A complete calculus for the multialgebraic and functional semantics of nondeterminism

The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity and consequently equate (nondeterministic) terms that one would intuitively not consider equal. Furthermore, existing models for nondeterminism depart radically from the standard models for (equational) specifications of deterministic operators. One would prefer that a specification language for nondeterministic operators be based on an extension of the standard model concepts, preferably in such a way that the reasoning system for (possibly nondeterministic) operators becomes the standard equational one whenever restricted to the deterministic operators—the objective should be to minimize the departure from the standard frameworks. In this article we define a specification language for nondeterministic operators and multialgebraic semantics. The first complete reasoning system for such specifications is introduced. We also define a transformation of specifications of nondeterministic operators into derived specifications of deterministic ones, obtaining a “computational” semantics of nondeterministic specification by adopting the standard semantics of the derived specification as the semantics of the original one. This semantics turns out to be a refinement of multialgebra semantics. The calculus is shown to be sound and complete also with respect to the new semantics.

[1]  P. A. Subrahmanyam.  Nondeterminism in Abstract Data Types , 1981, ICALP.

[2]  R. Lyndon.  PROPERTIES PRESERVED UNDER HOMOMORPHISM , 1959 .

[3]  C. A. R. Hoare,et al.  An Axiomatic Definition of the Programming Language PASCAL , 1973, Acta Informatica.

[4]  D. Kapur.  TOWARDS A THEORY FOR ABSTRACT DATA TYPES , 1980 .

[5]  Matthew Hennessy,et al.  The Semantics of Nondeterminism , 1976, ICALP.

[6]  Hartmut Ehrig,et al.  Fundamentals of Algebraic Specification 2 , 1990, EATCS Monographs on Theoretical Computer Science.

[7]  Jack Minker,et al.  Semantics of Disjunctive Deductive Databases , 1992, ICDT.

[8]  Michael J. O'Donnell,et al.  Computing in systems described by equations , 1977, Lecture Notes in Computer Science.

[9]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[10]  Heinrich Hussmann,et al.  Nondeterminism in Algebraic Specifications and Algebraic Programs , 1993 .

[11]  Wim H. Hesselink,et al.  A mathematical approach to nondeterminism in data types , 1988, TOPL.

[12]  Martin Wirsing,et al.  Algebraic Specification , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[13]  Michael Löwe,et al.  Beyond Conditional Equations: Quasi-Initial Semantics for Parametric Algebraic Specifications , 1992, CAAP.

[14]  Michal Walicki,et al.  Singular and Plural Nondeterministic Parameters , 1997, SIAM J. Comput..

[15]  Glynn Winskel,et al.  An Introduction to Event Structures , 1989 .

[16]  Heinrich Hußmann.  Nondeterministic algebraic specifications , 1991 .

[17]  Joseph A. Goguen,et al.  What Is Unification?: A Categorical View of Substitution, Equation and Solution , 1989 .

[18]  C. A. R. Hoare,et al.  Communicating sequential processes , 1978, CACM.

[19]  Heinrich Hußmann,et al.  Nondeterministic Algebraic Specifications and Nonconfluent Term Rewriting , 1992, J. Log. Program..

[20]  Peter D. Mosses.  Unified Algebras and Action Semantics , 1989, STACS.

[21]  Jan A. Bergstra,et al.  Process theory based on bisimulation semantics , 1988, REX Workshop.

[22]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[23]  José Meseguer,et al.  Conditional Rewriting Logic: Deduction, Models and Concurrency , 1990, CTRS.

[24]  Donald Sannella,et al.  On Observational Equivalence and Algebraic Specification , 1985, TAPSOFT, Vol.1.

[25]  Horst Reichel,et al.  On Behavioural Equivalence of Data Types , 1983, J. Inf. Process. Cybern..

[26]  Peter D. Mosses,et al.  Unified Algebras and Institutions , 1989 .

[27]  B. TRUMPY,et al.  University of Bergen , 1948, Nature.

[28]  Stéphane Kaplan.  Rewriting with a Nondeterministic Choice Operator , 1988, Theor. Comput. Sci..

[29]  Edsger W. Dijkstra,et al.  A Discipline of Programming , 1976 .

[30]  Michal Walicki,et al.  Initiality + Nondeterminism ⇒ Junk , 1993 .

[31]  Michal Walicki,et al.  Multialgebras, Power Algebras and Complete Calculi of Identities and Inclusions , 1994, COMPASS/ADT.

[32]  José Meseguer,et al.  Conditioned Rewriting Logic as a United Model of Concurrency , 1992, Theor. Comput. Sci..

[33]  G. Naudé,et al.  Universal Realization , 1979, J. Comput. Syst. Sci..

[34]  Jean-Pierre Jouannaud,et al.  Conditional Term Rewriting Systems: 1st International Workshop Orsay, France, July 8-10, 1987. Proceedings , 1988 .

[35]  Hans-Jörg Kreowski,et al.  Recent Trends in Data Type Specification , 1985, Informatik-Fachberichte.

[36]  Sigurd Meldal.  Allocations of Objects Considered as Nondeterministic Expressions - Towards a More Abstract Axiomatics of Access Types , 1987 .

[37]  MeseguerJosé.  Conditional rewriting logic as a unified model of concurrency , 1992 .

[38]  Michal Walicki,et al.  Reasoning and Rewriting with Set-Relations I: Ground Completeness , 1994, CSL.

[39]  G. Huet,et al.  Equations and rewrite rules: a survey , 1980 .

[40]  Friedrich L. Bauer,et al.  The Munich Project CIP: Volume I: The Wide Spectrum Language CIP-L , 1985 .

[41]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[42]  H. Pickett,et al.  Homomorphisms and subalgebras of multialgebras. , 1967 .

[43]  Nachum Dershowitz,et al.  Conditional rewriting , 1985, SOEN.

[44]  Hartmut Ehrig,et al.  Fundamentals of Algebraic Specification 2: Module Specifications and Constraints , 1990 .

[45]  José Meseguer,et al.  Remarks on remarks on many-sorted equational logic , 1987, SIGP.

[46]  Tobias Nipkow.  Observing Non-Deterministic Data Types , 1987, ADT.

[47]  José Meseguer,et al.  Universal Realization, Persistent Interconnection and Implementation of Abstract Modules , 1982, ICALP.

[48]  Robin Milner,et al.  A Calculus of Communicating Systems , 1980, Lecture Notes in Computer Science.

[49]  Peter D. Mosses,et al.  Unified Algebras and Action Semantics , 1988, STACS.

[50]  H. Arbeláez,et al.  Korth cm. International business : environment and management. Prentice hall, inc, englewood cliffs, 1985, 2a ed , 1985 .

[51]  Allen Goldberg,et al.  Referential opacity in nondeterministic data refinement , 1993, LOPL.

[52]  Matthew Hennessy.  Observing processes , 1988, REX Workshop.

[53]  Hartmut Ehrig,et al.  Fundamentals of Algebraic Specification 1 , 1985, EATCS Monographs on Theoretical Computer Science.

[54]  Grzegorz Rozenberg,et al.  Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency , 1988, Lecture Notes in Computer Science.

[55]  Gilles Kahn,et al.  The Semantics of a Simple Language for Parallel Programming , 1974, IFIP Congress.

[56]  Jan A. Bergstra,et al.  Algebra of Communicating Processes , 1995, Workshops in Computing.

[57]  J. A. Goguen,et al.  Completeness of many-sorted equational logic , 1981, SIGP.

[58]  G. Birkhoff,et al.  On the Structure of Abstract Algebras , 1935 .

[59]  Sigurd Meldal.  An abstract axiomatization of pointer types , 1989, [1989] Proceedings of the Twenty-Second Annual Hawaii International Conference on System Sciences. Volume II: Software Track.

[60]  Donald Sannella,et al.  On Observational Equivalence and Algebraic Specification , 1987, J. Comput. Syst. Sci..