Nondeterminism in algebraic specifications and algebraic programs.

263 Pages·2012·15.2 MB·English

by  Heinrich, Hussmann

Progress in Theoretical Computer Science
Editor
Ronald V. Book, University of California
Editorial Board
Erwin Engeler, ΕΤΗ Zentrum, Zurich, Switzerland
G6rard Huet, INRIA, Le Chesnay, France
Jean-Pierre Jouannaud, Universit6 de Paris-Sud, Orsay, France
Robin Milner, University of Edinburgh, Edinburgh, Scotland
Maurice Nivat, Universite de Paris VII, Paris, France
Martin Wirsing, Universität Passau, Passau, Germany
Heinrich Hussmann
Nondeterminism in
Algebraic Specifications
and Algebraic Programs
Birkhäuser
Boston · Basel · Berlin
Heinrich Hussmann
Technische Universität München
Institut für Informatik
D-8000 München 2
Germany
Library of Congress Cataloging-in-Publication Data
Hussmann, Heinrich, 1959-
Nondeterminism in algebraic specification and algebraic programs /
Heinrich Hussmann.
p. cm. -- (Progress in theoretical computer science)
Includes bibliographical references,
ISBN 0-8176-3700-1 (alk. paper)
1. Computer science—Mathematics. I. Title. II. Series.
QA76.9.M35H87 1993 93-9340
005.13'l~dc20 CIP
Printed on acid-free paper.
© Birkhäuser Boston 1993.
Copyright is not claimed for works of U.S. Government employees.
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ISBN 0-8176-3700-1
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Printed and bound by Quinn-Woodbine, Woodbine, NJ.
Printed in the U.S.A.
9 8 7 6 5 4 3 21
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-Additi vity 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
7.1. Communicating Processes (CP) 187
7.2. Semantics of CP 189
7.2.1. 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 Beizner, Alfons Geser,
Jürgen Knopp, Bernhard Möller, 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 Birkhäuser for their excellent support
in the technical production of the book.
Munich, December 22, 1992 Heinrich Hussmann
Chapter Ο
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.