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Temporal Structures

Published online by Cambridge University Press:  04 March 2009

Ross Casley ,
Roger F. Crew ,
José Meseguer  and
Vaughan Pratt
Ross Casley
Affiliation:
Dept. of Computer Science, Stanford University, Stanford, CA 94305
Roger F. Crew
Affiliation:
Dept. of Computer Science, Stanford University, Stanford, CA 94305
José Meseguer
Affiliation:
SRI International, Menlo Park, CA 94025 and Center for the Study of Language and Information, Stanford University, Stanford, CA 94305
Vaughan Pratt
Affiliation:
Dept. of Computer Science, Stanford University, Stanford, CA 94305
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Abstract

We combine the principles of the Floyd-Warshall-Kleene algorithm, enriched categories, and Birkhoff arithmetic, to yield a useful class of algebras of transitive vertex-labeled spaces. The motivating application is a uniform theory of abstract or parametrized time in which to any given notion of time there corresponds an algebra of concurrent behaviors and their operations, always the same operations but interpreted automatically and appropriately for that notion of time. An interesting side application is a language for succinctly naming a wide range of datatypes.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 1 , Issue 2 , July 1991 , pp. 179 - 213
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

Aho, A. V., Hopcroft, J. E. and Ullman, J.D. (1974) The Design and Analysis of Computer Algorithms, Addison-Welsely.Google Scholar
Arbib, M. and Manes, E. (1975). Arrows, Structures, and Functors: The Categorical Imperative, Academic Press.Google Scholar
Betti, R., Carboni, A., Street, R. and Walters, R. (1983) Variation through enrichment. J. Pure and Applied Algebra, 29:109–127.CrossRefGoogle Scholar
Birkhoff, G. (1937) An extended arithmetic. Duke Mathematical Journal, 3(2).CrossRefGoogle Scholar
Birkhoff, G. (1942) Generalized arithmetic. Duke Mathematical Journal, 9(2).CrossRefGoogle Scholar
Casley, R.T., Crew, R.F., Meseguer, J. and Pratt, V.R. (1989) Temporal structures. In Proc. Conf. on Category Theory and Computer Science, LNCS 389, Manchester. Springer-Verlag. Revised version to appear in Math. Structure in Comp. Sci. 1:1.Google Scholar
Conway, J.H. (1971) Regular Algebra and Finite Machines. Chapman and Hall, London.Google Scholar
Eilenberg, Samuel and Max Kelly, G. (1966) Closed categories. In Eilenberg, S., Harrison, D.K., MacLane, S. and Röhrl, H., editors, Proceedings of the Conference on Categorical Algebra, La Jolla, 1965, pages 421–562. Springer-Verlag.CrossRefGoogle Scholar
Floyd, R.W. (1962) Algorithm 97: shortest path. Communications of the ACM, 5(6):345.CrossRefGoogle Scholar
Gailfman, H. (1989) Modeling concurrency by partial orders and nonlinear transition systems. In Proc. REX School/Workshop on Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, pages 467–488, Noordwijkerhout, The Netherlands. Springer-Verlag.CrossRefGoogle Scholar
Girard, J.-Y. (1987) Linear logic. Theoretical Computer Science, 50:1–102.CrossRefGoogle Scholar
Gaifman, H. and Pratt, V.R. (1987) Partial order models of concurrency and the computation of functions. In Proc. 2nd Annual IEEE Symp. on Logic in Computer Science, pages 72–85, Ithaca, NY.Google Scholar
Kelly, G.M. (1982) Basic Concepts of Enriched Category Theory: London Math. Soc. Lecture Notes. 64. Cambridge University Press.Google Scholar
Kozen, D. (1980) A representation theorem for models of *-free PDL. In Proc. 7th Colloq. on Automata, Languages, and Programming, pages 351–362.CrossRefGoogle Scholar
Kozen, D. (1981) On induction vs. *-continuity. In Kozen, D., editor, Proc. Workshop on Logics of Programs 1981, LNCS 131, pages 167–176. Spring-Verlag.Google Scholar
Lawvere, W. (1973) Metric spaces, generalized logic, and closed categories. In Rendiconti del Seminario Matematica e Fisico di Milano, XLIII. Tipografia Fusi, Pavia.Google Scholar
Lewis, H. (1990) A logic of concrete time intervals. In Proc. 5th Annual IEEE Symp. on Logic in Computer Science, Philadelphia.Google Scholar
Mac Lane, S. (1971) Categories for the Working Mathematician. Springer-Verlag.CrossRefGoogle Scholar
Pratt, V.R. (1984) The pomset model of parallel processes: Unifying the temporal and the spatial. In Proc. CMU/SERC Workshop on Analysis of Concurrency, LNCS 197, pages 180–196, Pittsburgh. Springer-Verlag.Google Scholar
Pratt, V.R. (1986) Modeling concurrency with partial order. Int. J. of Parallel Programming, 15(1):33–71.CrossRefGoogle Scholar
Pratt, V.R. (1989) Enriched categories and the Floyd-Warshall connection. In Proc. First International Conference on Algebraic Methodology and Software Technology, Pages 177–180, lowa City.Google Scholar
Pratt, V.R. (1991) Event spaces and their linear logic. Submitted to a technical conference. Available by anonymous FTP from Boole Stanford.EDU as /pub/catl.tex. Stanford University.Google Scholar
Pratt, V.R. (1991a) Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages.CrossRefGoogle Scholar
Robert, P. and Ferland, J. (1968) Généralisation de l'algorithme de Warshall. Revue francaise d'Informatique et de Recherche Operationnelle, 2(7):13–25.Google Scholar
Roy, B. (1959) Transitivité et connexité. Comptes Rendues Acad. Sci., 249:216–218.Google Scholar
Tarlecki, Andrzej (1985) Bits and pieces of the theory of institutions. In Category Theory and Computer Programming, Lecture Notes in Computer Science 240: pages 334–363, Guildford, UK. Springer-Verlag.Google Scholar
Vickers, S. (1989) Topology via Logic. Cambridge University Press.Google Scholar
Warshall, S. (1962) A theorem on Boolean matrices. Journal of the ACM, 9(1):11–12.CrossRefGoogle Scholar