Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T08:18:58.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular algorithms for heterogeneous modal logics via multi-sorted coalgebra

Published online by Cambridge University Press:  25 March 2011

LUTZ SCHRÖDER  and
DIRK PATTINSON
LUTZ SCHRÖDER
Affiliation:
DFKI Bremen and Dept. of Comput. Sci., Univ. Bremen, Cartesium, Enrique-Schmidt-Str. 5, 28359 Bremen, Germany Email: [email protected]
DIRK PATTINSON
Affiliation:
Department of Computing, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. In this paper, we show that the combination of features can be reflected algorithmically, and we 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 algorithmic description with 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 this is also 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, for example, Segala systems and games with uncertain information.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Special Issue 2: Coalgebraic Logic , April 2011 , pp. 235 - 266
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adámek, J., Herrlich, H. and Strecker, G. E. (1990) Abstract and Concrete Categories, Wiley Interscience.Google Scholar
Alur, R., Henzinger, T. A. and Kupferman, O. (2002) Alternating-time temporal logic. Journal of the ACM 49 672–713.CrossRefGoogle Scholar
Barr, M. (1993) Terminal coalgebras in well-founded set theory. Theoretical Computer Science 114 299–315.Google Scholar
Bartels, F., Sokolova, A. and de Vink, E. (2003) A hierarchy of probabilistic system types. In: Gumm, H.-P. (ed.) Coalgebraic Methods in Computer Science, CMCS 03. Electronic Notes in Theoretical Computer Science 82.Google Scholar
Berghofer, S. and Wenzel, M. (1999) Inductive datatypes in HOL – lessons learned in formal-logic engineering. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C. and Théry, L. (eds.) Theorem Proving in Higher Order Logics, TPHOLs 1999. Springer-Verlag Lecture Notes in Computer Science 1690 19–36.CrossRefGoogle Scholar
Blackburn, P., de Rijke, M. and Venema, Y. (2001) Modal Logic, Cambridge University Press.Google Scholar
Calin, G., Myers, R., Pattinson, D. and Schröder, L. (2009) CoLoSS: The Coalgebraic Logic Satisfiability Solver (system description). In: Areces, C. and Demri, S. (eds.) Methods for Modalities, M4M-5, 2007. Electronic Notes in Theoretical Computer Science 231 41–54.Google Scholar
Chandra, A. and Stockmeyer, L. (1981) Alternation. Journal of the ACM 28 (1)114–133.Google Scholar
Chellas, B. (1980) Modal Logic, Cambridge University Press.CrossRefGoogle Scholar
Cîrstea, C. and Pattinson, D. (2007) Modular construction of modal logics. Theoretical Computer Science 388 (1-3)83–108.CrossRefGoogle Scholar
D'Agostino, G. and Visser, A. (2002) Finality regained: A coalgebraic study of Scott-sets and multisets. Archive for Mathematical Logic 41 267–298.Google Scholar
de Vink, E. and Rutten, J. (1999) Bisimulation for probabilistic transition systems: A coalgebraic approach. Theoretical Computer Science 221 271–293.Google Scholar
Fine, K. (1972) In so many possible worlds. Notre Dame Journal of Formal Logic 13 516–520.Google Scholar
Goré, R., Kupke, C. and Pattinson, D. (2010a) Optimal tableau algorithms for coalgebraic logics. In: Esparza, J. and Majumdar, R. (eds.) Tools and Algorithms for the Construction and Analysis of Systems, TACAS 10. Springer-Verlag Lecture Notes in Computer Science 6015 114–128.CrossRefGoogle Scholar
Goré, R., Kupke, C., Pattinson, D. and Schröder, L. (2010b) Global caching for coalgebraic description logics. In: Giesl, J. and Haehnle, R. (eds.) International Joint Conference on Automated Reasoning, IJCAR 2010. Springer-Verlag Lecture Notes in Computer Science 6173 46–60.CrossRefGoogle Scholar
Halpern, J. Y. (2003) Reasoning About Uncertainty, MIT Press.Google Scholar
Hansson, H. and Jonsson, B. (1990) A calculus for communicating systems with time and probabilities. In: Real-Time Systems, RTSS 90, IEEE 278–287.Google Scholar
Hansson, H. A. (1994) Time and Probability in Formal Design of Distributed Systems, Elsevier.Google Scholar
Heifetz, A. and Mongin, P. (2001) Probabilistic logic for type spaces. Games and Economic Behavior 35 31–53.Google Scholar
Hemaspaandra, E. (1994) Complexity transfer for modal logic. In: Abramsky, S. (ed.) Logic in Computer Science, LICS 94, IEEE 164–173.Google Scholar
Jacobs, B. (2001) Many-sorted coalgebraic modal logic: a model-theoretic study. Theoretical Informatics and Applications 35 31–59.Google Scholar
Jonsson, B., Yi, W. and Larsen, K. G. (2001) Probabilistic extensions of process algebras. In: Bergstra, J., Ponse, A. and Smolka, S. (eds.) Handbook of Process Algebra, Elsevier.Google Scholar
Kurucz, A. (2006) Combining modal logics. In: van Benthem, J., Blackburn, P. and Wolter, F. (eds.) Handbook of Modal Logic, Elsevier.Google Scholar
Kutz, O., Lutz, C., Wolter, F. and Zakharyaschev, M. (2004) ℰ-connections of abstract description systems. Artificial Intelligence 156 1–73.Google Scholar
Larsen, K. and Skou, A. (1991) Bisimulation through probabilistic testing. Information and Computation 94 1–28.CrossRefGoogle Scholar
Lewis, D. (1975) Intensional logics without iterative axioms. Journal of Philosophical Logic 3 457–466.Google Scholar
Moss, L. (1999) Coalgebraic logic. Annals of Pure and Applied Logic 96 277–317.Google Scholar
Mossakowski, T., Schröder, L., Roggenbach, M. and Reichel, H. (2006) Algebraic-coalgebraic specification in CoCASL. Journal of Logic and Algebraic Programming 67 146–197.Google Scholar
Myers, R., Pattinson, D. and Schröder, L. (2009) Coalgebraic hybrid logic. In: de Alfaro, L. (ed.) Foundations of Software Science and Computation Structures, FoSSaCS 2009. Springer-Verlag Lecture Notes in Computer Science 5504 137–151.Google Scholar
Olivetti, N., Pozzato, G. L. and Schwind, C. (2007) A sequent calculus and a theorem prover for standard conditional logics. ACM Transactions on Computational Logic 8.CrossRefGoogle Scholar
Pattinson, D. (2004) Expressive logics for coalgebras via terminal sequence induction. Notre Dame Journal of Formal Logic 45 19–33.Google Scholar
Pattinson, D. and Schröder, L. (2008) Beyond rank 1: Algebraic semantics and finite models for coalgebraic logics. In: Amadio, R. (ed.) Foundations of Software Science and Computation Structures, FoSSaCS 2008. Springer-Verlag Lecture Notes in Computer Science 4962 66–80.Google Scholar
Pauly, M. (2002) A modal logic for coalitional power in games. Journal of Logic and Computation 12 149–166.Google Scholar
Rutten, J. (2000) Universal coalgebra: A theory of systems. Theoretical Computer Science 249 3–80.CrossRefGoogle Scholar
Schröder, L. (2007) A finite model construction for coalgebraic modal logic. Journal of Logic and Algebraic Programming 73 97–110.Google Scholar
Schröder, L. (2008) Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science 390 230–247.Google Scholar
Schröder, L. and Pattinson, D. (2007) Modular algorithms for heterogeneous modal logics. In: Arge, L., Tarlecki, A. and Cachin, C. (eds.) Automata, Languages and Programming, ICALP 2007. Springer-Verlag Lecture Notes in Computer Science 4596 459–471.Google Scholar
Schröder, L. and Pattinson, D. (2008a) How many toes do I have? Parthood and number restrictions in description logics. In: Brewka, G. and Lang, J. (eds.) Principles of Knowledge Representation and Reasoning, KR 2008, AAAI Press 307–218.Google Scholar
Schröder, L. and Pattinson, D. (2008b) Shallow models for non-iterative modal logics. In: Dengel, A., Berns, K., Breuel, T., Bomarius, F. and Roth-Berghofer, T. (eds.) Advances in Artificial Intelligence, KI 2008. Springer-Verlag Lecture Notes in Artificial Intelligence 5243 324–331.Google Scholar
Schröder, L. and Pattinson, D. (2009) PSPACE bounds for rank-1 modal logics. ACM Transactions on Computational Logic 10 (2:13)1–33.CrossRefGoogle Scholar
Schröder, L. and Pattinson, D. (2010) Rank-1 modal logics are coalgebraic. Journal of Logic and Computation 20 (5)1113–1147.Google Scholar
Schröder, L., Pattinson, D. and Kupke, C. (2009) Nominals for everyone. In: Boutilier, C. (ed.) International Joint Conferences on Artificial Intelligence, IJCAI 09, AAAI Press 917–922.Google Scholar
Segala, R. (1995) Modelling and Verification of Randomized Distributed Real-Time Systems, Ph.D. thesis, Massachusetts Institute of Technology.Google Scholar
Segala, R. (2006) Probability and nondeterminism in operational models of concurrency. In: Baier, C. and Hermanns, H. (eds.) Concurrency Theory, CONCUR 2006. Springer-Verlag Lecture Notes in Computer Science 4137 64–78.Google Scholar
Segala, R. and Turrini, A. (2005) Comparative analysis of bisimulation relations on alternating and non-alternating probabilistic models. In: Quantitative Evaluation of Systems, QEST 2005, IEEE 44–53.Google Scholar
Sokolova, A., de Vink, E. and Woracek, H. (2009) Coalgebraic weak bisimulation for action-type systems. Scientific Annals of Computer Science 19 93–144.Google Scholar
Tobies, S. (2001) PSPACE reasoning for graded modal logics. Journal of Logic and Computation 11 85–106.Google Scholar
Vardi, M. (1989) On the complexity of epistemic reasoning. In: Logic in Computer Science, LICS 89, IEEE 243–251.Google Scholar
Wolter, F. (1998) Fusions of modal logics revisited. In: Kracht, M., de Rijke, M. and Wansing, H. (eds.) Advances in modal logic, AiML 98. CSLI Lecture Notes 1 361–379.Google Scholar