Nondeterminism in Algebraic Specifications and Algebraic Programs

259 Pages·1993·4.38 MB·English

by  Heinrich Hussmann (auth.)

Progress in Theoretical Computer Science
Editor
Ronald V. Book, University of California
Editorial Board
Erwin Engeler, ETH Zentrum, Zurich, Switzerland
Gerard Huet, INRIA, Le Chesnay, France
Jean-Pierre Jouannaud, Universite de Paris-Sud, Orsay, France
Robin Milner, University of Edinburgh, Edinburgh, Scotland
Maurice Nivat, Universite de Paris VII, Paris, France
Martin Wirsing, UniversiHit Passau, Passau, Germany
Heinrich Hussmann
N ondeterminism in
Algebraic Specifications
and Algebraic Programs
Birkhauser
Boston' Basel, Berlin
Heinrich Hussmann
Technische Universitat Munchen
Institut fUr Informatik
D-8000 Munchen 2
Germany
Library of Congress Cataloging-in-Publication Data
Hussmann, Heinrich, 1959-
Nondeterminism in algebraic specification and algebraic programs I
Heinrich Hussmann.
p. cm. --(Progress in theoretical computer science)
Includes bibliographical references,
ISBN-13: 978-1-4684-6836-6 e-ISBN-13: 978-1-4684-6834-2
DOl: 10.1 007/978-1-4684-6834-2
1. Computer science--Mathematics. I. Title. II. Series.
QA76.9.M35H87 1993 93-9340
005. 13'1--dc20 CIP
Printed on acid-free paper.
© Birkhiiuser Boston 1993.
Softcover reprint of the hardcover I st edition 1993
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ISBN -13: 978-1-4684-6836-6
Typset copy prepared by the Author.
9 8 7 6 5 4 3 2 1
Table of Contents
Chapter 0: Introduction ...................................................................... .1
0.1 Preview ........................................................................... .3
0.2 Historical Background ..........................................................9
0.3 Basic Notions .................................................................. 10
Chapter 1: Nondeterministic Algebraic Specifications ............................. 17
1.1 Nondeterministic Algebras .................................................. 17
1.1.1 A Discussion of Alternative Approaches ...................... 18
1.1.2 The Principle of Extensionality ................................. 22
1.1.3 The Notion of an Algebra ......................................... 24
1.2 Inclusion Rules as a Specification Language .......................... 28
1.2.1 Axioms and their Semantics ...................................... 28
1.2.2 The Calculus of Term Rewriting ................................ 31
1.2.3 Soundness: A Negative Result.. ................................. 34
1.2.4 Right-Linearity: A Special Case ................................ 38
Chapter 2: Specifications with a Deterministic Basis .............................. 43
2.1 Deterministic Basis ........................................................... 44
2.1.1 Soundness and Deterministic Basis ............................ .45
2.1.2 Determinacy Predicate .............................................. 46
2.1.3 Completeness: A Negative Result ............................. .49
2.2 Additive Specifications ...................................................... 53
2.2.1 DET-Completeness and DET-Additivity ...................... 54
2.2.2 Term Models and Completeness ................................. 57
2.3 Junk-Free Models ............................................................. 60
2.3.1 "Junk" in Nondeterministic Models ............................ 60
2.3.2 Breadth Induction .................................................... 63
2.3.4 DET- Generated Models ............................................. 66
2.3.5 Term-Generated Models ............................................ 67
2.4 Hierarchical Specifications .................................................. 71
Chapter 3: Structure of the Model Classes ............................................ 75
3.1 Homomorphisms and Extremal Algebras ............................... 76
3.2 Initial Models .................................................................. 80
3.3 Initial Models with Deterministic Basis ................................ 83
Chapter 4: Nondeterministic Specifications as a General Framework .......... 89
4.1 Equational Logic .............................................................. 89
4.2 Term Rewriting ................................................................ 91
vi CONTENTS
4.3 Conditional Axioms .......................................................... 93
4.4 Algebraic Programming ..................................................... 99
4.4.1 Constructor-Based Specifications ................................ 99
4.4.2. Narrowing without Confluence ................................ 107
4.5 Logic Programming ........................................................ 113
4.5.1. Narrowing Simulates Logic Programming ................. 114
4.5.2. Logic Programming Simulates Narrowing ................. 119
Chapter 5: Implementation and Examples ........................................... 125
5.1 Term Rewriting .............................................................. 125
5.1.1 Innermost Rewriting .............................................. 126
5.1.2 Search Strategies ................................................... 131
5.1.3 Optimizations ...................................................... 132
5.2 Graph Rewriting ............................................................. 136
5.2.1 Representation of Terms by Graphs .......................... 137
5.2.2 Rewriting of Term Graphs ...................................... 139
5.2.3 Soundness and Completeness .................................. 142
5.3 Examples ...................................................................... 147
5.3.1 Nondeterministic Finite State Automata .................... 147
5.3.2 Petri Nets ............................................................ 150
5.3.3 The Eight Queens Problem ..................................... 152
5.3.4 The Monkey-Banana Problem .................................. 154
5.3.5 Printer Scheduling ................................................. 156
Chapter 6: Partial Nondeterministic Specifications ............................... 161
6.1 Partial Operations ........................................................... 161
6.1.1 Undefined "Values" ................................................ 161
6.1.2 Partial Multi-Algebras ........................................... 164
6.2 Partiality and Term Rewriting ........................................... 167
6.2.1 A Calculus for Partial Specifications ........................ 168
6.2.2 Partial DET-Completeness and DET-Additivity ........... 170
6.3 Partial Specifications with Constructor Basis ....................... 172
6.4 Structure of the Model Classes .......................................... 176
6.4.1 Homomorphisms .................................................. 176
6.4.2 Initial Algebras ..................................................... 178
6.4.3 Terminal Algebras ................................................. 183
CONTENTS vii
Chapter 7: Communicating Processes: An Example ............................. 187
7.l. Communicating Processes (CP) ........................................ 187
7.2. Semantics of CP ............................................................ 189
7.2.l. Transition Semantics ............................................. 190
7.2.2. Trace Semantics .................................................... 195
7.2.3. Refusal Semantics ................................................. 197
7.3. Improvements and Applications ......................................... 200
Chapter 8: Concluding Remarks ....................................................... 201
8.1. Summary and Evaluation ................................................. 201
8.2. Future Work .................................................................. 202
References ................................................................................... 205
Appendix A: Proofs .......................................................................... 215
Appendix B: Experiments with RAP .................................................... 243
Preface
Algebraic specification, nondeterminism and term rewriting are three active
research areas aiming at concepts for the abstract description of software
systems: Algebraic specifications are well-suited for describing data structures
and sequential software systems in an abstract way. Term rewriting methods are
used in many prototyping systems and form the basis for executing specifi
cations. Nondeterminism plays a major role in formal language theory; in
programming it serves for delaying design decisions in program development and
occurs in a "natural" way in formalisations of distributed processes.
Heinrich Hussmann presents an elegant extension of equational specification and
term rewriting to include nondeterminism. Based on a clean modeltheoretic
semantics he considers term rewriting systems without confluence restrictions as
a specification language and shows that fundamental properties such as the
existence of initial models or the soundness and completeness of narrowing, the
basic mechanism for executing equational specifications, can be extended to
nondeterministic computations.
The work of Heinrich Hussmann is an excellent contribution to Algebraic
Programming; it gives a framework that admits a direct approach to program
verification, is suitable for describing concurrent and distributed processes, and it
can be executed as fast as Prolog.
Munich, January 1993 Martin Wirsing
Preface by the Author
This monograph is based on a Ph. D. thesis with the title "Nichtdetermini
stische Algebraische Spezifikationen" (in German), which was accepted by the
University of Passau in the winter term 1988/1989. The text has been
thoroughly revised and substantially extended in the meantime. The original
thesis was primarily aimed at the "core" theme of generalizing algebraic
specifications to nondeterminism. The monograph version now contains a
comparison with the established field of logic programming, a new chapter on
narrowing for nondeterministic specifications and a more detailed treatment of
graph rewriting techniques. I hope that the revision will extend the readership of
this book from specialists in algebraic specification to everybody who is
interested in the relationship between logic programming, term rewriting and
formal specification.
I would like to thank Prof. Broy for finding the topic of the thesis and for his
manyfold support, in particular for many fruitful discussions and for reading a
preliminary version. Many thanks also to Prof. Wirsing, for reading a draft, for
valuable remarks and for proposing the pUblication within this series of books.
The anonymous referee has given very helpful suggestions for the revision of
the manuscript, many thanks also to him or her.
Many collegues have contributed to this work by discussions and remarks; for
their particular interest I would like to thank Thomas Belzner, Alfons Geser,
Jiirgen Knopp, Bernhard Moller, Peter Mosses, Tobias Nipkow, Gert Smolka
and Michal Walicki.
Gabi Kohlmaier, now Gabi Hussmann, is entitled to special warm thanks for
moral support.
Finally, I would like to thank the staff of Birkhauser for their excellent support
in the technical production of the book.
Munich, December 22, 1992 Heinrich Hussmann
Chapter 0
Introduction
This monograph presents a generalization of the theory of equational algebraic
specifications, where the equational axioms are replaced by directed rewrite rules.
A model-theoretic semantics for such specifications is given, which provides a
rather general framework for studying
• the integration of nondeterminism into algebraic specifications, and
• model-oriented semantics for general (non-confluent) term rewriting.
The study of this central topic leads to interesting side results in the fields of
• relationships between algebraic and logic programming, and
• relationships between term rewriting and graph rewriting.
The starting point for this work is the observation that the available formal
specification languages for software are very much influenced by the concepts of
traditional mathematical logic. In particular, the notion of equality (which is a
symmetric operation) plays a central role in algebraic specifications. This
emphasis on symmetry does not correspond well to the fact that software
belongs to a computational paradigm, which is always directed. Every execution
of an algorithm consists in a directed evaluation of its formal descripton
(therefore leading to such problems as the question of termination). This kind of
directed evaluation transforms syntactical objects into semantically equal ones.
Classical (deterministic) evaluation gives a close connection between a non
symmetric relation between objects (the operational evaluation) and a symmetric
one (the semantical denotation). The theory of term rewriting is an ideal
framework for studying such connections.