Modular algorithms for heterogeneous modal logics via multi-sorted coalgebra

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.

[1]  Kit Fine,et al.  In so many possible worlds , 1972, Notre Dame J. Formal Log..

[2]  Bengt Jonsson,et al.  A calculus for communicating systems with time and probabilities , 1990, [1990] Proceedings 11th Real-Time Systems Symposium.

[3]  Lutz Schröder.  A Finite Model Construction for Coalgebraic Modal Logic , 2006, FoSSaCS.

[4]  Richard J. Boulton,et al.  Theorem Proving in Higher Order Logics , 2003, Lecture Notes in Computer Science.

[5]  Marc Pauly,et al.  A Modal Logic for Coalitional Power in Games , 2002, J. Log. Comput..

[6]  Albert Visser,et al.  Finality regained: A coalgebraic study of Scott-sets and multisets , 1999, Arch. Math. Log..

[7]  Dirk Pattinson,et al.  Modular Algorithms for Heterogeneous Modal Logics , 2007, ICALP.

[8]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[9]  Frank Wolter,et al.  Handbook of Modal Logic, Volume 3 (Studies in Logic and Practical Reasoning) , 2006 .

[10]  Roberto Segala.  Probability and Nondeterminism in Operational Models of Concurrency , 2006, CONCUR.

[11]  Frank Wolter,et al.  Fusions of Modal Logics Revisited , 1996, Advances in Modal Logic.

[12]  Erik P. de Vink,et al.  Coalgebraic Weak Bisimulation for Action-Type Systems , 2009, Sci. Ann. Comput. Sci..

[13]  Stephan Tobies.  PSPACE Reasoning for Graded Modal Logics , 2001, J. Log. Comput..

[14]  Roberto Segala,et al.  Modeling and verification of randomized distributed real-time systems , 1996 .

[15]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[16]  Lawrence S. Moss,et al.  Coalgebraic Logic , 1999, Ann. Pure Appl. Log..

[17]  Dirk Pattinson,et al.  CoLoSS : The Coalgebraic Logic Satisfiability Solver ( System Description ) , 2008 .

[18]  Erik P. de Vink,et al.  Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach , 1997, Theor. Comput. Sci..

[19]  Dirk Pattinson,et al.  Beyond Rank 1: Algebraic Semantics and Finite Models for Coalgebraic Logics , 2008, FoSSaCS.

[20]  Moshe Y. Vardi.  On the complexity of epistemic reasoning , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[21]  Michael Barr,et al.  Terminal Coalgebras in Well-Founded Set Theory , 1993, Theor. Comput. Sci..

[22]  Stefan Berghofer,et al.  Inductive Datatypes in HOL - Lessons Learned in Formal-Logic Engineering , 1999, TPHOLs.

[23]  Erik P. de Vink,et al.  A hierarchy of probabilistic system types , 2003, CMCS.

[24]  Dirk Pattinson,et al.  Coalgebraic Hybrid Logic , 2009, FoSSaCS.

[25]  Kim G. Larsen,et al.  Bisimulation through Probabilistic Testing , 1991, Inf. Comput..

[26]  Carsten Lutz,et al.  E-connections of abstract description systems , 2004, Artif. Intell..

[27]  Horst Herrlich,et al.  Abstract and concrete categories , 1990 .

[28]  Joseph A. Ziegler,et al.  On the space cost curve and industrial location: Comment , 1985 .

[29]  Lutz Schroder,et al.  Shallow Models for Non-Iterative Modal Logics , 2008 .

[30]  Dirk Pattinson.  Expressive Logics for Coalgebras via Terminal Sequence Induction , 2004, Notre Dame J. Formal Log..

[31]  Barbara König,et al.  Tools and Algorithms for the Construction and Analysis of Systems , 2012, Lecture Notes in Computer Science.

[32]  Brinch Hansen,et al.  Automata, Languages and Programming , 2005, Lecture Notes in Computer Science.

[33]  Corina Cîrstea,et al.  Modular construction of complete coalgebraic logics , 2007, Theor. Comput. Sci..

[34]  Dirk Pattinson,et al.  Nominals for Everyone , 2009, IJCAI.

[35]  Dirk Pattinson,et al.  Shallow Models for Non-iterative Modal Logics , 2008, KI.

[36]  David Lewis.  Intensional logics without interative axioms , 1974, J. Philos. Log..

[37]  H. S. ALLEN.  Time and Probability , 1938, Nature.

[38]  Lutz Schröder,et al.  Expressivity of coalgebraic modal logic: The limits and beyond , 2008, Theor. Comput. Sci..

[39]  Rajeev Goré,et al.  Global Caching for Coalgebraic Description Logics , 2010, IJCAR.

[40]  Rajeev Goré,et al.  Optimal Tableau Algorithms for Coalgebraic Logics , 2010, TACAS.

[41]  Aviad Heifetz,et al.  Probability Logic for Type Spaces , 2001, Games Econ. Behav..

[42]  Markus Roggenbach,et al.  Algebraic-coalgebraic specification in CoCasl , 2006, J. Log. Algebraic Methods Program..

[43]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[44]  Dirk Pattinson,et al.  CoLoSS: The Coalgebraic Logic Satisfiability Solver , 2009, M4M.

[45]  Hans A. Hansson.  Time and probability in formal design of distributed systems , 1991, DoCS.

[46]  Agi Kurucz,et al.  Combining modal logics , 2007, Handbook of Modal Logic.

[47]  Camilla Schwind,et al.  A sequent calculus and a theorem prover for standard conditional logics , 2007, TOCL.

[48]  Edith Hemaspaandra.  Complexity transfer for modal logic , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[49]  Jan J. M. M. Rutten,et al.  Universal coalgebra: a theory of systems , 2000, Theor. Comput. Sci..

[50]  Roberto Segala,et al.  Comparative analysis of bisimulation relations on alternating and non-alternating probabilistic models , 2005, Second International Conference on the Quantitative Evaluation of Systems (QEST'05).

[51]  Corina Cîrstea,et al.  Modular Construction of Modal Logics , 2004, CONCUR.

[52]  Thomas A. Henzinger,et al.  Alternating-time temporal logic , 1999 .

[53]  Dirk Pattinson,et al.  How Many Toes Do I Have? Parthood and Number Restrictions in Description Logics , 2008, KR.

[54]  Dirk Pattinson,et al.  Rank-1 Modal Logics are Coalgebraic , 2007, J. Log. Comput..

[55]  Bart Jacobs,et al.  Many-Sorted Coalgebraic Modal Logic: a Model-theoretic Study , 2001, RAIRO Theor. Informatics Appl..

[56]  Wang Yi,et al.  Probabilistic Extensions of Process Algebras , 2001, Handbook of Process Algebra.

[57]  Dirk Pattinson,et al.  PSPACE Bounds for Rank-1 Modal Logics , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).