no. 2
μ-bicomplete categories and parity games
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 2, pp. 195-227.

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.

DOI : 10.1051/ita:2002010
Classification : 18A30, 68Q65, 91A43
Mots clés : parity games, bicomplete categories, initial algebras, final coalgebras, inductive and coinductive types 
@article{ITA_2002__36_2_195_0,
     author = {Santocanale, Luigi},
     title = {$\mu $-bicomplete categories and parity games},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {195--227},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     doi = {10.1051/ita:2002010},
     mrnumber = {1948769},
     zbl = {1024.18001},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2002010/}
}
TY  - JOUR
AU  - Santocanale, Luigi
TI  - $\mu $-bicomplete categories and parity games
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2002
SP  - 195
EP  - 227
VL  - 36
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita:2002010/
DO  - 10.1051/ita:2002010
LA  - en
ID  - ITA_2002__36_2_195_0
ER  - 
%0 Journal Article
%A Santocanale, Luigi
%T $\mu $-bicomplete categories and parity games
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2002
%P 195-227
%V 36
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita:2002010/
%R 10.1051/ita:2002010
%G en
%F ITA_2002__36_2_195_0
Santocanale, Luigi. $\mu $-bicomplete categories and parity games. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 2, pp. 195-227. doi : 10.1051/ita:2002010. http://archive.numdam.org/articles/10.1051/ita:2002010/

[1] P. Aczel, Non-well-founded sets. Stanford University Center for the Study of Language and Information, Stanford, CA (1988). | MR | Zbl

[2] 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).

[3] J. Adámek and V. Koubek, Least fixed point of a functor. J. Comput. System Sci. 19 (1979) 163-178. | MR | Zbl

[4] J. Adámek and J. Rosický, Locally presentable and accessible categories. Cambridge University Press, Cambridge (1994). | MR | Zbl

[5] A. Arnold and D. Niwinski, Rudiments of mu-calculus. Elsevier, North-Holland, Stud. Logic Found. Math. 146 (2001). | MR | Zbl

[6] M. Barr, Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114 (1993) 299-315. | MR | Zbl

[7] S.L. Bloom and Z. Ésik, Iteration theories. Springer-Verlag, Berlin (1993). The equational logic of iterative processes. | MR | Zbl

[8] S.L. Bloom, Z. Ésik, A. Labella and E.G. Manes, Iteration 2-theories. Appl. Categ. Structures 9 (2001) 173-216. | MR | Zbl

[9] 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. | MR | Zbl

[10] J.R.B. Cockett and D. Spencer, Strong categorical datatypes. II. A term logic for categorical programming. Theoret. Comput. Sci. 139 (1995) 69-113. | MR | Zbl

[11] R. Cockett and T. Fukushima, About Charity, Yellow Series Report No. 92/480/18. Department of Computer Science, The University of Calgary (1992).

[12] 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.

[13] E.A. Emerson, C.S. Jutla and A.P. Sistla, On model checking for the μ-calculus and its fragments. Theoret. Comput. Sci. 258 (2001) 491-522. | MR | Zbl

[14] Z. Ésik and A. Labella, Equational properties of iteration in algebraically complete categories. Theoret. Comput. Sci. 195 (1998) 61-89. Mathematical foundations of computer science, Cracow (1996). | MR | Zbl

[15] P. Freyd, Algebraically complete categories, in Category theory (Como, 1990). Springer, Berlin (1991) 95-104. | MR | Zbl

[16] E. Giménez, A tutorial on recursive types in Coq. Technical Report 0221, INRIA (1998).

[17] J.M.E. Hyland, The effective topos, in The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981). North-Holland, Amsterdam (1982) 165-216. | MR | Zbl

[18] A. Joyal, Free bicomplete categories. C. R. Math. Rep. Acad. Sci. Canada 17 (1995) 219-224. | MR | Zbl

[19] A. Joyal, Free lattices, communication and money games, in Logic and scientific methods (Florence, 1995). Kluwer Acad. Publ., Dordrecht (1997) 29-68. | MR | Zbl

[20] G.M. Kelly, Elementary observations on 2-categorical limits. Bull. Austral. Math. Soc. 39 (1989) 301-317. | MR | Zbl

[21] J. Lambek, A fixpoint theorem for complete categories. Math. Z. 103 (1968) 151-161. | MR | Zbl

[22] D.J. Lehmann and M.B. Smyth, Algebraic specification of data types: A synthetic approach. Math. Systems Theory 14 (1981) 97-139. | MR | Zbl

[23] M. Makkai and R. Paré, Accessible categories: The foundations of categorical model theory. American Mathematical Society, Providence, RI (1989). | MR | Zbl

[24] R. Mcnaughton, Infinite games played on finite graphs. Ann. Pure Appl. Logic 65 (1993) 149-184. | MR | Zbl

[25] 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. | MR | Zbl

[26] D. Niwiński, Equational μ-calculus, in Computation theory (Zaborów, 1984). Springer, Berlin, Lecture Notes in Comput. Sci. 208 (1985) 169-176. | MR | Zbl

[27] C. Reutenauer, Ensembles libres de chemins dans un graphe. Bull. Soc. Math. France 114 (1986) 135-152. | Numdam | MR | Zbl

[28] J.J.M.M. Rutten, Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249 (2000) 3-80. | MR | Zbl

[29] L. Santocanale, 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). | MR | Zbl

[30] 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. | MR | Zbl

[31] L. Santocanale, Free μ-lattices. J. Pure Appl. Algebra 168 (2002) 227-264. Category theory 1999 (Coimbra). | MR | Zbl

[32] 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. | MR

[33] W. Thomas, Languages, automata, and logic edited by G. Rozenberg and A. Salomaa. Springer-Verlag, New York, Handbook of Formal Language Theory III (1996). | MR

[34] I. Walukiewicz, Pushdown processes: Games and model-checking. Inform. and Comput. 164 (2001) 234-263. FLOC '96 (New Brunswick, NJ). | Zbl

[35] W. Zielonka, Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoret. Comput. Sci. 200 (1998) 135-183. | MR | Zbl

Cité par Sources :