Intro duction to Functional Programming Lecture

Intro duction to

Functional Programming

John Harrison

University of Cambridge

Lecture

Intro duction and Overview

Topics covered

 Imp erative programming

 Functional programming

 The merits of functional programming

 Historical remarks

 Overview of the course

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Imp erative programming

Imp erative or pro cedural programs rely on

mo difying a state by using a sequence of

commands

The state is mainly mo died by the assignment

v E or v E command written

We can execute one command b efore another by

writing them in sequence p erhaps separated by a

C C semicolon

Commands can b e executed conditionally using

if and rep eatedly using while

Programs are a series of instructions on how to

mo dify the state

FORTRAN Algol C Imp erative languages eg

Mo dula supp ort this style of programming

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

An abstract view

We ignore inputoutput op erations and assume

that a program runs for a limited time pro ducing

a result

We can consider the execution in an abstract way

as

! ! !


!

n

The program is started with the computer in an

including the inputs to the initial state

program

The program nishes with the computer in a nal

containing the outputs of the program state

n

The state passes through a nite sequence of

changes to get from to in general each

n

command may mo dify the state

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Functional programming

A functional program is simply an expression and

executing the program means evaluating the

expression We can relate this to the imp erative

E view by writing

n

 There is no state ie there are no variables

 Therefore there is no assignment since theres

nothing to assign to

 And there is no sequencing and no rep etition

since one expression do es not aect another

But on the p ositive side

 We can have recursive functions giving

something comparable to rep etition

 Functions can b e used much more exibly

higher order functions eg we can have

Functional languages supp ort this style of

programming

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Example the factorial

The factorial function can b e written imp eratively

in C as follows

int factint n

int x

while n

x x n

n n

return x

whereas it would b e expressed in ML as a

recursive function

fun fact n

if n then

else n factn

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Why

At rst sight a language without variables

assignment and sequencing lo oks very impractical

We will show in this course how a lot of

interesting programming can b e done in the

functional style

Imp erative programming languages have arisen as

an abstraction of the hardware from machine

co de through assemblers and macro assemblers

to FORTRAN and b eyond

Perhaps this is the wrong approach and we should

approach the task from the human side Mayb e

functional languages are b etter suited to p eople

But what concrete reasons are there for preferring

functional languages

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Merits of functional programming

By avoiding variables and assignments we gain

the following advantages

 Clearer semantics Programs corresp ond more

directly to abstract mathematical ob jects

 More freedom in implementation eg

parallelizability

By the more exible use of functions we gain

 Conciseness and elegance

 Better parametrization and mo dularity of

programs

 Convenient ways of representing innite data

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Denotational semantics

We can identify our ML factorial function with an

abstract mathematical partial function Z ! Z

8

<

n if n 

factn

:

? otherwise

where ? denotes undenedness since for negative

arguments the program fails to terminate

Once we have a state this simple interpretation

no longer works Here is a C function that

do esnt corresp ond to any mathematical function

int randvoid

static int n

return n n

This gives dierent results on successive calls

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Semantics of imp erative programs

In order to give a corresp onding semantics to

imp erative programs we need to make the state

explicit For example we can mo del commands as

 Partial functions ! Strachey

 Relations on  Hoare

 Predicate transformers ie total functions

! bool ! ! bool Dijkstra

If we allow the goto statement even these are not

enough and we need a semantics based on

continuations Wadsworth Morris

All these metho ds are quite complicated

With functional programs we have a real chance

of proving their correctness or the correctness of

certain transformations or optimizations

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Problems with functional programs

Functional programming is not without its

deciencies Some things are harder to t into a

purely functional mo del eg

 Inputoutput

 Interactive or continuously running programs

eg editors pro cess controllers

However in many ways innite data structures

can b e used to accommo date these things

Functional languages also corresp ond less closely

to current hardware so they can b e less ecient

and it can b e hard to reason ab out their time and

space usage

ML is not a pure functional language so you can

use variables and assignments if required

However most of our work is in the pure

functional subset

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Historical remarks

Some of the ideas b ehind functional programming

go back a long way eg to lamb da calculus a

logical formalism due to Alonzo Church invented

in the s b efore electronic computers

The earliest real functional programming

language was LISP invented by McCarthy in the

s However this had a numb er of defects which

we will discuss later

The mo dern trend really b egins with ISWIM

invented by Peter Landin in the s

The ML family started with Robin Milners

theorem prover Edinburgh LCF in the late s

The language we shall study is essentially core

Standard ML but there are other imp ortant

dialects notably CAML and Ob jective CAML

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Overview of the course

 Practicalities of interacting with ML

 Key functional concepts eg evaluation

strategy higher order functions

 Polymorphic typ es

 Recursive functions and recursive structures

 Hints for eective programming

 Exceptions references and other imp erative

features

 Proving programs correct

John Harrison University of Cambridge January

Intro duction to Functional Programming Lecture

Overview of the course

We want to show the p ower of ML so well nish

with more substantial examples that illustrate

some of the p ossibilities

 Symb olic dierentiation

 Recursive descent parsing

 A Prolog interpreter

 A theorem prover

The co de for these examples will b e made

available on Thor

John Harrison University of Cambridge January