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IFIP - The International Federation for Information Processing
IFIP was founded in 1960 under the auspices of UNESCO, following the First World
Computer Congress held in Paris the previous year. An umbrella organization for societies
working in information processing, IFIP's aim is two-fold: to support information processing
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encourages and assists in the development, exploitation and application of information
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through a number of technical committees, which organize events and publications. IFIP's
events range from an international congress to local .seminars, but the most important are:
• the IFIP World Computer Congress, held every second year;
• open conferences;
• working conferences.
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Algorithmic
Languages
and Calculi
IFIP TC2 WG2.1 International Workshop
on Algorithmic Languages and Calculi
17-22 February 1997, Alsace, France
Edited by
Richard S. Bird
Computing Laboratory
Oxford University
Oxford
UK
and
Lambert Meertens
CWI
Amsterdam
The Netherlands
/UII/
SPRINGER INTERNATIONAL PUBLISHING, CHAM
First edition 1997
© 1997 IFIP International Federation for Information Processing
Originally published by Chapman & Hall in 1997
Softcover reprint of the hardcover 1st edition 1997 978-0-41282050-2
ISBN 978-1-5041-2877-3 ISBN 978-0-387-35264-0 (eBook)
DOI 10.1007/978-0-387-35264-0
Apart from any fair dealing for the purposes of research or private study. or criticism or
review. as permined under the UK Copyright Designs and Patents Act. 1988. this publication
may not be reproduced. stored. or transmitted, in any form or by any means. without the prior
permission in writing of the publishers. or in the case of reprographic reproduction only in
accordance with the terms of the licences issued by the Copyright Licensing Agency in the
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Rights Organization outside the UK. Enquiries concerning reproduction outside the terms
stated here should be sent to the publishers at the London address printed on this page.
The publisher makes no representation. express or implied. with regard to the accuracy of
the information contained in this book and cannot accept any legal responsibility or liability
for any errors or omissions that may be made.
A catalogue record for this book is available from the British Library
@ Printed on permanent acid-free text paper. manufactured in accordance with
ANSIINISO Z39.48-1992 and ANSIINISO Z39.48-1984 (Permanence of Paper).
CONTENTS
Preface
VII
Programme Committee viii
1 Dynamic programming: a different perspective
S. Curtis 1
2 Calculating with pointer structures
B. Moller 24
3 Deriving a systolic regular language recognizer
M. Vaccari and R. Backhouse 49
Discussion 73
4 A calculational fusion system HYLO
Y. Onoue, Z. Hu, H. Iwasaki and M. Takeichi 76
5 Refinement of data parallel programs in PEl
E. Violard, S. Genaud and G.-R. Perrin 107
6 Solving large systems of differential equations in parallel
using covers and skeletons
M. Siidholt, C. Piepenbrock, K. Obermayer and P. Pepper 132
Discussion 160
7 Refinement of parameterized algebraic specifications
Y. V. Srinivas 164
8 Towards a compositional interpretation of object diagrams
J.C. Bicarregui, K.C. Lano and T.S.E. Maibaum 187
9 Defining and detecting feature interactions
M. Frappier; A. Mili and J. Desharnais 212
10 Synthesising interconnections
J.L. Fiadeiro, A. Lopes and T.S.E. Maibaum 240
II A powerful transformation rule, its applications and variants
K. Achatz and H. Partsch 265
12 Program derivation via list introduction
A. Pettorossi and M. Proietti 296
13 Towards a calculus for abstract syntax trees
D.S. Wile 324
Discussion 354
14 Principled strength reduction
Y.A. Liu 357
vi Conte11ts
15 The fonna1 reconstruction and speedup of the linear time fragment
of Willard's relational calculus subset
D. Goyal and R. Paige 382
16 Predicative semantics of loops
T.S. Norvell 415
Discussion 438
Index of contributors 441
Keyword index 443
PREFACE
This book contains the papers presented at the IFIP Working Conference
on Algorithmic Languages and Calculi, held in Bisschoffsheim, Alsace, on
February 17-19, 1997. The conference was sponsored by IFIP Technical Com
mittee 2 and organised in cooperation with Working Group 2.1 on Algorithmic
Languages and Calculi, whose 50th meeting was held immediately afterwards.
For a number of years the inspiration ofWG 2.1 has been to seek to develop
a theory of algorithmics, comprising notations and calculi appropriate for rig
orous yet elegant methods of program development. The following papers
reflect this inspiration in a variety of ways. Together, they provide a state-of
the-art overview of ongoing research in the field, as well as its application to
the production of practical software. There are derivations both of individual
programs and classes of algorithms, a number of which are conducted in a
point-free style using equational and inequational reasoning in a relational
calculus. There are presentations of general program-transformation schemes,
descriptions of implemented systems for carrying out transformations auto
matically (or semi-implemented systems for semi-automatic transformations),
and explorations of the use of categorical concepts in the specification and de
sign of general systems. Overall, the papers attest to continuing progress in
our ability to improve conceptual clarity, to increase notational simplicity, and
to tackle progressively harder problems. All this is made possible by striving
for mathematical abstraction and unification.
The papers are recorded in the order they were presented at the conference,
and are interleaved with edited transcripts of four short discussion sessions.
The editors are grateful to the members of the Program Committee and
the reviewers for their careful work, and to the Organising Committee for
providing a stimulating environment for the conference. They would also like
to thank Doug Smith, chairman of WG 2.1, for providing much useful infor
mation, and Aileen Parlane of Chapman & Hall for general support in the
preparation of this book.
Amsterdam, April 1997 Richard Bird
Lambert Meertens
PROGRAMME COMMITTEE
D.R. Smith, Palo Alto, USA (Chair)
J-P. Finance, Nancy, France (Co-Chair)
R. Back, Turku, Finland
R. Backhouse, Eindhoven, The Netherlands
T. Coquand, Goteborg, Sweden
H. Ehrig, Berlin, Germany
M.S. Feather, Pasadena, USA
J. Gibbons, Auckland, New Zealand
D. Gries, Ithaca, USA
A.M. Haeberer, Rio de Janeiro, Brazil
R. Hebner, Toronto, Canada
L.G.L.T. Meertens, Amsterdam, The Netherlands
B. Moller, Augsburg, Germany
C. Morgan, Oxford, UK
F. Orejas, Barcelona, Spain
R. Paige, New York, USA
H. Partsch, Ulm, Germany
P. Pepper, Berlin, Germany
D. Sannella, Edinburgh, UK
M. Sintzoff, Louvain-la-Neuve, Belgium
W. Turski, Warsaw, Poland
1
Dynamic programming: a
different perspective
Sharon Curtis
Oxford University Computing Laboratory
Parks Road, Oxford, UK. OX1 3QD
sharon~comlab.ox.ac.uk
Abstract
Dynamic programming has long been used as an algorithm design technique, with
various mathematical theories proposed to model it. Here we take a different per
spective, using a relational calculus to model both. problems and their solutions
obtained from dynamic programming. This approach serves to shed new light on
the different styles of dynamic programming, representing them by different search
strategies of the tree-like space of partial solutions.
Keywords
dynamic programming, optimization problems, relational calculus
1 INTRODUCTION AND HISTORY
Dynamic programming is an algorithm design technique for solving many differ
ent types of optimization problem, applicable to such diverse fields as operations
research (Ecker and Kupferschmid, 1988) and ileutron transport theory (Bellman,
Kagiwada and Kalaba, 1967).
The mathematical theory of the subject dates back to 1957, when Richard Bell
man (Bellman, 1957) first popularized the idea, producing a mathematical theory
to model multi-stage decision processes and to solve related optimization problems.
He was also the first to introduce the idea of the Principle of Optimality:
"An optimal policy has the property that whatever the initial state and initial
decision are, the remaining decisions must constitute an optimal policy with regard to
the state resulting from the first decision."
This he presented as the essential condition that such an optimization problem had
to satisfy for the dynamic programming technique to be applicable. (Morin (1982)
subsequently showed that this is a sufficient but not necessary condition.)
The early standard formalizations of dynamic programming as applied to com
binatorial optimization problems concerned discrete decision processes, which are
a set of strings of decisions (or policies) and a cost function on policies. These
were modelled as automata by Shreider (1961), and further extended to sequen
tial decision processes (discrete decision processes with an added cost structure
imposed) by Karp, Held and others (Karp and Held, 1962 and 1967; Elmaghraby,
1970; lbaraki, 1973). These structures model the problem itself, then the solution
is obtained by checking that the principle of optimality applies, and producing the
relevant functional equation, a recursion equation giving a recipe for the actual
computer program. The method of solution was either via tabulation of' the values
of the function concerned, or by direct application of the functional equation, using
the technique of memoization to avoid computing the same subproblem twice.
©lAP 1997. Published by Chapman & Hall
