Abstract
We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for two natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, contrasting it with both coalgebraic modal logic and existing first-order proposals for special classes of Set-coalgebras (apart for relational structures, also neighbourhood frames and topological spaces). The semantic characterization of expressivity is based on the fact that our language inherits a coalgebraic variant of the Van Benthem-Rosen Theorem. Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes which allow for completeness—and in some cases beyond that.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49, 672–713 (2002)
van Benthem, J.: Modal Correspondence Theory. Ph.D. thesis, Department of Mathematics, University of Amsterdam (1976)
ten Cate, B., Gabelaia, D., Sustretov, D.: Modal languages for topology: Expressivity and definability. Ann. Pure Appl. Logic 159(1-2), 146–170 (2009)
Chang, C.: Modal model theory. In: Cambridge Summer School in Mathematical Logic. LNM, vol. 337, pp. 599–617. Springer (1973)
Chellas, B.: Modal Logic. Cambridge University Press (1980)
Cirstea, C., Kurz, A., Pattinson, D., Schröder, L., Venema, Y.: Modal logics are coalgebraic. The Computer J. 54, 31–41 (2011)
Demri, S., Lugiez, D.: Presburger Modal Logic Is PSPACE-Complete. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 541–556. Springer, Heidelberg (2006)
Enderton, H.B.: A mathematical introduction to logic. Academic Press (1972)
Fine, K.: In so many possible worlds. Notre Dame J. Formal Logic 13, 516–520 (1972)
Goldblatt, R.: An abstract setting for Henkin proofs. CSLI Lecture Notes, pp. 191–212. CSLI Publications (1993)
Halpern, J.Y.: An analysis of first-order logics of probability. Artif. Intell. 46(3), 311–350 (1990)
Hansen, H.H., Kupke, C., Pacuit, E.: Neighbourhood structures: Bisimilarity and basic model theory. Log. Methods Comput. Sci. 5 (2009)
Hodkinson, I.: Hybrid formulas and elementarily generated modal logics. Notre Dame J. Formal Logic 47, 443–478 (2006)
Jacobs, B.: Predicate logic for functors and monads (2010)
Kupke, C., Kurz, A., Pattinson, D.: Ultrafilter Extensions for Coalgebras. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 263–277. Springer, Heidelberg (2005)
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94, 1–28 (1991)
Makowsky, J.A., Marcja, A.: Completeness theorems for modal model theory with the Montague-Chang semantics I. Math. Logic Quarterly 23, 97–104 (1977)
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. 208, 1447–1468 (2010)
Pauly, M.: A modal logic for coalitional power in games. J. Log. Comput. 12, 149–166 (2002)
Rosen, E.: Modal logic over finite structures. J. Logic, Language and Information 6(4), 427–439 (1997)
Rossman, B.: Homomorphism preservation theorems. J. ACM 55, 15:1–15:53 (2008)
Schröder, L.: A finite model construction for coalgebraic modal logic. J. Log. Algebr. Prog. 73, 97–110 (2007)
Schröder, L., Pattinson, D.: Coalgebraic Correspondence Theory. In: Ong, L. (ed.) FOSSACS 2010. LNCS, vol. 6014, pp. 328–342. Springer, Heidelberg (2010)
Schröder, L., Pattinson, D.: Named models in coalgebraic hybrid logic. In: Symposium on Theoretical Aspects of Computer Science, STACS 2010. Leibniz Int. Proceedings in Informatics, vol. 5, pp. 645–656. Schloss Dagstuhl – Leibniz-Center of Informatics (2010)
Schröder, L., Pattinson, D.: Rank-1 modal logics are coalgebraic. J. Log. Comput. 20, 1113–1147 (2010)
Seligman, J., Liu, F., Girard, P.: Logic in the Community. In: Banerjee, M., Seth, A. (eds.) ICLA 2011. LNCS, vol. 6521, pp. 178–188. Springer, Heidelberg (2011)
Sgro, J.: The interior operator logic and product topologies. Trans. AMS 258, 99–112 (1980)
Staton, S.: Relating coalgebraic notions of bisimulation. Log. Methods Comput. Sci. 7 (2011)
Ziegler, A.: Topological model theory. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics. Springer (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Litak, T., Pattinson, D., Sano, K., Schröder, L. (2012). Coalgebraic Predicate Logic. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31585-5_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-31585-5_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31584-8
Online ISBN: 978-3-642-31585-5
eBook Packages: Computer ScienceComputer Science (R0)