lecture 17 – 4 / 20
Monads
The Maybe and MaybeOK monads
The obvious ways to define the monad operations for the Maybe and
MaybeOK type constructors are these (MaybeOK is not in the library):
-- monad ops for Maybe
data Maybe t = Just t
| Nothing
return x = Just x
(Just x) >>= f = f x
Nothing >>= _ = Nothing
-- monad ops for MaybeOK
data MaybeOK t = OK t
| Error String
return x = OK x
(OK x) >>= f = f x
(Error m) >>= _ = Error m
In a sequence of calls to »=, as long as all invocations of f succeed,
(returning Just x or OK x), you keep going.
Once you get a failure indication, (returning Nothing or Error m), you
keep that failure indication and perform no further operations.
COMP90048 Declarative Programming Lecture 17 – 5 / 20
Monads
Why you may want these monads
Suppose you want to encode a sequence of operations that each may fail.
Here are two such operations:
maybe_head :: [a] -> MaybeOK a
maybe_head [] = Error "head of empty list"
maybe_head (x:_) = OK x
maybe_sqrt :: Int -> MaybeOK Double
maybe_sqrt x =
if x >= 0 then
OK (sqrt (fromIntegral x))
else
Error "sqrt of negative number"
How can you encode a sequence of operations such as taking the head of a
list and computing its square root?
NOTE
The fromIntegral function can convert an integer to any other numeric
type.
COMP90048 Declarative Programming Lecture 17 – 6 / 20
Monads
Simplifying code with monads
maybe_head :: [a] -> MaybeOK a
maybe_sqrt :: Int -> MaybeOK Double
maybe_sqrt_of_head :: [Int] -> MaybeOK Double
-- definition not using monads
maybe_sqrt_of_head l =
case maybe_head l of
Error msg -> Error msg
OK h -> maybe_sqrt h
-- simpler definition using monads
maybe_sqrt_of_head l =
maybe_head l >>= maybe_sqrt
NOTE
The monadic version is simpler because in that version, the work of
checking for failure and handling it if found is done once, in the MaybeOK
monad’s sequence operator.
In the version without monads, this code would have to be repeated for
every step in a sequence of possibly-failing operations. If this sequence has
ten operations, you will be grow bored of writing the pattern-match code
way before writing the tenth copy. The steady increase in indentation
required by the offside rule also makes the code ugly, since the code of the
tenth iteration would have to be squashed up against the right margin.
Longer sequences of operations may even require the use of more columns
than your screen actually has.
Note that the two occurrences of Error m in the first definition are of
different types; the first is type MaybeOK Int, while the second is of type
MaybeOK Double. The two occurrences of Error m in the definition of
the sequence operation for the MaybeOK monad are similarly of different
types, MaybeOK a for the first and MaybeOK b for the second.
COMP90048 Declarative Programming Lecture 17 – 7 / 20
Monads
I/O actions in Haskell
Haskell has a type constructor called IO. A function that returns a value of
type IO t for some t will return a value of type t, but can also do input
and/or output. Such functions can be called I/O functions or I/O actions.
Haskell has several functions for reading input, including
getChar :: IO Char
getLine :: IO String
Haskell has several functions for writing output, including
putChar :: Char -> IO ()
putStr :: String -> IO ()
putStrLn :: String -> IO ()
print :: (Show a) => a -> IO ()
The type (), called unit, is the type of 0-tuples (tuples containing zero
values). This is similar to the void type in C or Java. There is only one
value of this type, the empty tuple, which is also denoted ().
NOTE
The notion that a function that returns a value of type IO t for some t
actually does input and/or output is only approximately correct; we will
get to the fully correct notion in a few slides.
Since there is only one value of type (), variables of this type carry no
information.
Haskell represents a computation that takes a value of type a and
computes a value of type b as a function of type a -> b.
Haskell represents a computation that takes a value of type a and
computes a value of type b while also possibly performing input and/or
output (subject to the caveat above) as a function of type a -> IO b.
All the functions listed on this slide are defined in the prelude.
getChar returns the next character in the input.
getLine returns the next line in the input, without the final ’\n’.
putChar prints the given character.
putStr prints the given string.
putStrLn prints the given string, and appends a newline character.
print uses the show function (which is defined for every type in the Show
type class) to turn the given value into a string; it then prints that string
and appends a newline character.
COMP90048 Declarative Programming Lecture 17 – 9 / 20
Monads
Operations of the I/O monad
The type constructor IO is a monad.
The identity operation: return val just returns val (inside IO) without
doing any I/O.
The sequencing operation: f »= g
1 calls f, which may do I/O, and which will return a value rf that may
be meaningful or may be (),
2 calls g rf (passing the return value of f to g), which may do I/O,
and which will return a value rg that also may be meaningful or may
be (),
3 returns rg (inside IO) as the result of f »= g.
You can use the sequencing operation to create a chain of any number of
I/O actions.
COMP90048 Declarative Programming Lecture 17 – 11 / 20
Monads
Example of monadic I/O: hello world
hello :: IO ()
hello =
putStr "Hello, "
>>=
\_ -> putStrLn "world!"
This code has two I/O actions connected with »=.
1 The first is a call to putStr. This prints the first half of the message,
and returns ().
2 The second is an anonymous function. It takes as argument and
ignores the result of the first action, and then calls putStrLn to print
the second half of the message, adding a newline at the end.
The result of the action sequence is the result of the last action.
NOTE
In this case, the result of the last action will always be ().
Actually, there is a third monad operation besides return and »=: the
operation ». This is identical to »=, with the exception that it ignores the
return value of its first operand, and does not pass it to its second
operand. This is useful when the second operand would otherwise have to
explicitly ignore its argument, as in this case.
With », the code of this function could be slightly simpler:
hello :: IO ()
hello =
putStr "Hello, "
>>
putStrLn "world!"
COMP90048 Declarative Programming Lecture 17 – 12 / 20
Monads
Example of monadic I/O: greetings
greet :: IO ()
greet =
putStr "Greetings! What is your name? "
>>=
\_ -> getLine
>>=
\name -> (
putStr "Where are you from? "
>>=
\_ -> getLine
>>=
\town ->
let msg = "Welcome, " ++ name ++
" from " ++ town
in putStrLn msg
)
NOTE
This example shows how each call to getLine is followed by a lambda
expression that captures the value of the string returned getLine: the
name for the first call, and the town for the second. In the case of the first
call, the return value is not used immediately; we do not want to pass it to
the next call putStr. That is why we need to make sure that the scope of
the lambda expression that takes name as its argument includes all the
following actions, or at least all the following actions that need access to
the value of name (which in this case is the same thing).
COMP90048 Declarative Programming Lecture 17 – 13 / 20
Monads
do blocks
Code written using monad operations is often ugly, and writing it is usually
tedious. To address both concerns, Haskell provides do blocks. These are
merely syntactic sugar for sequences of monad operations, but they make
the code much more readable and easier to write.
A do block starts with the keyword do, like this:
hello = do
putStr "Hello, "
putStrLn "world"
COMP90048 Declarative Programming Lecture 17 – 14 / 20
Monads
do block components
Each element of a do block can be
an I/O action that returns an ignored value, usually of type (), such
as the calls to putStr and putStrLn below (just call the function);
an I/O action whose return value is used to bind a variable, (use
var <- expr to bind the variable);
bind a variable to a non-monadic value (use let var = expr (no in)).
greet :: IO ()
greet = do
putStr "Greetings! What is your name? "
name <- getLine
putStr "Where are you from? "
town <- getLine
let msg = "Welcome, " ++ name ++ " from " ++ town
putStrLn msg
NOTE
Each do block element has access to all the variables defined by previous
actions but not later ones.
Let clauses in do blocks do not end with the keyword in, since the variable
defined by such a let clause is available in all later operations in the
block, not just the immediately following opeation.
This notation is so much more convenient to use than raw monad
operations that raw monad operations occur in real Haskell programs only
rarely.
COMP90048 Declarative Programming Lecture 17 – 15 / 20
Monads
Operator priority problem
Unfortunately, the following line of code does not work:
putStrLn "Welcome, " ++ name ++ " from " ++ town
The reason is that due to its system of operator priorities, Haskell thinks
that the main function being invoked here is not putStrLn but ++, with
its left argument being putStrLn "Welcome, ".
This is also the reason why Haskell accepts only the second of the
following equations. It parses the left hand side of the first equation as
(len x):xs, not as len (x:xs).
len x:xs = 1 + len xs
len (x:xs) = 1 + len xs
COMP90048 Declarative Programming Lecture 17 – 17 / 20
Monads
Working around the operator priority problem
There are two main ways to fix this problem:
putStrLn ("Welcome, " ++ name ++ " from " ++ town)
putStrLn $ "Welcome, " ++ name ++ " from " ++ town
The first simply uses parentheses to delimit the possible scope of the ++
operator.
The second uses another operator, $, which has lower priority than ++, and
thus binds less tightly.
The main function invoked on the line is thus $. Its first argument is its
left operand: the function putStrLn, which is of type String -> IO ().
Its second argument is its right operand: the expression "Welcome, " ++
name ++ " from " ++ town, which is of type String.
$ is of type (a -> b) -> a -> b. It applies its first argument to its
second argument, so in this case it invokes putStrLn with the result of
the concatenation.
COMP90048 Declarative Programming Lecture 17 – 18 / 20
Monads
return
If a function does I/O and returns a value, and the code that computes
the return value does not do I/O, you will need to invoke the return
monad operation as the last operation in the do block.
main :: IO ()
main = do
putStrLn "Please input a string"
len <- readlen
putStrLn $ "The length of that string is " ++ show len
readlen :: IO Int
readlen = do
str <- getLine
return (length str)
COMP90048 Declarative Programming Lecture 17 – 20 / 20
More on monads
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 18
More on monads
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
More on monads
I/O actions as descriptions
Haskell programmers usually think of functions that return values of type
IO t as doing I/O as well as returning a value of type t. While this is
usually correct, there are some situations in which it is not accurate
enough.
The correct way to think about such functions is that they return two
things:
a value of type t, and
a description of an I/O operation.
The monadic operator »= can then be understood as taking descriptions of
two I/O operations, and returning a description of those two operations
being executed in order.
The monadic operator return simply associates a description of a
do-nothing I/O operation with a value.
COMP90048 Declarative Programming Lecture 18 – 1 / 18
More on monads
Description to execution: theory
Every complete Haskell program must have a function named main, whose
signature should be
main :: IO ()
As in C, this is where the program starts execution. Conceptually,
the OS starts the program by invoking the Haskell runtime system;
the runtime system calls main, which returns a description of a
sequence of I/O operations; and
the runtime system executes the described sequence of I/O
operations.
COMP90048 Declarative Programming Lecture 18 – 2 / 18
More on monads
Description to execution: practice
In actuality, the compiler and the runtime system together ensure that each
I/O operation is executed as soon as its description has been computed,
provided that the description is created in a context which guarantees
that the description will end up in the list of operation descriptions
returned by main, and
provided that all the previous operations in that list have also been
executed.
The provisions are necessary since
you don’t want to execute an I/O operation that the program does
not actually call for, and
you don’t want to execute I/O operations out of order.
COMP90048 Declarative Programming Lecture 18 – 3 / 18
More on monads
Example: printing a table of squares directly
main ::IO ()
main = do
putStrLn "Table of squares:"
print_table 1 10
print_table :: Int -> Int -> IO ()
print_table cur max
| cur > max = return ()
| otherwise = do
putStrLn (table_entry cur)
print_table (cur+1) max
table_entry :: Int -> String
table_entry n = (show n) ++ "^2 = " ++ (show (n*n))
NOTE
The definition of print_table uses guards. If cur > max, then the
applicable right hand side is the one that follows that guard expression; if
cur <= max, then the applicable right hand side is the one that follows
the keyword otherwise.
COMP90048 Declarative Programming Lecture 18 – 4 / 18
More on monads
Non-immediate execution of I/O actions
Just because you have created a description of an I/O action, does not
mean that this I/O action will eventually be executed.
Haskell programs can pass around descriptions of I/O operations. They
cannot peer into a description of an I/O operation, but they can
nevertheless do things with them, such as
build up lists of I/O actions, and
put I/O actions into binary search trees as values.
Those lists and trees can then be processed further, and programmers can,
if they wish, take the descriptions of I/O actions out of those data
structures, and have them executed by including them in the list of actions
returned by main.
COMP90048 Declarative Programming Lecture 18 – 5 / 18
More on monads
Example: printing a table of squares indirectly
main = do
putStrLn "Table of squares:"
let row_actions = map show_entry [1..15]
execute_actions (take 10 row_actions)
table_entry :: Int -> String
table_entry n = (show n) ++ "^2 = " ++ (show (n*n))
show_entry :: Int -> IO ()
show_entry n = do putStrLn (table_entry n)
execute_actions :: [IO ()] -> IO ()
execute_actions [] = return ()
execute_actions (x:xs) = do
x
execute_actions xs
NOTE
The take function from the Haskell prelude returns the first few elements
of a list; the number of elements it should return is specified by the value
of its first argument.
COMP90048 Declarative Programming Lecture 18 – 7 / 18
More on monads
Input, process, output
A typical batch program reads in its input, does the required processing,
and prints its output.
A typical interactive program goes through the same three stages once for
each interaction.
In most programs, the vast majority of the code is in the middle
(processing) stage.
In programs written in imperative languages like C, Java, and Python, the
type of a function (or procedure, subroutine or method) does not tell you
whether the function does I/O.
In Haskell, it does.
COMP90048 Declarative Programming Lecture 18 – 9 / 18
More on monads
I/O in Haskell programs
In most Haskell programs, the vast majority of the functions are not I/O
functions and they do no input or output. They merely build, access and
transform data structures, and do calculations. The code that does I/O is
a thin veneer on top of this bulk.
This approach has several advantages.
A unit test for a non-IO function is a record of the values of the
arguments and the expected value of the result. The test driver can
read in those values, invoke the function, and check whether the
result matches. The test driver can be a human.
Code that does no I/O can be rearranged. Several optimizations
exploit this fact.
Calls to functions that do no I/O can be done in parallel. Selecting
the best calls to parallelize is an active research area.
COMP90048 Declarative Programming Lecture 18 – 10 / 18
More on monads
Debugging printfs
One standard approach for debugging a program written in C is to edit
your code to insert debugging printfs to show you what input your buggy
function is called with and what results it computes.
In a program written in Haskell, you can’t just insert printing code into
functions not already in the IO monad.
Debugging printfs are only used for debugging, so you’re not concerned
with where the output from debugging printfs appears relative to other
output. This is where the function unsafePerformIO comes in: it allows
you to perform IO anywhere, but the order of output will probably be
wrong.
Do not use unsafePerformIO in real code, but it is useful for debugging.
COMP90048 Declarative Programming Lecture 18 – 11 / 18
More on monads
unsafePerformIO
The type of unsafePerformIO is IO t -> t.
You give it as argument an I/O operation, which means a function of
type IO t. unsafePerformIO calls this function.
The function will return a value of type t and a description of an I/O
operation.
unsafePerformIO executes the described I/O operation and returns
the value.
Here is an example:
sum :: Int -> Int -> Int
sum x y = unsafePerformIO $ do
putStrLn ("summing " ++ (show x) ++ " and " ++ (show y))
return (x + y)
NOTE
As its name indicates, the use of unsafePerformIO is unsafe in general.
This is because you the program does not indicate how the I/O operations
performed by an invocation of unsafePerformIO fit into the sequence of
operations executed by main. This is why unsafePerformIO is not
defined in the prelude. This means that your code will not be able to call
it unless you import the module that defines it, which is GHC.IOBase.
COMP90048 Declarative Programming Lecture 18 – 12 / 18
More on monads
The State monad
The State monad is useful for computations that need to thread
information throughout the computation. It allows such information to be
transparently passed around a computation, and accessed and replaced
when needed. That is, it allows an imperative style of programming
without losing Haskell’s declarative semantics.
This code adds 1 to each element of a tree, and does not need a monad:
data Tree a = Empty | Node (Tree a) a (Tree a)
deriving Show
type IntTree = Tree Int
incTree :: IntTree -> IntTree
incTree Empty = Empty
incTree (Node l e r) =
Node (incTree l) (e + 1) (incTree r)
COMP90048 Declarative Programming Lecture 18 – 13 / 18
More on monads
Threading state
If we instead wanted to add 1 to the leftmost element, 2 to the next
element, and so on, we would need to pass an integer into our function
saying what to add, but also we need to pass an integer out, saying what
to add to the next element. This requires more complex code:
incTree1 :: IntTree -> IntTree
incTree1 tree = fst (incTree1’ tree 1)
incTree1’ :: IntTree -> Int -> (IntTree, Int)
incTree1’ Empty n = (Empty, n)
incTree1’ (Node l e r) n =
let (newl, n1) = incTree1’ l n
(newr, n2) = incTree1’ r (n1 + 1)
in (Node newl (e+n1) newr, n2)
NOTE
Given a tuple of two elements, the fst builtin function returns its first
element, and the snd builtin function returns its second element.
COMP90048 Declarative Programming Lecture 18 – 14 / 18
More on monads
Introducing the State monad
The State monad abstracts the type s -> (v,s), hiding away the s part.
Haskell’s do notation allows us to focus on the v part of the computation
while ignoring the s part where not relevant.
incTree2 :: IntTree -> IntTree
incTree2 tree = fst (runState (incTree2’ tree) 1)
incTree2’ :: IntTree -> State Int IntTree
incTree2’ Empty = return Empty
incTree2’ (Node l e r) = do
newl <- incTree2’ l
n <- get -- gets the current state
put (n + 1) -- sets the current state
newr <- incTree2’ r
return (Node newl (e+n) newr)
COMP90048 Declarative Programming Lecture 18 – 15 / 18
More on monads
Abstracting the state operations
In this case, we do not need the full generality of being able to update the
integer state in arbitrary ways; the only update operation we need is an
increment. We can therefore provide a version of the state monad that is
specialized for this task. Such specialization provides useful
documentation, and makes the code more robust.
type Counter = State Int
withCounter :: Int -> Counter a -> a
withCounter init f = fst (runState f init)
nextCount :: Counter Int
nextCount = do
n <- get
put (n + 1)
return n
COMP90048 Declarative Programming Lecture 18 – 16 / 18
More on monads
Using the counter
Now the code that uses the monad is even simpler:
incTree3 :: IntTree -> IntTree
incTree3 tree = withCounter 1 (incTree3’ tree)
incTree3’ :: IntTree -> Counter IntTree
incTree3’ Empty = return Empty
incTree3’ (Node l e r) = do
newl <- incTree3’ l
n <- nextCount
newr <- incTree3’ r
return (Node newl (e+n) newr)
COMP90048 Declarative Programming Lecture 18 – 17 / 18
Laziness
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 19
Laziness
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Laziness
Eager vs lazy evaluation
In a programming language that uses eager evaluation, each expression is
evaluated as soon as it gets bound to a variable, either explicitly in an
assignment statement, or implicitly during a call. (A call implicitly assigns
each actual parameter expression to the corresponding formal parameter
variable.)
In a programming language that uses lazy evaluation, an expression is not
evaluated until its value is actually needed. Typically, this will be when
the program wants the value as input to an arithmetic operation, or
the program wants to match the value against a pattern, or
the program wants to output the value.
Almost all programming languages use eager evaluation. Haskell uses lazy
evaluation.
COMP90048 Declarative Programming Lecture 19 – 1 / 18
Laziness
Laziness and infinite data structures
Laziness allows a program to work with data structures that are
conceptually infinite, as long as the program looks at only a finite part of
the infinite data structure.
For example, [1..] is a list of all the positive numbers. If you attempt to
print it out, the printout will be infinite, and will take infinite time, unless
you interrupt it.
On the other hand, if you want to print only the first n positive numbers,
you can do that with take n [1..].
Even though the second argument of the call to take is infinite in size, the
call takes finite time to execute.
NOTE
The expression [1..] has the same value as all_ints_from 1, given the
definition
all_ints_from :: Integer -> [Integer]
all_ints_from n = n:(all_ints_from (n+1))
COMP90048 Declarative Programming Lecture 19 – 2 / 18
Laziness
The sieve of Eratosthenes
-- returns the (infinite) list of all primes
all_primes :: [Integer]
all_primes = prime_filter [2..]
prime_filter :: [Integer] -> [Integer]
prime_filter [] = []
prime_filter (x:xs) =
x:prime_filter (filter (not . (‘divisibleBy‘ x)) xs)
-- n ‘divisibleBy‘ d means n is divisible by d
divisibleBy n d = n ‘mod‘ d == 0
NOTE
The input to prime_filter is a list of integers which share the property
that they are not evenly divisible by any prime that is smaller than the first
element of the list.
The invariant is trivially true for the list [2..], since there are no primes
smaller than 2.
Since the smallest element of such a list cannot be evenly divisible by any
number smaller than itself, it must be a prime. Therefore filtering
multiples of the first element out of the input sequence before giving the
filtered sequence as input to the recursive call to prime_filter maintains
the invariant.
COMP90048 Declarative Programming Lecture 19 – 4 / 18
Laziness
Using all_primes
To find the first n primes:
take n all_primes
To find all primes up to n:
takeWhile (<= n) all_primes
Laziness allows the programmer of all_primes to concentrate on the
function’s task, without having to also pay attention to exactly how the
program wants to decide how many primes are enough.
Haskell automatically interleaves the computation of the primes with the
code that determines how many primes to compute.
COMP90048 Declarative Programming Lecture 19 – 5 / 18
Laziness
Representing unevaluated expressions
In a lazy programming language, expressions are not evaluated until you
need their value. However, until then, you do need to remember the code
whose execution will compute that value.
In Haskell implementations that compile Haskell to C (this includes GHC),
the data structure you need for that is a pointer to a C function, together
with all the arguments you will need to give to that C function.
This representation is sometimes called a suspension, since it represents a
computation whose evaluation is temporarily suspended.
It can also be called a promise, since it also represents a promise to carry
out a computation if its result is needed.
Historically inclined people can also call it a thunk, because that was the
name of this construct in the first programming language implementation
that used it. That language was Algol-60.
NOTE
The word “thunk” can actually refer to several programming language
implementation constructs, of which this is only one. Think of “thunk” as
the compiler writer’s equivalent of the mechanical engineer’s word
“gadget”: they can both be used to refer to anything small and clever.
COMP90048 Declarative Programming Lecture 19 – 7 / 18
Laziness
Parametric polymorphism
Parametric polymorphism is the name for the form of polymorphism in
which types like [a] and Tree k v, and functions like length and
insert_bst, include type variables, and the types and functions work
identically regardless of what types the type variables stand for.
The implementation of parametric polymorphism requires that the values
of all types be representable in the same amount of memory. Without this,
the code of e.g. length wouldn’t be able to handle lists with elements of
all types.
That “same amount of memory” will typically be the word size of the
machine, which is the size of a pointer. Anything that does not fit into
one word is represented by a pointer to a chunk of memory on the heap.
Given this fact, the arguments of the function in a suspension can be
stored in an array of words, and we can arrange for all functions in
suspensions to take their arguments from a single array of words.
NOTE
Parametric polymorphism is the form of polymorphism on which the type
systems of languages like Haskell are based. Object-oriented languages like
Java are based on a different form of polymorphism, which is usually called
“inclusion polymorphism” or “subtype polymorphism”.
The “same amount of memory” obviously does not include the memory
needed by the pointed-to heap cells, or the cells they point to directly or
indirectly. A value such as [1, 2, 3, 4, 5] will require several heap
cells; in this case, these will be five cons cells and (depending on the
details of the implementation) maybe one cell for the nil at the end. (The
nonempty list constructor : is usually pronounced “cons”, while the empty
list constructor [] is usually pronounced “nil”.) However, the whole value
can be represented by a pointer to the first cons cell.
COMP90048 Declarative Programming Lecture 19 – 8 / 18
Laziness
Evaluating lazy values only once
Many functions use the values of some variables more than once. This
includes takeWhile, which uses x twice:
takeWhile _ [] = []
takeWhile p (x:xs)
| p x = x : takeWhile p xs
| otherwise = []
You need to know the value of x to do the test p x, which requires calling
the function in the suspension representing x; if the test succeeds, you will
again need to know the value of x to put it at the front of the output list.
To avoid redundant work, you want the first call to x’s suspension to
record the result of the call, and you want all references to x after the first
to get its value from this record.
Therefore once you know the result of the call, you don’t need the
function and its arguments anymore.
COMP90048 Declarative Programming Lecture 19 – 9 / 18
Laziness
Call by need
Operations such as printing, arithmetic and pattern matching start by
ensuring their argument is at least partially evaluated.
They will make sure that at least the top level data constructor of the
value is determined. However, the arguments of that data constructor may
remain suspensions.
For example, consider the match of the second argument of takeWhile
against the patterns [] and (p:ps). If the original second argument is a
suspension, it must be evaluated enough to ensure its top-level constructor
is determined. If it is x:xs, then the first argument must be applied to x.
Whether x needs to be evaluated will depend on what the first argument
(function) does.
This is called “call by need”, because function arguments (and other
expressions) are evaluated only when their value is needed.
COMP90048 Declarative Programming Lecture 19 – 11 / 18
Laziness
Control structures and functions
(a) ... if (x < y) f(x); else g(y); ...
(b) ... ite(x < y, f(x), g(y)); ...
int ite(bool c, int t, int e)
{ if (c) then return t; else return e; }
In C, (a) will generate a call to only one of f and g, but (b) will generate a
call to both.
(c) ... if x < y then f x else g y ...
(d) ... ite (x < y) (f x) (g y) ...
ite :: Bool -> a -> a -> a
ite c t e = if c then t else e
In Haskell, (c) will execute a call to only one of f and g, and thanks to
laziness, this is also true for (d).
NOTE
The Haskell implementation of if-then-else calls evaluate_suspension
on the suspension representing the condition. If the condition’s value is
True, it will then call evaluate_suspension on the suspension
representing then part, and return its value; if the condition’s value is
False, it will call evaluate_suspension on the suspension representing
the else part, and return its value. In each case, the suspension for the
other part will remain unevaluated. Roughly speaking, both (c) and (d)
are implemented this way.
COMP90048 Declarative Programming Lecture 19 – 12 / 18
Laziness
Implementing control structures as functions
Without laziness, using a function instead of explicit code such as a
sequence of if-then-elses could get unnecessary non-termination or at least
unnecessary slowdowns.
Laziness’ guarantee that an expression will not be evaluated if its value is
not needed allows programmers to define their own control structures as
functions.
For example, you can define a control structure that returns the value of
one of three expressions, with the expression chosen based on whether an
expression is less than, equal to or greater than zero like this:
ite3 :: (Ord a, Num a) => a -> b -> b -> b -> b
ite3 x lt eq gt
| x < 0 = lt
| x == 0 = eq
| x > 0 = gt
NOTE
This ability to define new control structures can come in handy if you find
yourself repeatedly writing code with the same nontrivial decision-making
structure. However, such situations are pretty rare.
The kind of three-way branch shown by this example was the main way to
implement choice in the first widely-used high level programming
language, the original dialect of Fortran. That version of the if statement
specified three labels; which one control jumped to next depended on the
value of the control expression.
COMP90048 Declarative Programming Lecture 19 – 13 / 18
Laziness
Using laziness to avoid unnecessary work
minimum = head . sort
On the surface, this looks like a very wasteful method for computing the
minimum, since sorting is usually done with an O(n2) or O(n logn)
algorithm, and min should be doable with an O(n) algorithm.
However, in this case, the evaluation of the sorted list can stop after the
materialization of the first element.
If sort is implemented using selection sort, this is just a somewhat higher
overhead version of the direct code for min.
COMP90048 Declarative Programming Lecture 19 – 14 / 18
Laziness
Multiple passes
output_prog chars = do
let anno_chars = annotate_chars 1 1 chars
let tokens = scan anno_chars
let prog = parse tokens
let prog_str = show prog
putStrLn prog_str
This function takes as input one data structure (chars) and calls for the
construction of four more (anno_chars, tokens, prog and prog_str).
This kind of pass structure occurs frequently in real programs.
COMP90048 Declarative Programming Lecture 19 – 16 / 18
Laziness
The effect of laziness on multiple passes
With eager evaluation, you would completely construct each data structure
before starting construction of the next.
The maximum memory needed at any one time will be the size of the
largest data structure (say pass n), plus the size of any part of the previous
data structure (pass n − 1) needed to compute the last part of pass n. All
other memory can be garbage collected before then.
With lazy evaluation, execution is driven by putStrLn, which needs to
know what the next character to print (if any) should be. For each
character to be printed, the program will materialize the parts of those
data structures needed to figure that out.
The memory demand at a given time will be given by the tree of
suspensions from earlier passes that you need to materialize the rest of the
string to be printed. The maximum memory demand can be significantly
less than with eager evaluation.
NOTE
The memory requirements analysis above assumes that the code that
constructs pass n’s data structure uses only data from pass n − 1, and
does not need access to data from pass n − 2, n − 3 etc.
If the last pass builds a data structure instead of printing out the data
structure built by the second-last pass, then you will need to add the
memory needed by the part of the last data structure constructed so far to
the memory needed at any point in time. This will increase the maximum
memory demand with lazy evaluation, but it can still be less than with
eager evaluation.
COMP90048 Declarative Programming Lecture 19 – 17 / 18
Laziness
Lazy input
In Haskell, even input is implemented lazily.
Given a filename, readFile returns the contents of the file as a string,
but it returns the string lazily: it reads the next character from the file
only when the rest of the program needs that character.
parse_prog_file filename = do
fs <- readFile filename
let tokens = scan (annotate_chars 1 1 fs)
return (parse_prog [] tokens)
When the main module calls parse_prog_file, it gets back a tree of
suspensions.
Only when those suspensions start being forced will the input file be read,
and each call to evaluate_suspension on that tree will cause only as
much to be read as is needed to figure out the value of the forced data
constructor.
NOTE
The readFile function is defined in the prelude.
COMP90048 Declarative Programming Lecture 19 – 18 / 18
Performance
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 20
Performance
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Performance
Effect of laziness on performance
Laziness adds two sorts of overhead that slow down programs.
The execution of a Haskell program creates a lot of suspensions, and
most of them are evaluated, so eventually they also need to be
unpacked.
Every access to a value must first check whether the value has been
materialized yet.
However, laziness can also speed up programs by avoiding the execution of
computations that take a long time, or do not terminate at all.
Whether the dominant effect is the slowdown or the speedup will depend
on the program and what kind of input it typically gets.
The usual effect is something like lotto: in most cases you lose a bit, but
sometimes you win a little, and in some rare cases you win a lot.
COMP90048 Declarative Programming Lecture 20 – 1 / 18
Performance
Strictness
Theory calls the value of an expression whose evaluation loops infinitely or
throws an exception “bottom”, denoted by the symbol ⊥.
A function is strict if it always needs the values of all its arguments.
In formal terms, this means that if any of its arguments is ⊥, then its
result will also be ⊥.
The addition function + is strict. The function ite from earlier in the last
lecture is nonstrict.
Some Haskell compilers including GHC include strictness analysis, which is
a compiler pass whose job is to analyze the code of the program and figure
out which of its functions are strict and which are nonstrict.
When the Haskell code generator sees a call to a strict function, instead of
generating code that creates a suspension, it can generate the code that
an imperative language compiler would generate: code that evaluates all
the arguments, and then calls the function.
NOTE
Eager evaluation is also called strict evaluation, while lazy evaluation is
also called nonstrict evaluation.
COMP90048 Declarative Programming Lecture 20 – 2 / 18
Performance
Unpredictability
Besides generating a slowdown for most programs, laziness also makes it
harder for the programmer to understand where the program is spending
most of its time and what parts of the program allocate most of its
memory.
This is because small changes in exactly where and when the program
demands a particular value can cause great changes in what parts of a
suspension tree are evaluated, and can therefore cause great changes in
the time and space complexity of the program. (Lazy evaluation is also
called demand driven computation.)
The main problem is that it is very hard for programmers to be
simultaneous aware of all the relevant details in the program.
Modern Haskell implementations come with sophisticated profilers to help
programmers understand the behavior of their programs. There are
profilers for both time and for memory consumption.
COMP90048 Declarative Programming Lecture 20 – 3 / 18
Performance
Memory efficiency
(Revised) BST insertion code:
insert_bst :: Ord k => Tree k v -> k -> v -> Tree k v
insert_bst Leaf ik iv = Node ik iv Leaf Leaf
insert_bst (Node k v l r) ik iv
| k == ik = Node ik iv l r
| k > ik = Node k v (insert_bst l ik iv) r
| otherwise = Node k v l (insert_bst r ik iv)
As discussed earlier, this creates new data structures instead of
destructively modifying the old structure.
The advantage of this is that the old structure can still be used.
The disadvantage is new memory is allocated and written. This takes
time, and creates garbage that must be collected.
COMP90048 Declarative Programming Lecture 20 – 4 / 18
Performance
Memory efficiency
Insertion into a BST replaces one node on each level of the tree: the node
on the path from the root to the insertion site.
In (mostly) balanced trees with n nodes, the height of the tree tends to be
about log2(n).
Therefore the number of nodes allocated during an insertion tends to be
logarithmic in the size of the tree.
If the old version of the tree is not needed, imperative code can do
better: it must allocate only the new node.
If the old version of the tree is needed, imperative code will do worse:
it must copy the entire tree, since without that, later updates to the
new version would update the old one as well.
COMP90048 Declarative Programming Lecture 20 – 5 / 18
Performance
Reusing memory
When insert_bst inserts a new node into the tree, it allocates new
versions of every node on the path from the root to the insertion point.
However, every other node in the tree will become part of the new tree as
well as the old one.
This shows what happens when you insert the key "h" into a binary search
tree that already contains "a" to "g".
”d” 4 • •
”b” 2 • •
”a” 1 / / ”c” 3 / /
”f” 6 • •
”e” 5 / / ”g” 7 / /
”d” 4 • •
”f” 6 • •
”g” 7 / •
”h” 8 / /
NOTE
In this small example, the old and the new tree share the entire subtree
rooted at the node whose key is “b”, and the node whose key is “e”.
COMP90048 Declarative Programming Lecture 20 – 6 / 18
Performance
Deforestation
As we discussed earlier, many Haskell programs have code that follows this
pattern:
You start with the first data structure, ds1.
You traverse ds1, generating another data structure, ds2.
You traverse ds2, generating yet another data structure, ds3.
If the programmer can restructure the code to compute ds3 directly from
ds1, this should speed up the program, for two reasons:
the new version does not need to create ds2, and
the new version does one traversal instead of two.
Since the eliminated intermediate data structures are often trees of one
kind or another, this optimization idea is usually called deforestation.
COMP90048 Declarative Programming Lecture 20 – 7 / 18
Performance
Simple Deforestation
In some cases, you can deforest your own code with minimal effort. For
example, you can always deforest two calls to map:
map (+1) $ map (2*) list
is equivalent to
map ((+1) . (2*)) list
The second one is more succinct, more elegant, and more efficient.
You can combine two calls to filter in a similar way:
filter (>=0) $ filter (<10) list
is always the same as
filter (\x -> x >= 0 & x < 10) list
COMP90048 Declarative Programming Lecture 20 – 8 / 18
Performance
filter_map
filter_map :: (a -> Bool) -> (a -> b) -> [a] -> [b]
filter_map _ _ [] = []
filter_map f m (x:xs) =
let newxs = filter_map f m xs in
if f x then (m x):newxs else newxs
one_pass xs = filter_map is_even triple xs
two_pass xs = map triple (filter is_even xs)
The one_pass function performs exactly the same task as the two_pass
function, but it does the job with one list traversal, not two, and does not
create an intermediate list.
One can also write similarly deforested combinations of many other pairs
of higher order functions, such as map and foldl.
COMP90048 Declarative Programming Lecture 20 – 9 / 18
Performance
Computing standard deviations
four_pass_stddev :: [Double] -> Double
four_pass_stddev xs =
let
count = fromIntegral (length xs)
sum = foldl (+) 0 xs
sumsq = foldl (+) 0 (map square xs)
in
(sqrt (count * sumsq - sum * sum)) / count
square :: Double -> Double
square x = x * x
This is the simplest approach to writing code that computes the standard
deviation of a list. However, it traverses the input list three times, and it
also traverses a list of that same length (the list of squares) once.
COMP90048 Declarative Programming Lecture 20 – 10 / 18
Performance
Computing standard deviations in one pass
data StddevData = SD Double Double Double
one_pass_stddev :: [Double] -> Double
one_pass_stddev xs =
let
init_sd = SD 0.0 0.0 0.0
update_sd (SD c s sq) x =
SD (c + 1.0) (s + x) (sq + x*x)
SD count sum sumsq = foldl update_sd init_sd xs
in
(sqrt (count * sumsq - sum * sum)) / count
NOTE
This is an example of a call to foldl in which the base is of one type
(StddevData) while the list elements are of another type (Double).
It is also an example of a let clause that defines an auxiliary function, in
this case update_sd, and one in which the last part picks up the values of
the arguments of the SD data constructor in three variables.
COMP90048 Declarative Programming Lecture 20 – 11 / 18
Performance
Cords
Repeated appends to the end of a list take time that is quadratic in the
final length of the list.
In imperative languages, you would avoid this quadratic behavior by
keeping a pointer to the tail of the list, and destructively updating that tail.
In declarative languages, the usual solution is to switch from lists to a data
structure that supports appends in constant time. These are usually called
cords. This is one possible cord design; there are several.
data Cord a = Nil | Leaf a | Branch (Cord a) (Cord a)
append_cords :: Cord a -> Cord a -> Cord a
append_cords a b = Branch a b
COMP90048 Declarative Programming Lecture 20 – 12 / 18
Performance
Converting cords to lists
The obvious algorithm to convert a cord to a list is
cord_to_list :: Cord a -> [a]
cord_to_list Nil = []
cord_to_list (Leaf x) = [x]
cord_to_list (Branch a b) =
(cord_to_list a) ++ (cord_to_list b)
Unfortunately, it suffers from the exact same performance problem that
cords were designed to avoid.
The cord Branch (Leaf 1) (Leaf 2) that the last equation converts to
a list may itself be one branch of a bigger cord, such as Branch (Branch
(Leaf 1) (Leaf 2)) (Leaf 3).
The list [1], converted from Leaf 1, will be copied twice by ++, once for
each Branch data constructor in whose first operand it appears.
COMP90048 Declarative Programming Lecture 20 – 13 / 18
Performance
Accumulators
With one exception, all leaves in a cord are followed by another item, but
the second equation puts an empty list behind all leaves, which is why all
but one of the lists it creates will have to be copied again. The other two
equations make the same mistake for empty and branch cords.
Fixing the performance problem requires telling the conversion function
what list of items follows the cord currently being converted. This is easy
to arrange using an accumulator.
cord_to_list :: Cord a -> [a]
cord_to_list c = cord_to_list’ c []
cord_to_list’ :: Cord a -> [a] -> [a]
cord_to_list’ Nil rest = rest
cord_to_list’ (Leaf x) rest = x:rest
cord_to_list’ (Branch a b) rest =
cord_to_list’ a (cord_to_list’ b rest)
COMP90048 Declarative Programming Lecture 20 – 14 / 18
Performance
Sortedness check
The obvious way to write code that checks whether a list is sorted:
sorted1 :: (Ord a) => [a] -> Bool
sorted1 [] = True
sorted1 [_] = True
sorted1 (x1:x2:xs) = x1 <= x2 && sorted1 (x2:xs)
However, the code that looks at each list element handles three
alternatives (lists of length zero, one and more).
It does this because each sortedness comparison needs two list elements,
not one.
COMP90048 Declarative Programming Lecture 20 – 15 / 18
Performance
A better sortedness check
sorted2 :: (Ord a) => [a] -> Bool
sorted2 [] = True
sorted2 (x:xs) = sorted_lag x xs
sorted_lag :: (Ord a) => a -> [a] -> Bool
sorted_lag _ [] = True
sorted_lag x1 (x2:xs) = x1 <= x2 && sorted_lag x2 xs
In this version, the code that looks at each list element handles only two
alternatives. The value of the previous element, the element that the
current element should be compared with, is supplied separately.
COMP90048 Declarative Programming Lecture 20 – 16 / 18
Performance
Optimisation
You can use :set +s in GHCi to time execution.
Prelude> :l sorted
[1 of 1] Compiling Sorted ( sorted.hs, interpreted )
Ok, modules loaded: Sorted.
*Sorted> :set +s
*Sorted> sorted1 [1..100000000]
True
(50.11 secs, 32,811,594,352 bytes)
*Sorted> sorted2 [1..100000000]
True
(40.76 secs, 25,602,349,392 bytes)
The sorted2 version is about 20% faster and uses 22% less memory.
COMP90048 Declarative Programming Lecture 20 – 17 / 18
Performance
Optimisation
However, the Haskell compiler is very sophisticated. After doing
ghc -dynamic -c -O3 sorted.hs, we get this:
Prelude> :l sorted
Ok, modules loaded: Sorted.
Prelude Sorted> :set +s
Prelude Sorted> sorted1 [1..100000000]
True
(2.89 secs, 8,015,369,944 bytes)
Prelude Sorted> sorted2 [1..100000000]
True
(2.91 secs, 8,002,262,840 bytes)
Compilation gives a factor of 17 speedup and a factor of 3 memory
savings. It also removes the difference between sorted1 and sorted2.
Always benchmark your compiled code when trying to speed it up.
COMP90048 Declarative Programming Lecture 20 – 18 / 18
Interfacing with foreign languages
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 21
Interfacing with foreign languages
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Interfacing with foreign languages
Foreign language interface
Many applications involve code written in a number of different languages;
declarative languages are no different in this respect. There are many
reasons for this:
to interface to existing code (especially libraries) written in another
language;
to write performance-critical code in a lower-level language (typically
C or C++);
to write each part of an application in the most appropriate language;
as a way to gracefully translate an application from one language to
another, by replacing one piece at a time.
Any language that hopes to be successful must be able to work with other
languages. This is generally done through what is called a foreign language
interface or foreign function interface.
COMP90048 Declarative Programming Lecture 21 – 1 / 23
Interfacing with foreign languages
Application binary interfaces
In computer science, a platform is a combination of an instruction set
architecture (ISA) and an operating system, such as x86/Windows 10,
x86/Linux or SPARC/Solaris.
Each platform typically has an application binary interface, or ABI, which
dictates such things as where the callers of functions should put the
function parameters and where the callee function should put the result.
By compiling different files to the same ABI, functions in one file can call
functions in a separately compiled file, even if compiled with different
compilers.
The traditional way to interface two languages, such as C and Fortran, or
Ada and Java, is for the compilers of both languages to generate code that
follows the ABI.
COMP90048 Declarative Programming Lecture 21 – 2 / 23
Interfacing with foreign languages
Beyond C
ABIs are typically designed around C’s simple calling pattern, where each
function is compiled to machine language, and each function call passes
some number of inputs, calls another known function, possibly returns one
result, and is then finished.
This model does not work for lazy languages like Haskell, languages like
Prolog or Mercury that support nondeterminism, languages like Prolog,
Python, and Java that are implemented through an abstract machine, or
even languages like C++ where function (method) calls may invoke
different code each time they are executed.
In such languages, code is not compiled to the normal ABI. Then it
becomes necessary to provide a mechanism to call code written in other
languages. Typically, calling C code through the normal ABI is supported,
but interfacing to other languages may also be supported.
COMP90048 Declarative Programming Lecture 21 – 3 / 23
Interfacing with foreign languages
Boolean functions
One application of a foreign interface is to use specialised data structures
and algorithms that would be difficult or inefficient to implement in the
host language.
Some applicatations need to be able to efficiently manipulate Boolean
formulas (Boolean functions). This includes the following primitive values
and operations:
true, false
Boolean variables: eg: a, b, c, . . .
Operations: and (∧), or (∨), not (¬), implies (→), iff (↔), etc.
Tests: satisfiability (is there any binding for the variables that makes
a formula true?), equivalence (are two formulas the same for every set
of variable bindings?)
For example, is a↔ b equivalent to ¬((a ∧ ¬b) ∨ (¬a ∧ b))?
COMP90048 Declarative Programming Lecture 21 – 4 / 23
Interfacing with foreign languages
Binary Decision Diagrams
Deciding satisfiability or equivalence of Boolean functions is NP-complete,
so we need an efficient implementation.
BDDs are decision graphs, based on if-then-else (ite) nodes, where each
node is labeled by a boolean variable and each leaf is a truth value.
With a truth assignment for each variable, the value of the formula can be
determined by traversing from the root, following then branch for true
variables and else branch for false variables.
a
true b
true false
a b the BDD
T T T
T F T
F T T
F F F
COMP90048 Declarative Programming Lecture 21 – 5 / 23
Interfacing with foreign languages
BDDs in Haskell
We could represent BDDs in Haskell with this type:
data BDD label = BTrue | BFalse
| Ite label (BDD label) (BDD label)
The meaning of a BDD is given by:
meaning BTrue = true
meaning BFalse = false
meaning (Ite v t e) = (v ∧meaning t) ∨ (¬v ∧meaning e)
So for example, meaning (Ite a BTrue (Ite b BTrue BFalse))
= (a ∧ true) ∨ (¬a ∧ (b ∧ true ∨ (¬b ∧ false)))
= a ∨ (¬a ∧ b ∨ false)
= a ∨ b
COMP90048 Declarative Programming Lecture 21 – 6 / 23
Interfacing with foreign languages
ROBDDs
Reduced Ordered Binary Decision Diagrams (ROBDDs) are BDDs where
labels are in increasing order from root to leaf, no node has two identical
children, and no two distinct nodes have the same semantics.
a
b
c
false true
false
a b c the BDD
T T T F
T T F T
T F T F
T F F T
F T T F
F T F T
F F T F
F F F F
By sharing the c node, the ROBDD is smaller than it would be if it were a
tree. For larger ROBDDs, this can be a big savings.
COMP90048 Declarative Programming Lecture 21 – 7 / 23
Interfacing with foreign languages
Object Identity
ROBDD algorithms traverse DAGs, and often meet the same subgraphs
repeatedly. They greatly benefit from caching : recording results of past
operations to reuse without repeating the computation.
Caching requires efficiently recognizing when a node is seen again. Haskell
does not have the concept of object identity, so it cannot distinguish
a
b
c
false true
false
from
a
c
false true
b
c
false true
false
COMP90048 Declarative Programming Lecture 21 – 8 / 23
Interfacing with foreign languages
Structural Hashing
Building a new ROBDD node ite(v , thn, els) must ensure that:
1 the two children of every node are different; and
2 for any Boolean formula, there is only one ROBDD node with that
semantics.
The first is achieved by checking if thn = els, and if so, returning thn
The second is achieved by structural hashing (AKA hash-consing):
maintaining a hash table of past calls ite(v , thn, els) and their results, and
always returning the past result when a call is repeated.
Because of point 1 above, satisfiability of an ROBDD can be tested in
constant time (meaning r is satisfiable iff r 6= false)
Because of point 2 above, equality of two ROBDDs is also a constant time
test (meaning r1 = meaning r2 iff r1 = r2)
COMP90048 Declarative Programming Lecture 21 – 9 / 23
Interfacing with foreign languages
Impure implementation of pure operations
(We haven’t cheated NP-completeness: we have only shifted the cost to
building ROBDDs.)
Yet ROBDD operations are purely declarative. Constructing ROBDDs,
conjunction, disjunction, negation, implication, checking satisfiability and
equality, etc., are all pure.
This is an example of using impurity to build purely declarative code.
In fact, all declarative code running on a commodity computer does that:
these CPUs work through impurity. Even adding two numbers on a
modern CPU works by destructively adding one register to another.
If you are required to work in an imperative or object-oriented language,
you can still use such languages to build declarative abstractions, and work
with them instead of working directly with impure constructs.
COMP90048 Declarative Programming Lecture 21 – 10 / 23
Interfacing with foreign languages
robdd.h: C interface for ROBDD implementation
extern robdd *true_rep(void); /* ROBDD true */
extern robdd *false_rep(void); /* ROBDD false */
extern robdd *variable_rep(int var); /* ROBDD for var */
extern robdd *conjoin(robdd *a, robdd *b);
extern robdd *disjoin(robdd *a, robdd *b);
extern robdd *negate(robdd *a);
extern robdd *implies(robdd *a, robdd *b);
extern int is_true(robdd *f); /* is ROBDD == true? */
extern int is_false(robdd *f); /* is ROBDD == false? */
extern int robdd_label(robdd *f); /* label of f */
extern robdd *robdd_then(robdd *f); /* then branch of f */
extern robdd *robdd_else(robdd *f); /* else branch of f */
COMP90048 Declarative Programming Lecture 21 – 11 / 23
Interfacing with foreign languages
Interfacing Haskell to C
For simple cases, the Haskell foreign function interface is fairly simple. You
can interface to a C function with a declaration of the form:
foreign import ccall "C name" Haskell name :: Haskell type
But how shall we represent an ROBDD in Haskell?
C primitive types convert to and from natural Haskell types, e.g.,
C int ←→ Haskell Int
In Haskell, we want to treat ROBDDs as an opaque type: a type we
cannot peer inside, we can only pass it to, and receive it as output from,
foreign functions.
The Haskell Word type represents a word of memory, much like an Int.
However Word is not opaque, as we can confuse it with an integer, or any
Word type.
COMP90048 Declarative Programming Lecture 21 – 12 / 23
Interfacing with foreign languages
newtype
Declaring type BoolFn = Word would not make BoolFn opaque; it
would just be an alias for Word, and could be passed as a Word.
We can make it opaque with a data BoolFn = BoolFn Word
declaration. We could convert a Word w to a BoolFn with BoolFn w, and
convert a BoolFn b to a Word with
let BoolFn w = b in . . .
But this would box the word, adding an extra indirection to operations.
Instead we declare:
newtype BoolFn = BoolFn Word deriving Eq
We can only use newtype to declare types with only one constructor, with
exactly one argument. This avoids the indirection, makes the type opaque,
and allows it to be used in the foreign interface.
COMP90048 Declarative Programming Lecture 21 – 13 / 23
Interfacing with foreign languages
The interface
foreign import ccall "true_rep" true :: BoolFn
foreign import ccall "false_rep" false :: BoolFn
foreign import ccall "variable_rep" variable :: Int->BoolFn
foreign import ccall "is_true" isTrue :: BoolFn->Bool
foreign import ccall "is_false" isFalse :: BoolFn->Bool
foreign import ccall "robdd_label" minVar :: BoolFn->Int
foreign import ccall "robdd_then" minThen :: BoolFn->BoolFn
foreign import ccall "robdd_else" minElse :: BoolFn->BoolFn
type BoolBinOp = BoolFn -> BoolFn -> BoolFn
foreign import ccall "conjoin" conjoin :: BoolBinOp
foreign import ccall "disjoin" disjoin :: BoolBinOp
foreign import ccall "negate" negation:: BoolFn->BoolFn
foreign import ccall "implies" implies :: BoolBinOp
COMP90048 Declarative Programming Lecture 21 – 14 / 23
Interfacing with foreign languages
Using it
To make C code available, compile it and pass the object file on the ghc
or ghci command line.
(There is also code to show BoolFns in disjunctive normal form; all code is
in BoolFn.hs, robdd.c and robdd.h in the examples directory.)
nomad% gcc -c -Wall robdd.c
nomad% ghci robdd.o
GHCi, version 8.4.3: http://www.haskell.org/ghc/
Prelude> :l BoolFn.hs
[1 of 1] Compiling BoolFn ( BoolFn.hs, interpreted )
Ok, one module loaded.
*BoolFn> (variable 1) ‘disjoin‘ (variable 2)
((1) | (~1 & 2))
*BoolFn> it ‘conjoin‘ (negation $ variable 3)
((1 & ~3) | (~1 & 2 & ~3))
COMP90048 Declarative Programming Lecture 21 – 15 / 23
Interfacing with foreign languages
Interfacing to Prolog
The Prolog standard does not standardise a foreign language interface.
Each Prolog system has its own approach.
The SWI Prolog approach does most of the work on the C side, rather
than in Prolog. This is powerful, but inconvenient.
In an SWI Prolog source file, the declaration
:- use_foreign_library(swi_robdd).
will load a compiled C library file that links the code in swi_robdd.c,
which forms the interface to Prolog, and the robdd.c file.
These are compiled and linked with the shell command:
swipl-ld -shared -o swi_robdd swi_robdd.c robdd.c
COMP90048 Declarative Programming Lecture 21 – 16 / 23
Interfacing with foreign languages
Connecting C code to Prolog
The swi_robdd.c file contains C code to interface to Prolog:
install_t install_swi_robdd() {
PL_register_foreign("boolfn_node", 4, pl_bdd_node, 0);
PL_register_foreign("boolfn_true", 1, pl_bdd_true, 0);
PL_register_foreign("boolfn_false", 1, pl_bdd_false, 0);
PL_register_foreign("boolfn_conjoin", 3, pl_bdd_and, 0);
PL_register_foreign("boolfn_disjoin", 3, pl_bdd_or, 0);
PL_register_foreign("boolfn_negation", 2, pl_bdd_negate, 0);
PL_register_foreign("boolfn_implies", 3, pl_bdd_implies, 0);
}
This tells Prolog that a call to boolfn_node/4 is implemented as a call to
the C function pl_bdd_node, etc.
COMP90048 Declarative Programming Lecture 21 – 17 / 23
Interfacing with foreign languages
Marshalling data
A C function that implements a Prolog predicate needs to convert between
Prolog terms and C data structures. This is called marshalling data.
static foreign_t
pl_bdd_and(term_t f, term_t g, term_t result_term) {
void *f_nd, *g_nd;
if (PL_is_integer(f)
&& PL_is_integer(g)
&& PL_get_pointer(f, &f_nd)
&& PL_get_pointer(g, &g_nd)) {
robdd *result = conjoin((robdd *)f_nd, (robdd *)g_nd);
return PL_unify_pointer(result_term, (void *)result);
} else {
PL_fail;
}
}
COMP90048 Declarative Programming Lecture 21 – 18 / 23
Interfacing with foreign languages
Making Boolean functions abstract in Prolog
To keep Prolog code from confusing an ROBDD (address) from a number,
we wrap the address in a boolfn/1 term, much like we did in Haskell. We
must do this manually; it is most easily done in Prolog code.
% conjoin(+BFn1, +BFn2, -BFn)
% BFn is the conjunction of BFn1 and BFn2.
conjoin(boolfn(F), boolfn(G), boolfn(FG)) :-
boolfn_conjoin(F, G, FG).
We can make Prolog print BDDs (or anything) nicely by adding a clause
for user:portray/1:
:- multifile(user:portray/1).
user:portray(boolfn(BDD)) :- !,
% definition is in boolfn.pl ...
COMP90048 Declarative Programming Lecture 21 – 19 / 23
Interfacing with foreign languages
Using it
nomad% swipl-ld -shared -o swi_robdd swi_robdd.c robdd.c
nomad% pl
Welcome to SWI-Prolog (threaded, 64 bits, version 7.7.19)
1 ?- [boolfn].
true.
2 ?- variable(1,A), variable(2,B), variable(3,C),
| disjoin(A,B,AB), negation(C,NotC), conjoin(AB,NotC,X).
A = ((1)),
B ((2)),
C = ((3)),
AB = ((1) | (~1 & 2)),
NotC = ((~3)),
X = ((1 & ~3) | (~1 & 2 & ~3)).
COMP90048 Declarative Programming Lecture 21 – 20 / 23
Interfacing with foreign languages
Impedance mismatch
Declarative languages like Haskell and Prolog typically use different
representations for similar data. For example, what would be represented
as a list in Haskell or Prolog would most likely be represented as an array
in C or Java.
The consequence of this is that in each language (declarative and
imperative) it is difficult to write code that works on data structures
defined in the other language.
This problem, usually called impedance mismatch, is the reason why most
cross-language interfaces are low level, and operate only or mostly on
values of primitive types.
NOTE
Impedance mismatch is the name of a problem in electrical engineering
that has some similarity to this situation. In both cases, difficulties arise
from trying to connect two systems that make incompatible assumptions.
COMP90048 Declarative Programming Lecture 21 – 21 / 23
Interfacing with foreign languages
Comparative strengths of declarative languages
Programmers can be significantly more productive because they can
work at a significantly higher level of abstraction. They can focus on
the big picture without getting lost in details, such as whose
responsibility it is to free a data structure.
Processing of symbolic data is significantly easier due to the presence
of algebraic data types and parametric polymorphism.
Programs can be significantly more reliable, because
you cannot make a mistake in an aspect of programming that the
language automates (e.g. memory allocation), and
the compiler can catch many more kinds of errors.
What debugging is still needed is easier because you can jump
backward in time.
Maintenance is significantly easier, because
the type system helps to locate what needs to be changed, and
the typeclass system helps avoid unwanted coupling in the first place.
You can automatically parallelize declarative programs.
COMP90048 Declarative Programming Lecture 21 – 22 / 23
Interfacing with foreign languages
Comparative strengths of imperative languages
If you are willing to put in the programming time, you can make the
final program significantly faster.
Most existing software libraries are written in imperative languages.
Using them in declarative languages is harder than using them in
another imperative language (due to dissimilarity of basic concepts),
while using them is easiest in the language they are written in. If the
bulk of a program interfaces to an existing library, this argues for
writing the program in the language of the library:
Java for Swing
Visual Basic.NET or C# for ASP.NET
There is a much greater variety of programming tools to choose from
(debuggers, profilers, IDEs etc).
It is much easier to find programmers who know or can quickly learn
the language.
NOTE
ASP.NET is a .NET application, which means that it is native not to a
single programming language but to the .NET ecosystem, of which the
two principal languages are Visual Basic.NET and C#.
COMP90048 Declarative Programming Lecture 21 – 23 / 23
Parsing
The University of Melbourne
School of Computing and Information Systems
COMP90048
Declarative Programming
Lecture 22
Parsing
Copyright c© 2023 The University of Melbourne
COMP90048 Declarative Programming
Parsing
Parsing
A parser is a program that extracts a structure from a linear sequence of
elements.
For example, a parser is responsible for taking input like:
3+ 4 ∗ 5
and producing a data structure representing the intended structure of the
input:
+
3 ∗
4 5
as opposed to
∗
+
3 4
5
COMP90048 Declarative Programming Lecture 22 – 1 / 21
Parsing
Using an existing parser
The simplest parsing technique is to use an existing parser. Since every
programming language must have a parser to parse the program, it may
also give the programmer access to its parser.
A Domain Specific Language (DSL) is a small programming language
intended for a narrow domain. Often these are embedded in existing
languages, extending the host language with new capabilities, but using
the host language for functionality outside that domain.
If a DSL can be parsed by extending the host language parser, that makes
the DSL more convenient to use, since it just adds new constructs to the
language.
Prolog handles that quite nicely, as we saw earlier. The read/1 built-in
predicate reads a term. You can use op/3 declarations to extend the
language.
COMP90048 Declarative Programming Lecture 22 – 2 / 21
Parsing
Operator precedence
Operator precedence parsing is a simple technique based on operators:
precedence which operators bind tightest;
associativity whether repeated infix operators associate to the left, right,
or neither (e.g., whether a − b − c is (a − b)− c or
a − (b − c) or an error); and
fixity whether an operator is infix, prefix, or postfix.
In Prolog, the op/3 predicate declares an operator:
:- op(precedence,fixity,operator)
where precedence is a precedence number (larger number is lower
precedence; 1000 is precedence of goals), fixity is a two or three letter
symbol giving fixity and associativity (f indicates the operator, x indicates
subterm at lower precedence, y indicates subterm at same precedence),
and operator is the operator to declare.
COMP90048 Declarative Programming Lecture 22 – 3 / 21
Parsing
Example: Prolog imperative for loop
:- op(950, fx, for). :- op(940, xfx, in).
:- op(600, xfx, ’..’). :- op(1050, xfy, do).
for Generator do Body :-
( call(Generator),
call(Body),
fail
; true
).
Var in Low .. High :-
between(Low, High, Var).
Var in [H|T] :-
member(Var, [H|T]).
COMP90048 Declarative Programming Lecture 22 – 4 / 21
Parsing
Example: Prolog imperative for loop
?- for X in 1 .. 4 do format(’~t~d ~6|^ 2 = ~d~n’, [X, X^2]).
1 ^ 2 = 1
2 ^ 2 = 4
3 ^ 2 = 9
4 ^ 2 = 16
true.
?- for X in [2,3,5,7,11] do
| ( X mod 2 =:= 0 -> Parity = even
| ; Parity = odd
| ),
| format(’~t~d~6| is ~w~n’, [X,Parity]).
2 is even
3 is odd
5 is odd
7 is odd
11 is odd
true.
COMP90048 Declarative Programming Lecture 22 – 5 / 21
Parsing
Haskell operators
Haskell operators are simpler, but more limited. Haskell does not support
prefix or postfix operators, only infix.
Declare an infix operator with:
associativity precedence operator
where associativity is one of:
infixl left associative infix operator
infixr right associative infix operator
infix non-associative infix operator
and precedence is an integer 1–9, where lower numbers are lower (looser)
precedence.
COMP90048 Declarative Programming Lecture 22 – 6 / 21
Parsing
Haskell example
This code defines % as a synonym for mod in Haskell:
infixl 7 %
(%) :: Integral a => a -> a -> a
a % b = a ‘mod‘ b
COMP90048 Declarative Programming Lecture 22 – 7 / 21
Parsing
Grammars
Parsing is based on a grammar that specifies the language to be parsed.
Grammars are defined in terms of terminals, which are the symbols of the
language, and non-terminals, which each specify a linguistic category.
Grammars are defined by a set of rules of the form:
(nonterminal ∪ terminal)∗ → (nonterminal ∪ terminal)∗
where ∪ denotes set union, ∗ (Kleene star) denotes a sequence of zero or
more repetitions, and the part on the left of the arrow must contain at
least one non-terminal. Most commonly, the left side of the arrow is just a
single non-terminal :
expression→ expression ’+’ expression
expression→ expression ’-’ expression
expression→ expression ’*’ expression
expression→ expression ’/’ expression
expression→ number
Here we denote terminals by enclosing them in quotes.
COMP90048 Declarative Programming Lecture 22 – 8 / 21
Parsing
Definite Clause Grammars
Prolog directly supports grammars, called definite clause grammars
(DCG), which are written using a very similar syntax:
Nonterminals are written using a syntax like ordinary Prolog goals.
Terminals are written between backquotes.
The left and right sides are separated with –> (instead of :-).
Parts on the right side are separated with commas.
Empty terminal is written as [] or “
For example, the above grammar can be written as a Prolog DCG:
expr --> expr, ‘+‘, expr.
expr --> expr, ‘-‘, expr.
expr --> expr, ‘*‘, expr.
expr --> expr, ‘/‘, expr.
expr --> number.
COMP90048 Declarative Programming Lecture 22 – 9 / 21
Parsing
Producing a parse tree
A grammar like this one can only be used to test if a string is in the
defined language; usually we want to produce a data structure (a parse
tree) that represents the linguistic structure of the input.
This is done very easily in a DCG by adding arguments, ordinary Prolog
terms, to the nonterminals.
expr(E1+E2) --> expr(E1), ‘+‘, expr(E2).
expr(E1-E2) --> expr(E1), ‘-‘, expr(E2).
expr(E1*E2) --> expr(E1), ‘*‘, expr(E2).
expr(E1/E2) --> expr(E1), ‘/‘, expr(E2).
expr(N) --> number(N).
We will see a little later how to define the number nonterminal.
COMP90048 Declarative Programming Lecture 22 – 10 / 21
Parsing
Recursive descent parsing
DCGs map each nonterminal to a Prolog predicate that
nondeterministically parses one instance of that nonterminal. This is called
recursive descent parsing.
To use a grammar in Prolog, use the built-in phrase/2 predicate:
phrase(nonterminal,string). For example:
?- phrase(expr(Expr), ‘3+4*5‘).
ERROR: Stack limit (1.0Gb) exceeded
This exposes a weakness of recursive descent parsing: it cannot handle left
recursion.
COMP90048 Declarative Programming Lecture 22 – 11 / 21
Parsing
Left recursion
A grammar rule like
expr(E1+E2) --> expr(E1), ‘+‘, expr(E2).
is left recursive, meaning that the first thing it does, before parsing any
terminals, is to call itself recursively. Since DCGs are transformed into
similar ordinary Prolog code, this turns into a clause that calls itself
recursively without consuming any input, so it is an infinite recursion.
But we can transform our grammar to remove left recursion:
1 Rename left recursive rules to A_rest and remove their first
nonterminal.
2 Add a rule for A_rest that matches the empty input.
3 Then add A_rest to the end of the non-left recursive rules.
COMP90048 Declarative Programming Lecture 22 – 12 / 21
Parsing
Left recursion removal
This is a little harder for DCGs with arguments: you also need to
transform the arguments.
Replace the argument of non-left recursive rules with a fresh variable, use
the original argument of the rule as the first argument of the _rest added
nonterminal, and that fresh variable as the second. So:
expr(N) --> number(N).
would be transformed to:
expr(E) --> number(N), expr_rest(N, E).
COMP90048 Declarative Programming Lecture 22 – 13 / 21
Parsing
Left recursion removal
For non-recursive rules, use the argument of the left-recursive nonterminal
as the first head argument and a fresh variable as the second. Then use
the original argument of the head as the first argument of the _tail call
and a fresh variable as the second argument of the head and of the _tail
call. So:
expr(E1+E2) --> expr(E1), ‘+‘, expr(E2).
would be transformed to:
expr_rest(E1, R) --> ‘+‘, expr(E2), expr_rest(E1+E2, R).
COMP90048 Declarative Programming Lecture 22 – 14 / 21
Parsing
Ambiguity
With left recursion removed, this grammar no longer loops:
?- phrase(expr3(Expr), ‘3-4-5‘).
Expr = 3-(4-5) ;
Expr = 3-4-5 ;
false.
?- phrase(expr3(Expr), ‘3+4*5‘).
Expr = 3+4*5 ;
Expr = (3+4)*5 ;
false.
Unfortunately, this grammar is ambiguous: negation can be either left- or
right-associative, and it’s ambiguous whether + or * has higher precedence.
COMP90048 Declarative Programming Lecture 22 – 15 / 21
Parsing
Disambiguating a grammar
The ambiguity originated in the original grammar: a rule like
expr(E1-E2) --> expr(E1), ‘-‘, expr(E2).
applied to input “3-4-5” allows the first expr to match “3-4”, or the
second to match “4-5”.
The solution is to ensure only the desired one is possible by splitting the
ambiguous nonterminal into separate nonterminals, one for each
precedence level. The above rule should become:
expr(E-F) --> expr(E), ‘-‘ factor(F)
before eliminating left recursion.
COMP90048 Declarative Programming Lecture 22 – 16 / 21
Parsing
Disambiguating a grammar
This finally gives us:
expr(E) --> factor(F), expr_rest(F,E).
expr_rest(F1,E) --> ‘+‘, factor(F2), expr_rest(F1+F2,E).
expr_rest(F1,E) --> ‘-‘, factor(F2), expr_rest(F1-F2,E).
expr_rest(F,F) --> [].
factor(F) --> number(N), factor_rest(N,F).
factor_rest(N1,F) --> ‘*‘, number(N2), factor_rest(N1*N2,F).
factor_rest(N1,F) --> ‘/‘, number(N2), factor_rest(N1/N2,F).
factor_rest(N,N) --> [].
COMP90048 Declarative Programming Lecture 22 – 17 / 21
Parsing
Handling terminals
The “terminals” in a grammar can be whatever you like. Traditionally,
syntax analysis is divided into lexical analysis (also called tokenising) and
parsing.
Lexical analysis uses a simpler class of grammar to group characters into
tokens, while eliminating meaningless text, like whitespace and comments.
Tools are available for writing tokenisers, but you can write them by hand
or use the same grammar tool as you used for parsing, such as a DCG.
We will take that approach.
COMP90048 Declarative Programming Lecture 22 – 18 / 21
Parsing
DCG for parsing numbers
In addition to allowing literal ‘strings‘ as terminals DCGs allow you to
write lists as terminals (in fact, ‘strings‘ are just lists of ASCII codes).
DCGs also allow you to write ordinary Prolog code enclosed in {curley
braces}. If this code fails, the rule will fail. We can also use if->then;else
in DCGs.
number(N) -->
[C], { ‘0‘ =< C, C =< ‘9‘ },
{ N0 is C -‘0‘ },
number_rest(N0,N).
number_rest(N0,N) -->
( [C], { ‘0‘ =< C, C =< ‘9‘ }
-> { N1 is N0 * 10 + C - ‘0‘ },
number_rest(N1,N)
; { N = N0 }
).
COMP90048 Declarative Programming Lecture 22 – 19 / 21
Parsing
Demo
Finally, we have a working parser.
?- phrase(expr(E), ‘3-4-5‘), Value is E.
E = 3-4-5,
Value = -6 ;
false.
?- phrase(expr(E), ‘3+4*5‘), Value is E.
E = 3+4*5,
Value = 23 ;
false.
?- phrase(expr(E), ‘3*4+5‘), Value is E.
E = 3*4+5,
Value = 17 ;
false.
COMP90048 Declarative Programming Lecture 22 – 20 / 21
Parsing
Going beyond
This is just the beginning. Take Programming Language Implementation
to go further with parsing and compilation. A few final comments:
DCGs can run backwards, generating text from structure:
flatten(empty) --> []
flatten(node(L,E,R)) -->
flatten(L),
[E],
flatten(R).
Haskell has several ways to build parsers:
Read type class for parsing Haskell expressions; opposite of
Show.
ReadP More general, more efficient string parser.
Parsec Full-fledged file parsing.
COMP90048 Declarative Programming Lecture 22 – 21 / 21