Mathematical Methods in Program Development

536 Pages·1997·21.26 MB·English

by  David Gries (auth.), Manfred Broy, Birgit Schieder (eds.)

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Mathematical Methods
in Program Development
Edited by
Manfred Broy
Birgit Schieder
Institut fur Informatik
Technische Universitat Munchen
D-80290 Munchen, Germany
Springer
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on Mathematical
Methods in Program Development, held in Marktoberdorf, Germany,
July 30 - August 11, 1996
Library of Congress Cataloging-In-Publication Data
Mathematical methods in program development/ edited by Manfred Broy.
Birgit Schieder.
p. cm. -- (NATO ASI series. Series F. Computer and systems
sciences; vol. 158)
"Proceedings of the NATO Advances Study Institute on MathelUtical
Methods in Program Development. held in Marktoberdorf. Germany. July
3D-August 11. 1996"--T.p. verso.
Includes bibliographical references.
ISBN-13: 978-3-642-645884 e-ISBN-13: 978-3-642-60858-2
D01:1 0.1 007/978-3-642-60858-2
1. Electronic digital computers--Programming. 2. Computer
software--Development--Mathematical methods. I. Broy. M. • 1949-
II. Schieder. Birgit. III. NATO Advances Study Institute on
Mathematical Methods in Program Development (1996 : Marktoberdorf.
Germany) IV. Series.
OA76.6.M3648 1997
005.1'01'51--dc21 97-22450
CIP
CR Subject Classification (1991):
D.2.1, D.2.2, D.2.4, F.3.1,F.4.1, 1.2.2
ISBN-13: 978-3-642-64588-4 Springer-Verlag Berlin Heidelberg New York
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© Springer-Verlag Berlin Heidelberg 1997
Softcover reprint of the hardcover 1st edition 1997
Typesetting: Camera-ready by editor
Printed on acid-free paper
SPIN: 10552724 45/3142 - 5 43210
Preface
Program development is one of the most fascinating challenges of modern
technology. Still, the field is in an immature state. Only step by step is the
scientific foundation being discovered and understood in more detail. The
Summer School on Mathematical Methods in Program Development was held
in Marktoberdorf from July 30 to August 11, 1996. More than 100 people,
counting participants, lecturers, and staff, attended.
The Summer School centred around the techniques of the mathemat
ics of program development, including programming paradigms treated by
Kathleen Fisher, John C. Mitchell, and Gert Smolka, with their contri
butions on new programming paradigms, such as object-oriented program
ming and higher-order concurrent constraint programming. An excellent sur
vey over the history of programming languages, including many interesting
side-remarks explaining why the history went the way it did, is found in
F.L. Bauer's contribution on the genesis of algorithmic languages. Trying
to abstract away all the superficial differences in programming style are the
contributions by C.A.R. Hoare on unified theories of programming and by
Manfred Broy with respect to mathematical models used in early phases of
program development.
The contributions by J .-R. Abrial and L. Mussat and by B. von K;yger
concentrated on case studies, demonstrating the development of protocols by
applying well-founded mathematical techniques in the development process.
The rest of the contributions are devoted to the aspect of proving programs
correct. This includes questions of proof and program design and calcula
tionallogic, as in the contributions by Edsger W. Dijkstra, by W.H.J. Feijen
and A.J .M. van Gasteren, and by David Gries. Finally the use of machine
based verification assistance is explained by Larry Paulson with the system
Isabelle and by N. Shankar introducing PVS. The Summer School treated the
topics in a very comprehensive, but nevertheless harmonious way. It turned
out to be quite a surprise how closely all the aspects that were treated are
related and how well the lectures fitted together.
With people from more than 20 countries all over the world together
discussing, learning, and further developing this exciting topic of the math
ematics of program development, the Summer School was a unique event
both from a professional and from a social point of view. It is my privi
lege to thank all those people who helped to make the Summer School a big
success. These are the participants, the lecturers, our hosts in Marktober
dorf, the people helping in the staff, and especially Dr. Ralf Steinbriiggen
VI
and Frau Ingrid Luhn, who took care perfectly of all the many questions of
organization. Finally, I thank Dr. Hans W6ssner from Springer-Verlag, who
again was a very pleasant partner to work with when preparing these lecture
notes.
Munich, April 1997 Manfred Broy
Table of Contents
D. Gries
Eliminating the Chaff-Again:
Banquet Speech at Marktoberdorf 1996 1
C.A.R. Hoare
Envoi at Marktoberdorf 1996 9
o.
Part Designing Proofs and Programs 11
E. W. Dijkstm
Heuristics for a Very Simple Euclidean Proof (EWD1180) 13
Complete DAGs (EWD1204) ......................................... 17
A Result of Rabbit Removal (EWD1208) ............................. 19
For the Record: Painting the Squared Plane (EWD1212) ........... . 25
The Marriage Agency (EWD1214) .................................. . 27
Courtesy Dr. Birgit Schieder (EWD1215) ............................ 29
WLOG, or the Misery of the Unordered Pair (EWD1223) ............ 33
My Simplest Theorem (EWD1232) ................................... 35
The Transitive Closure of a Wellfounded Relation (EWD1241) ........ 37
The Formula for sin.(a + (3) (EWD1244a) ............................ 41
A Kind of Converse of Leibniz's Principle (EWD1245) 43
The Strengths of the Academic Enterprise (EWD1175) 45
W.H.J. Feijen, A.J.M. van Gaster-en
On a Method for the Formal Design of Multiprograms 53
D. Gries
Foundations for Calculational Logic 83
\/111
Part 1. Examples of Protocol Design ........................... 127
J.-R. Abrial, L. Mussat
Specification and Design of a Transmission Protocol
by Successive Refinements Using B ................................... 129
B. von Karger
Formal Derivation of a Protocol in CSP 201
Part 2. Languages and Models for Program Design 213
F.L. Bauer
Genesis of Algorithmic Languages .................................... 215
M. Bray
Mathematical Methods in System and Software Engineering .......... 271
C.A.R. Hoare
Unified Theories of Programming 313
Part 3. Programming Paradigms ............................... 369
K. Fisher, J. C. Mitchell
On the Relationship Between Classes, Objects
and Data Abstraction ................................................ 371
G. Smolka
The Oz Programming Model 409
A Foundation for Higher-order
Concurrent Constraint Programming ................................. 433
Part 4. Mechanical Proof Assistance 459
L.C. Paulson
Tool Support for Logics of Programs ................................. 461
N. Shankar
Machine-Assisted Verification Using Theorem Proving
and Model Checking ................................................. 499
Eliminating the Chaff -Again
Banquet Speech at Marktoberdorf 1996
David Gries
Computer Science, Cornell University
Ithaca, New York, 14853 USA
I was asked to speak at this banquet at a rather late date. In order to make preparation
easy, I did what any self-respecting software engineer does: reuse an already-produced
component -with changes to fit the new context. At Marktoberdorf 1978, the formal
development of algorithms was a hot topic, discussed with religious fervor by those for
and against. Yesterday, traces of that fervor cropped up in the discussion of operational
reasoning versus formal proof development. At Marktoberdorf 1978, I delivered a sermon
on the new religion of formal algorithm development, called "Eliminating the Chaff". This
sermon is the component that I will reuse. "Chaff", by the way, is what you get when you
thresh wheat to get the kernels -the chaff is the garbage that is thrown away.
Within this sermon, various authors are appropriated quoted (or misquoted, as the case
may be). From time to time, I will give an aside (footnote) to give you historical information
that you, at your age, may not know and to let you know when to laugh.
The reading for the day
The reading for this m3o'r. ning's sermon is taken from Knuth's first epistlel to the Structurians
(page 6, pars. 2 and This part of the epistle concerns Knuth's attempt to develop an
algorithm discussed in the first book of the old testament2 and the problems he faced:
Whenever I'm trying to write a program without goto statements, I waste an
inordinate amount of time deciding what type of iterative clause to use (while
or repeat, etc.) ... I know in my head what I want to do, but I have to trans'late
it painstakingly into a notation that often isn't well-suited to the mental concept
[I wrote the program while] ... I was in bed with a pad of paper, ... , at about
one AM; I expect I finished 15 or 20 minutes later. About 2 minutes were wasted
trying to think of a suitable iteration statement.
Here endeth the reading for the day.
'Knuth, D.E. A review of Structured Programming. STAN-CS-73-371, Computer Science Department,
Startford University, June 1973.
2Dahl, 0.-1., E.W. Dijkstra, and C.A.R. Hoare. Structu.red Programming. Academic Press, 1972.
M. Broy et al. (eds.), Mathematical Methods in Program Development
© Springer-Verlag Berlin Heidelberg 1997
2
The technical lesson
Keeping in mind the reading for the day, let us develop a divine little binary search algorithm
in the calculational way and see what we can learn from it. We ardently seek an algorithm
that, given x and an array segment b[O .. n - 1] that satisfies Q, stores in variable i to
truthify R:
Q: b[O] :::; x < b[n -1] A 2:::; n
R: 0 :::; i < n - 1 A b[i]:::; x < b[i + 1]
We note with satisfaction that for the segment b shown below, the divine little algorithm
will store in i the rightmost position of x -or if x is not in the array segment, the position
after which x belongs.
o 1 2 3 4 5 6
bl244477 8 1
We reject with horror specifications that call for x to be found or that fail if x is not in
the array. Such heathenistic suggestions, usually proposed by operational people who hide
their shame in unreadable handwriting, must be suppressed! Nay, eradicated! Even if the
ink is not tasteful!
We now seek simple ways to truthify R -or parts of it. We note that i:= 0 truthifies
all except the conjunct x < b[i + 1]. Hence, using a method passed down to us by one of
our leaders, we create a fresh variable j and weaken R to get a possible loop invariant P:
P : 0 :::; i < n - 1 A j < n A b[i]:::; x < bfj]
which is truthified by .i, j := 0, n - 1. In short order, we:
'*
(a) Calculate the loop condition B using the fact that it must satisfy P A..,B R.
(b) Divine the bound function j - i - 1 , using the fact that i starts out small, j starts
out large, and termination occurs when j = i + 1 .
(c) Strengthen the loop invariant to include i < j.
(d) Develop the loop body -in the calculational way- to decrease the bound function
and maintain invariant P, thus giving us the following algorithm:
i,j:= O,n -1;
{invariant P : 0 :::; i < j :::; n - 1 A b[i]:::; x < bfj]}
{bound function: j - i-I}
do i + 1 '" j -t var e := (i + j) ..;. 2;
{i < e < j}
if b[e] :::; x -t i := e
o x < b[e] -t j := e
fi
od