Abstract
Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the μ-calculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the μ-calculus. The family of such flat fixed point logics includes, e.g., CTL, the *-nesting-free fragment of PDL, and the logic of common knowledge. Here, we extend this notion to the generic semantic framework of coalgebraic logic, thus covering a wide range of logics beyond the standard μ-calculus including, e.g., flat fragments of the graded μ-calculus and the alternating-time μ-calculus (such as ATL), as well as probabilistic and monotone fixed point logics. Our main results are completeness of the Kozen-Park axiomatization and a timed-out tableaux method that matches ExpTime upper bounds inherited from the coalgebraic μ-calculus but avoids using automata.
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Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49, 672–713 (2002)
Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook. Cambridge University Press, Cambridge (2003)
Calin, G., Myers, R., Pattinson, D., Schröder, L.: CoLoSS: The Coalgebraic Logic Satisfiability Solver (system description). In: Methods for Modalities (M4M-5, 2007). ENTCS, vol. 231, pp. 41–54. Elsevier, Amsterdam (2009)
Cîrstea, C., Kupke, C., Pattinson, D.: EXPTIME tableaux for the coalgebraic μ-calculus. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 179–193. Springer, Heidelberg (2009)
D’Agostino, G., Visser, A.: Finality regained: A coalgebraic study of Scott-sets and multisets. Arch. Math. Logic 41, 267–298 (2002)
Dam, M.: CTL* and ECTL* as fragments of the modal mu-calculus. Theoret. Comput. Sci. 126, 77–96 (1994)
Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Sci. Comput. Program 2(3), 241–266 (1982)
Emerson, E.A., Halpern, J.Y.: Decision procedures and expressiveness in the temporal logic of branching time. J. Comput. System Sci. 30, 1–24 (1985)
Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM J. Comput. 29, 132–158 (1999)
Emerson, E.A., Lei, C.-L.: Efficient model checking in fragments of the propositional mu-calculus. In: Logic in Computer Science, LICS 1986, pp. 267–278. IEEE, Los Alamitos (1986)
Fine, K.: In so many possible worlds. Notre Dame J. Formal Logic 13, 516–520 (1972)
Goranko, V., van Drimmelen, G.: Complete axiomatization and decidability of alternating-time temporal logic. Theoret. Comput. Sci. 353, 93–117 (2006)
Goré, R., Kupke, C., Pattinson, D.: Optimal tableau algorithms for coalgebraic logics. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 114–128. Springer, Heidelberg (2010)
Kozen, D.: Results on the propositional μ-calculus. Theoret. Comput. Sci. 27, 333–354 (1983)
Kozen, D., Parikh, R.: An elementary proof of the completeness of PDL. Theoret. Comput. Sci. 14, 113–118 (1981)
Kupferman, O., Sattler, U., Vardi, M.Y.: The complexity of the graded μ-calculus. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 423–437. Springer, Heidelberg (2002)
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94, 1–28 (1991)
Laver, R.: Well-quasi-orders and sets of finite sequences. Math. Proc. Cambridge Philos. Soc. 79, 1–10 (1976)
Lewis, D.: Convention, A Philosophical Study. Harvard University Press, Cambridge (1969)
Parikh, R.: The logic of games and its applications. Annals of Discrete Mathematics 24, 111–140 (1985)
Pattinson, D.: Expressive logics for coalgebras via terminal sequence induction. Notre Dame J. Formal Logic 45, 19–33 (2004)
Pattinson, D., Schröder, L.: Cut elimination in coalgebraic logics. Inf. Comput. (to appear)
Pauly, M.: A modal logic for coalitional power in games. J. Log. Comput. 12, 149–166 (2002)
Peleg, D.: Concurrent dynamic logic. J. ACM 34, 450–479 (1987)
Pratt, V.R.: Semantical considerations on Floyd-Hoare logic. In: Foundations of Computer Science, FOCS 1976, pp. 109–121. IEEE, Los Alamitos (1976)
Santocanale, L.: Completions of μ-algebras. Ann. Pure Appl. Logic 154, 27–50 (2008)
Santocanale, L., Venema, Y.: Completeness for flat modal fixpoint logics. Annals Pure Appl. Log. (preprint under) (to appear), http://arxiv.org/abs/0812.2390
Schröder, L.: A finite model construction for coalgebraic modal logic. J. Log. Algebr. Prog. 73, 97–110 (2007)
Schröder, L., Pattinson, D.: PSPACE bounds for rank-1 modal logics. ACM Trans. Comput. Log. 10, 13:1–13:33 (2009)
Schröder, L., Pattinson, D.: Strong completeness of coalgebraic modal logics. In: Theoretical Aspects of Computer Science, STACS 2009, pp. 673–684. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl (2009)
Schröder, L., Pattinson, D., Kupke, C.: Nominals for everyone. In: International Joint Conferences on Artificial Intelligence, IJCAI 2009, pp. 917–922. AAAI Press, Menlo Park (2009)
Segerberg, K.: A completeness theorem in the modal logic of programs. In: Universal Algebra and Applications. Banach Centre Publications, vol. 9, pp. 31–46. PWN (1982)
Walukiewicz, I.: Completeness of Kozen’s axiomatisation of the propositional μ-calculus. Inf. Comput. 157, 142–182 (2000)
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Schröder, L., Venema, Y. (2010). Flat Coalgebraic Fixed Point Logics. In: Gastin, P., Laroussinie, F. (eds) CONCUR 2010 - Concurrency Theory. CONCUR 2010. Lecture Notes in Computer Science, vol 6269. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15375-4_36
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DOI: https://doi.org/10.1007/978-3-642-15375-4_36
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