Abstract
State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying state-based systems as multi-sorted coalgebras and associating both a logical and an algorithmic description to a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided that this is the case for the components. By instantiating the general framework to concrete cases, we obtain PSPACE decision procedures for a wide variety of structurally different logics, describing e.g. Segala systems and games with uncertain information.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bartels, F., Sokolova, A., de Vink, E.: A hierarchy of probabilistic system types. In: Gumm, H.-P. (ed.) Coalgebraic Methods in Computer Science. ENTCS, vol. 82, Elsevier, Amsterdam (2003)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Chellas, B.: Modal Logic. Cambridge University Press, Cambridge (1980)
Cîrstea, C., Pattinson, D.: Modular construction of modal logics. Theoret. Copmut. Sci. (to appear). Earlier version In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 258–275. Springer, Heidelberg (2004)
D’Agostino, G., Visser, A.: Finality regained: A coalgebraic study of Scott-sets and multisets. Arch. Math. Logic 41, 267–298 (2002)
Fine, K.: In so many possible worlds. Notre Dame J. Formal Logic 13, 516–520 (1972)
Halpern, J.Y.: Reasoning About Uncertainty. MIT Press, Cambridge (2003)
Hansson, H., Jonsson, B.: A calculus for communicating systems with time and probabilities. In: Real-Time Systems, RTSS 90, pp. 278–287. IEEE Computer Society Press, Los Alamitos (1990)
Heifetz, A., Mongin, P.: Probabilistic logic for type spaces. Games and Economic Behavior 35, 31–53 (2001)
Hemaspaandra, E.: Complexity transfer for modal logic. In: Abramsy, S. (ed.) LICS 1994. Logic in Computer Science, pp. 164–173. IEEE Computer Society Press, Los Alamitos (1994)
Jacobs, B.: Many-sorted coalgebraic modal logic: a model-theoretic study. Theor. Inform. Appl. 35, 31–59 (2001)
Jonsson, B., Yi, W., Larsen, K.G.: Probabilistic extensions of process algebras. In: Bergstra, J., Ponse, A., Smolka, S.M. (eds.) Handbook of Process Algebra, Elsevier, Amsterdam (2001)
Kurucz, A.: Combining modal logics. In: van Benthem, J., Blackburn, P., Wolter, F. (eds.) Handbook of Modal Logic, Elsevier, Amsterdam (2006)
Kutz, O., Lutz, C., Wolter, F., Zakharyaschev, M.: \(\mathcal E\)-connections of abstract description systems. Artificial Intelligence 156, 1–73 (2004)
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inform. Comput. 94, 1–28 (1991)
Mossakowski, T., Schröder, L., Roggenbach, M., Reichel, H.: Algebraic-coalgebraic specification in CoCASL. J. Logic Algebraic Programming 67, 146–197 (2006)
Pattinson, D.: Expressive logics for coalgebras via terminal sequence induction. Notre Dame J. Formal Logic 45, 19–33 (2004)
Pauly, M.: A modal logic for coalitional power in games. J. Logic Comput. 12, 149–166 (2002)
Rutten, J.: Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249, 3–80 (2000)
Schröder, L.: Expressivity of coalgebraic modal logic: the limits and beyond. Theoret. Comput. Sci. Earlier version In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 440–454. Springer, Heidelberg (2005) (to appear)
Schröder, L.: A semantic PSPACE criterion for the next 700 rank 0-1 modal logics, available at http://www.informatik.uni-bremen.de/~lschrode/papers/rank01pspace.pdf
Schröder, L.: A finite model construction for coalgebraic modal logic. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006 and ETAPS 2006. LNCS, vol. 3921, pp. 157–171. Springer, Heidelberg (2006)
Schröder, L., Pattinson, D.: PSPACE reasoning for rank-1 modal logics. In: Alur, R. (ed.) LICS 2006. Logic in Computer Science, pp. 231–240. IEEE Computer Society Press, Los Alamitos (2006)
Segala, R.: Modelling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology (1995)
Tobies, S.: PSPACE reasoning for graded modal logics. J. Logic Comput. 11, 85–106 (2001)
Wolter, F.: Fusions of modal logics revisited. In: Zakharyaschev, M., Segerberg, K., de Rijke, M., Wansing, H. (eds.) Advances in modal logic. CSLI Lect. Notes, vol. 1, pp. 361–379. CSLI, Stanford (1998)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schröder, L., Pattinson, D. (2007). Modular Algorithms for Heterogeneous Modal Logics. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_41
Download citation
DOI: https://doi.org/10.1007/978-3-540-73420-8_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73419-2
Online ISBN: 978-3-540-73420-8
eBook Packages: Computer ScienceComputer Science (R0)