Lambda-calculus, Combinators and Functional Programming

194 Pages·1988·6.634 MB·English

by  G. E. Revesz

Lambda-Calculus, Combinators, and Functional Programming
Cambridge Tracts in Theoretical Computer Science
Managing Editor: Professor C. J. van Rijsbergen, Computing Science Department,
University of Glasgow, Glasgow, Scotland
Editorial Board:
S. Abramsky, Department of Computing Science, Imperial College of Science and
Technology, London
P. H. Aczel, Department of Computer Science, Manchester
J. W. de Bakker, Centrum voor Wiskunde en Informatica, Amsterdam
J. A. Goguen, Computing Laboratory, University of Oxford
J. V. Tucker, Department of Computer Science, University of Swansea
Titles in the Series
I. G. J. Chaitin Algorithmic Information Theory
2. L. C. Paulson Logic and Computation
3. J. M. Spivey Understanding Z
4. G. E. Revesz Lambda Calculus, Combinators and Functional Programming
5. S. Vickers Topology via Logic
6. A. M. Ramsay Formal Methods in Artificial Intelligence
7. J. Y. Girard Proofs and Types
LAMBDA-CALCULUS,
COMBINATORS, AND
FUNCTIONAL
PROGRAMMING
G. E. REVESZ
IBM Thomas J. Watson Research Center, Yorktown Heights, New York
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CAMBRIDGE UNIVERSITY PRESS
Cambridge
New York Port Chester
JVfelbourne Sydney
Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 I RP
40 West 20th Street, New York, NY 10011, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
© Cambridge University Press 1988
First published 1988
Reprinted 1989
Printed in Great Britain by Billings & Sons Limited, Worcester
British Library cataloguing in publication data
Revesz, Gyorgy E.
Lambda-calculus, combinators
and functional programming.-(Cambridge tracts
in theoretical computer science; v. 4).
I. Lambda calculus
I. Title
511.3 QA9.5
ISBN 0 521 34589 8
CONTENTS
Preface Vll
1. Introduction
1.1 Variables and functions in mathematics and in programming
languages
1.2 Domains, types, and higher-order functions 6
1.3 Polymorphic functions and Currying 10
2. Type-free lambda-calculus
2.1 Syntactic and semantic considerations 14
2.2 Renaming, a-congruence, and substitution 18
2.3 Beta-reduction and equality 23
2.4 The Church-Rosser theorem 25
2.5 Beta-reduction revisited 28
3. Combinators and constant symbols
3.1 Lambda-expressions without free variables 35
3.2 Arithmetic and other constants and combinators 39
3.3 Recursive definitions and theY combinator 43
3.4 Elimination of bound variables: bracket abstraction 49
4. List manipulation in lambda-calculus
4.1 An extension of the syntax of .\-expressions 59
4.2 Additional axioms for list manipulation 63
4.3 List manipulating functions 66
vi CONTENTS
4.4 Dealing with mutual recursion 72
4.5 Computing with infinite lists 76
5. Rule-based semantics of .\-expressions
5.1 Program transformation as a computation 84
5.2 Controlled reduction 89
5.3 The functional programming system FP 94
5.4 Translation of functional programs to .\-calculus 99
5.5 List comprehension in Miranda 103
6. Outlines of a reduction machine
6.1 Graph representation of .\-expressions 113
6.2 The instructions of a reduction machine 118
6.3 Implementation of primitive functions 124
6.4 Demand-driven evaluation 128
7. Towards a parallel graph-reduction
7.1 Harnessing the implicit parallelism 135
7.2 On-the-fly garbage collection 138
7.3 Control of parallelism 147
Appendix A
A proof of the Church-Rosser theorem 152
Appendix B
Introduction to typed .\-calculus 162
Bibliographical notes 171
References 176
PREFACE
There is a growing interest in lambda-calculus and combinators among
computer scientists. The appeal of these abstract mathematical theories is
due to their elementary yet quite universal nature. They seem to have suc
cessfully captured the most general formal properties of the notion of a
mathematical function, which, in turn, is one of the most powerful con
cepts of modern mathematics.
The relevance of lambda-calculus to computer science is quite obvious
in many cases. For instance, the design of the programming language LISP
has been largely influenced by lambda-calculus. Also, the call by name
mechanism of parameter correspondence used in ALGOL-like languages
is closely related to the operation of formal substitution as defined pre
cisely in lambda-calculus. The same is true for the textual substitution
performed by macro generators. More recently John Backus and other
proponents of functional style programming have strongly emphasized the
importance of the function concept in computer science. Corresponding
research efforts to develop efficient implementation techniques for func
tional languages have produced many interesting results in the theory of
combinators as well as in lambda-calculus.
An explicit and systematic use of lambda-calculus in computer science
was initiated by Peter Landin, Christopher Strachey, and a few others who
started the development of a formal theory of semantics for programming
languages based directly on lambda-calculus. Their approach is now called
denotationa/ semantics and it is widely accepted as a relevant theory. At
first, however, denotational semantics was thought to be flawed by some
Vlll PREFACE
experts, because of its naive use of the type-free lambda-calculus, which
did not seem to have any consistent mathematical model at that time. The
first mathematical model for the type-free lambda-calculus was discovered
by Dana Scott only in 1969 when he was trying to prove the nonexistence
of such models. Since then, lambda-calculus and the theory of combinators
have experienced a vigorous revival, which must have surprised even their
originators, Alonso Church and Haske! B. Curry, who started their devel
opment in the early 30s.
Lambda-calculus may also play a more significant role in the theory
of parallel algorithms in the future. It represents an abstract mathematical
model of computation which is equivalent to the Turing machine. But,
while the Turing machine is basically a sequential device, lambda-calculus
can retain the implicit parallelism that is normally present in a mathematical
expression. In pure mathematics, there is no such thing as the 'side effect'
of a computation and thus, independent subexpressions may be computed
in parallel. We can take advantage of this type of parallelism when using
lambda-calculus.
This book approaches lambda-calculus and the theory of combinators
from a computer scientist's point of view. It presents only their funda
mental facts and some of their applications to functional programming in
a self-contained manner. But, in contrast with the usual treatment of
lambda-calculus in computer science texts, we present formal proofs for
most of the results which are relevant to our discussion. At the same time,
lambda-calculus is treated here as a functional language implemented by
software on a conventional computer. It can also be implemented by hard
ware in a nonconventional computer whose machine code language would
be lambda-calculus by itself. As a matter of fact, there already exist ex
perimental machines whose instruction sets are fashioned after lambda
calculus and/ or combinators.
As functional programming is coming of age, several more new hard
ware designs aimed at a direct execution of functional programs are likely
to appear in the future. Some variant of lambda-calculus will certainly be
used in most of them. Future generations of computers will probably have
the speed and the necessary sophistication built in the hardware that may
make them capable of running functional programs more efficiently, with
out placing the burden of controlling the parallelism on the programmer's
shoulders.