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  • Mathematical Structures in Computer Science, Volume 4, Issue 3
  • September 1994, pp. 363-392

An Oxford survey of order sorted algebra

  • Joseph Goguen (a1) and Răzvan Diaconescu (a1)
  • DOI:
  • Published online: 01 March 2009

This paper surveys several different variants of order sorted algebra (abbreviated OSA), comparing some of the main approaches (overloaded OSA, universe OSA, unified algebra, term declaration algebra, etc.), emphasising motivation and intuitions, and pointing out features that distinguish the original ‘overloaded’ OSA approach from some later developments. These features include sort constraints and retracts; the latter is particularly useful for handling multiple data representations (including automatic coercions among them). Many examples are given, for most of which, runs are shown on the OBJ3 system.

This paper also significantly generalises overloaded OSA by dropping the regularity and monotonicity assumptions, and by adding signatures of non-monotonicities, which support simple semantics for some aspects of object oriented programming. A number of new results for this generalisation are proved, including initiality, variety, and quasi-variety theorems. Axiomatisability results à la Birkhoff are also proved for unified algebras.

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J. Goguen and R. Burstall (1985) Institutions: Abstract Model Theory for Specification and Programming. Journal of the Association for Computing Machinery 39 (1) 95146. (Draft appears as Report ECS-LFCS-90–106, Computer Science Department, University of Edinburgh, January 1990. An early ancestor is ‘Introducing Institutions’ in: Clarke, E. and Kozen, D. (eds.) (1984) Proceedings, Logics of Programming Workshop, Springer-Verlag Lecture Notes in Computer Science 164 221–256.)

J. Goguen , J.-P. Jouannaud and J. Meseguer (1985) Operational Semantics of Order-Sorted Algebra. In: W. Brauer (ed.) Proceedings, 1985 International Conference on Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 194.

J. Goguen and J. Meseguer (1982) Universal Realization, Persistent Interconnection and Implementation of Abstract Modules. In: M. Nielsen and E. M. Schmidt (eds.) Proceedings, 9th International Conference on Automata, Languages and Programming, Aarhus, Denmark. Springer-Verlag Lecture Notes in Computer Science 140 265281.

J. Goguen and J. Meseguer (1987a) Models and Equality for Logical Programming. In: H. Ehrig , G. Levi , R. Kowalski and U. Montanari (eds.) Proceedings, 1987 TAPSOFT, Pisa, Italy. Springer- Verlag Lecture Notes in Computer Science 250 122. (Also, Report CSLI-87–91 (1987) Center for the Study of Language and Information, Stanford University; reprinted in Mathematical Logic in Programming, Mir (Moscow) (1991) 274–310 (in Russian).)

J. Goguen and J. Meseguer (1992) Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations. Theoretical Computer Science 105 (2) 217273. (Also: Programming Research Group Technical Monograph PRG-80 (1989) Oxford University, and Technical Report SRI-CSL-89–10 (1989) SRI International, Computer Science Lab; originally given as lecture at Seminar on Types, Carnegie-Mellon University, June 1983; many draft versions exist, from as early as 1985.)

J. Goguen , A. Stevens , K. Hobley and H. Hilberdink (1992) 2OBJ, a Metalogical Framework based on Equational Logic. Philosophical Transactions of the Royal Society Series A 339 6986. (Also in Hoare, C. A. R. and Gordon, M. J. C. (eds.) (1992) Mechanized Reasoning and Hardware Design, Prentice-Hall, 69–86.)

J. Goguen , J. Thatcher , E. Wagner and J. Wright (1977) Initial Algebra Semantics and Continuous Algebras. Journal of the Association for Computing Machinery 24 (1) 6895. An early version is ‘Initial Algebra Semantics’, Technical Report RC 4865 (1974), IBM T. J. Watson Research Center.)

P. J. Higgins (1963) Algebras with a Scheme of Operators. Mathematische Nachrichten 27 115132.

F. W. Lawvere (1963) Functorial Semantics of Algebraic Theories. Proceedings, National Academy of Sciences, U.S.A. 50 869872. (Summary of Ph.D. Thesis, Columbia University.)

P. Mosses (1989a) Unified Algebras and Action Semantics. Proceedings, Symposium on Theoretical Aspects of Computer Science. Springer-Verlag Lecture Notes in Computer Science 349.

A. Poigné (1984) Another Look at Parametrization using Algerbaic Specifications with Subsorts. Proceedings, Mathematical Foundations of Computer Science ‘84. Springer-Verlag Lecture Notes in Computer Science 176.

U. Waldmann (1992) Semantics of order-sorted specifications. Theoretical Computer Science 94 (1) 135.

C. Walther (1985) A Mechanical Solution of Schubert's Steamroller by Many-sorted Resolution. Artificial Intelligence 26 (2) 217–214.

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