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μ-Bicomplete Categories and Parity Games

Published online by Cambridge University Press:  15 December 2002

Luigi Santocanale
Luigi Santocanale*
Affiliation:
LaBRI, Université Bordeaux 1, Domaine Universitaire, 351 cours de la Libération, 33405 Talence Cedex, France; [email protected].
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Abstract

For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by μ-terms. We call the category μ-bicomplete if every μ-term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved through parity games: we associate to each game an algebraic expression and turn the game into a term of a categorical theory. We show that μ-terms and parity games are equivalent, meaning that they define the same property of being μ-bicomplete. Finally, the interpretation of a parity game in the category of sets is shown to be the set of deterministic winning strategies for a chosen player.

Keywords

Parity gamesbicomplete categoriesinitial algebrasfinal coalgebrasinductive and coinductive types.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 36 , Issue 2: Fixed Points in Computer Science (FICS'01) , April 2002 , pp. 195 - 227
Copyright
© EDP Sciences, 2002

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References

P. Aczel, Non-well-founded sets. Stanford University Center for the Study of Language and Information, Stanford, CA (1988).
P. Aczel, J. Adámek and J. Velebil, A coalgebraic view of infinite trees and iteration, edited by M.L.A. Corradini and U. Montanari. Elsevier Science Publishers, Electron. Notes in Theoret. Comput. Sci. 44 (2001).
Adámek, J. and Koubek, V., Least fixed point of a functor. J. Comput. System Sci. 19 (1979) 163-178. CrossRef
J. Adámek and J. Rosický, Locally presentable and accessible categories. Cambridge University Press, Cambridge (1994).
A. Arnold and D. Niwinski, Rudiments of mu-calculus. Elsevier, North-Holland, Stud. Logic Found. Math. 146 (2001).
Barr, M., Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114 (1993) 299-315. CrossRef
S.L. Bloom and Z. Ésik, Iteration theories. Springer-Verlag, Berlin (1993). The equational logic of iterative processes.
Bloom, S.L., Ésik, Z., Labella, A. and Manes, E.G., Iteration 2-theories. Appl. Categ. Structures 9 (2001) 173-216. CrossRef
J.R.B. Cockett and D. Spencer, Strong categorical datatypes. I, in Category theory 1991 (Montreal, PQ, 1991). Providence, RI, Amer. Math. Soc. (1992) 141-169.
Cockett, J.R.B. and Spencer, D., Strong categorical datatypes. II. A term logic for categorical programming. Theoret. Comput. Sci. 139 (1995) 69-113. CrossRef
R. Cockett and T. Fukushima, About Charity, Yellow Series Report No. 92/480/18. Department of Computer Science, The University of Calgary (1992).
E.A. Emerson and C.S. Jutla, Tree automata, mu-calculus and determinacy (extended abstract), in 32nd Annual Symposium on Foundations of Computer Science. IEEE (1991) 368-377.
Emerson, E.A., Jutla, C.S. and Sistla, A.P., On model checking for the µ-calculus and its fragments. Theoret. Comput. Sci. 258 (2001) 491-522. CrossRef
Ésik, Z. and Labella, A., Equational properties of iteration in algebraically complete categories. Theoret. Comput. Sci. 195 (1998) 61-89. Mathematical foundations of computer science, Cracow (1996). CrossRef
P. Freyd, Algebraically complete categories, in Category theory (Como, 1990). Springer, Berlin (1991) 95-104.
E. Giménez, A tutorial on recursive types in Coq. Technical Report 0221, INRIA (1998).
J.M.E. Hyland, The effective topos, in The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981). North-Holland, Amsterdam (1982) 165-216.
Joyal, A., Free bicomplete categories. C. R. Math. Rep. Acad. Sci. Canada 17 (1995) 219-224.
A. Joyal, Free lattices, communication and money games, in Logic and scientific methods (Florence, 1995). Kluwer Acad. Publ., Dordrecht (1997) 29-68.
Kelly, G.M., Elementary observations on 2-categorical limits. Bull. Austral. Math. Soc. 39 (1989) 301-317. CrossRef
Lambek, J., A fixpoint theorem for complete categories. Math. Z. 103 (1968) 151-161. CrossRef
Lehmann, D.J. and Smyth, M.B., Algebraic specification of data types: A synthetic approach. Math. Systems Theory 14 (1981) 97-139. CrossRef
M. Makkai and R. Paré, Accessible categories: The foundations of categorical model theory. American Mathematical Society, Providence, RI (1989).
McNaughton, R., Infinite games played on finite graphs. Ann. Pure Appl. Logic 65 (1993) 149-184. CrossRef
A.W. Mostowski, Regular expressions for infinite trees and a standard form of automata, in Computation theory (Zaborów, 1984). Springer, Berlin, Lecture Notes in Comput. Sci. 208 (1985) 157-168.
D. Niwinski, Equational µ-calculus, in Computation theory (Zaborów, 1984). Springer, Berlin, Lecture Notes in Comput. Sci. 208 (1985) 169-176.
Reutenauer, C., Ensembles libres de chemins dans un graphe. Bull. Soc. Math. France 114 (1986) 135-152. CrossRef
Rutten, J.J.M.M., Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249 (2000) 3-80. CrossRef
Santocanale, L., The alternation hierarchy for the theory of µ-lattices. Theory Appl. Categ. 9 (2002) 166-197. A special volume of articles from the Category Theory 2000 Conference (CT2000).
L. Santocanale, A calculus of circular proofs and its categorical semantics, in FOSSACS02, Foundations of Software Science and Computation Structures. Springer-Verlag, Lecture Notes Comput. Sci. 2303 (2002) 357-371.
Santocanale, L., Free µ-lattices. J. Pure Appl. Algebra 168 (2002) 227-264. Category theory 1999 (Coimbra). CrossRef
A.K. Simpson and G.D. Plotkin, Complete axioms for categorical fixed-point operators, in Proc. of 15th Annual Symposium on Logic in Computer Science (2000) 30-41.
W. Thomas, Languages, automata, and logic edited by G. Rozenberg and A. Salomaa. Springer-Verlag, New York, Handbook of Formal Language Theory III (1996).
Walukiewicz, I., Pushdown processes: Games and model-checking. Inform. and Comput. 164 (2001) 234-263. FLOC '96 (New Brunswick, NJ). CrossRef
Zielonka, W., Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoret. Comput. Sci. 200 (1998) 135-183. CrossRef