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EATCS Monographs on Theoretical Computer Science Volume 6 Editors: W. Brauer G. Rozenberg A. Salomaa Advisory Board: G. Ausiello S. Even M. Nivat Chr. Papadimitriou A. L. Rosenberg D. Scott H. Ehrig B. Mahr Fundamentals of Algebraic Specification 1 Equations and Initial Semantics Springer-Verlag Berlin Heidelberg New York Tokyo Authors Professor Dr. Hartmut Ehrig Priv. Doz. Dr. Bernd Mahr Technische UniversiHit Berlin FB 20, Informatik, Sekr. FR 6-1 Franklinstrasse 28/29, 1000 Berlin 10, Germany Editors Professor Dr. Wilfried Brauer FB Informatik der Universitat Hamburg Rothenbaum-Chaussee 67-69, 2000 Hamburg 13, Germany Professor Dr. Grzegorz Rozenberg Institut of Applied Mathematics and Computer Science University of Leiden, Wassenaarseweg 80, P.O. Box 9512 2300 RA Leiden, The Netherlands Professor Dr. Arto Salomaa Department of Mathematics, University of Turku 20500 Turku 50, Finland ISBN-13: 978-3-642-69964-1 e-ISBN-13: 978-3-642-69962-7 DOl: 10.1007/978-3-642-69962-7 This work is subject to copyright. All rights are reserved, whether the whole or part of material is concerned, specifically those of translation, reprinting, re·use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin Heidelberg 1985 Softcover reprint of the hardcover 1st edition 1985 The use of registered names, trademarks, etc. in the publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2145/3140-543210 Preface Since the early seventies concepts of specification became central in the whole area of computer science. Especially algebraic specification techniques for abstract data types and software systems have gained considerable importance in recent years, and are now playing a central role in the theory of data type specification. The aim of this book is to present fundamentals of algebraic specifica tion for teaching, research, and development with respect to the following three aspects: 1. Fundamentals in the sense of a carefully motivated introduction to algebraic specifications, which is easy to understand for computer scientists and mathematicians. 2. Fundamentals in the sense of mathematical theories which are the basis for precise definitions, constructions, results, and correctness proofs. 3. Fundamentals in the sense of concepts from computer science, which are introduced on a conceptual level and formalized in mathematical terms. We have organized the material in two volumes: Volume 1: Equations and Initial Semantics Volume 2: Formal Requirements and Modules Volume 1 includes equational specifications, algebras, specifications of abstract data types with initial semantics, characterization of equational classes, the equational calculus, term rewriting, correct ness and extension of specifications, parameterized specifications, functors, parameter passing, and in an appendix concepts, syntax and semantics of an algebraic specification language. Volume 2 will include first order logical-especially Universal Horn specifications, institutions, behavioral abstraction, algebraic con- VI Preface straints, and specification of modules and modular systems. These two volumes are concerning formal specification techniques from logical requirements to algebraic design specifications. It is planned that a third volume on algebraic specifications is published in this EATCS-monograph series by another author, where the connec tions between specifications, interpretations, implementations, and programs in programming languages are studied. Volume 1 is organized in a main part of eight chapters and an appendix of two chapters. The chapters are numbered using arabic numerals. Each chapter consists of several sections which are not numbered because concepts, definitions, facts, theorems and exam ples within one chapter are numbered consecutively. Each of the chapters of the main part and the appendix contains a separate section with bibliographical notes where also further extensions of the theory are mentioned. Chapters 1 and 2 are intended to serve as an introduction to algebraic specifications with equational axioms in general and to the specifica tion of abstract data types with initial semantics in particular. A more concrete level of presentation is intentionally chosen to avoid abstract notions and universal constructions in the beginning. The correspond ing abstract concepts "initiality" and "freeness" are introduced and studied in Chapter 3 and they are frequently used in the remaining chapters. Chapters 4 and 5 are not prerequisite for the following ones but they can be considered as interfaces of equational algebraic specifications to universal algebra and logic (including term rewriting) respectively. Chapter 6 continues a discussion on correctness proofs in Chapter 2. Though it uses results from Chapters 3 and 5, main parts can be read, immediately after 2. Chapters 7 and 8 are introducing and using concepts from category theory in order to state the semantical aspects and to prove the results of parameterized specifications, and parameter passing properly. The corresponding syntactical aspects are given in the first section of Chapter 7 and the first two sections of Chapter 8. These sections together with Chapters 1 and 2 are sufficient to understand main parts of the appendix. The first two chapters may serve as a textbook for undergraduate teaching, while the other ones are providing more advanced and abstract material for graduate level courses and further research and development. A partial overlap in contents between chapters present ing the theory on different levels of abstraction is intended to improve reading of later chapters without too much prerequisites of earlier ones. This volume is based on lectures of both authors at the computer science department of the Technical University in Berlin. It also presents main results of the DFG project ACT (Algebraic Specifica tion Techniques for Correct and Trusty Software Systems) and of common research projects with J. Thatcher, E. Wagner and J. Wright, IBM Yorktown Heights, and especially with H.-J. Kreowski now at the University of Bremen. Preface VII We are most grateful to these and several other colleagues for common research and stimulating discussions on the subject of this book. We are also indepted to G. Rozenberg because of his very engaged editorial guidance. The special attention given to the referee ing process for publications in the EAT CS-monograph series has definitely shown its merits. In this regard we want to express our gratitude to the referee for carefully studying our handwritten draft version and for most valuable comments leading to a considerably improved final version. The manuscript was excellently typed in world record time by H. Barnewitz. She did a fantastic job. Proof reading was done by P. Boehm, W. Fey, U. Grude, H. Hansen, K. P. Hasler, U. Hummert, R. Kutsche, A. Langen, F. Niirnberg, and K. J. Werner. As members resp. cooperators of the ACT group they are also involved in teaching, research, and development of algebraic specification techni ques and languages in Berlin. Many thanks to all of them and also to our families for great patience during the preparation of the book. Berlin, March 1985 Hartmut Ehrig Bernd Mahr Contents Introduction 1 Historical Remarks 1 The Chapters of this Volume 4 Further Topics 8 1 Equational Specifications and Algebras . 9 Basic Examples . . . . . . . . . . . . . . 10 Signatures, Terms, and Algebras ..... 14 Equational Specifications and Derivations 24 Bibliographic Notes for Chapter 1 31 2 Specifications of Abstract Data Types . 32 Abstract Data Types 33 Quotient Term Algebras 36 Semantics and Correctness of Specifications 46 Bibliographic Notes for Chapter 2 . 61 3 Initial Semantics of Specifications . 62 Facts about Homomorphisms . . . . 63 Term Algebras, Initial and Free Algebras 67 Congruences and Quotients . . . . 76 Initial and Free SPEC-Algebras .. 81 Bibliographic Notes for Chapter 3 88 X Contents 4 Specifiability and Characterization of Equational Classes 89 Equational Theory and Equational Classes 90 Subalgebras . . . . . . 92 Products ....... 95 Homomorphic Images . 97 Birkhoff-Characterizations of Equational Classes 101 Bibliographic Notes for Chapter 4 ....... 107 5 Equational Calculus and Term Rewriting 108 Formal Proofs by Equations . . . . . . . . 109 Correctness and Completeness of the Equational Calculus 115 Term Rewriting with Equations . . . . . . . . . . . . . . 124 Equivalence of Proofs and Term Rewriting with Equations 130 Bibliographic Notes for Chapter 5 ............ 137 6 Correctness and Extension of Specifications 138 Initial Correctness of Specifications . . . . . . 139 Correctness of Specifications with Hidden Functions 145 Stepwise Specification by Extensions . 151 Correctness of Example Specifications 158 Bibliographic Notes for Chapter 6 167 7 Parameterized Specifications and Functors 168 Parameterized Specifications . . . . 171 Categories and Functors . . . . . . . 176 Free Constructions and Free Functors 181 Semantics and Correctness of Parameterized Specifications 191 Bibliographic Notes for Chapter 7 ............ 199 8 Parameter Passing .. 201 Specification Morphisms 202 Standard Parameter Passing and Pushouts . 206 Amalgamation .............. 215 Persistency and Extension of Functors . . . 220 Correctness of Standard Parameter Passing 226 Parameterized Parameter Passing . 231 Iterated Parameter Passing . . . . 238 Bibliographic Notes for Chapter 8 242 Contents XI Appendix . ............. . 244 9 Concepts of a Specification Language 245 Basic Specifications 246 Combination 250 Renaming . . . 256 Actualization . 258 Modularization 263 Syntax of the Specification Language ACT ONE 268 10 Semantics ofthe Specification Language ACT ONE 273 First Level of Semantics . . . . . . . . . . . . 275 Second Level of Semantics .......... 282 Semantical Properties and Context-Conditions 290 Initial Algebra Semantics . . . . . . 298 Bibliographic Notes for the Appendix 305 Bibliography 307 SubjectIndex 316 Introduction The theory of algebraic specifications has its origins in the mid seventies and evolved from different sources and work independently done by different authors. Conceptually it is based on notions and ideas of classical and universal algebra in pure mathematics, and on concepts of abstract data types and software system speci fication in computer science. The rapidly growing interest in algebraic specifications has led to a well-developed theory and a great variety of problem areas presently being explored by different groups of researchers. The theory of algebraic specifications, which grew out from research in program and software development methodology, has now come to a state which is that of a well-established field equally interesting for theoreticians and practitioners in computer science and software development. Historical Remarks The concept of abstract data types was developed about 15 years ago starting with the debacles of large software systems in the late 60's. Since that time a large number of papers on algebraic specification techniques have appeared studying speci fication problems from the theoretical and the applications point of view. Algebraic specification techniques and languages have been successfully applied in research and development for the specification of systems ranging from basic data types up to highly sophisticated software systems. The basic data types are including numeral types like natural numbers, integers, rationals, and boolean values and abstract data types like stacks, queues, strings, sets, arrays, trees, and graphs, for example. Basic data types have been combined to construct larger data types like symbol tables,