PSPACE Bounds for Rank-1 Modal Logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatization, in PSPACE. This leads not only to a unified derivation of (known) tight PSPACE-bounds for a number of logics including K, coalition logic, and graded modal logic (and to a new algorithm in the latter case), but also to a previously unknown tight PSPACE-bound for probabilistic modal logic, with rational probabilities coded in binary. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

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

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

[3]  Bart Jacobs,et al.  The Coalgebraic Class Specification Language CCSL , 2001, J. Univers. Comput. Sci..

[4]  Dirk Pattinson,et al.  Coalgebraic modal logic: soundness, completeness and decidability of local consequence , 2003, Theor. Comput. Sci..

[5]  Bart Jacobs,et al.  Towards a Duality Result in Coalgebraic Modal Logic , 2000, CMCS.

[6]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[7]  Ullrich Hustadt,et al.  Resolution-Based Methods for Modal Logics , 2000, Log. J. IGPL.

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

[9]  Azaria Paz,et al.  Probabilistic automata , 2003 .

[10]  Helle Hvid Hansen,et al.  A Coalgebraic Perspective on Monotone Modal Logic , 2004, CMCS.

[11]  Jack W. Carlyle,et al.  Realizations by Stochastic Finite Automata , 1971, J. Comput. Syst. Sci..

[12]  John-Jules Ch. Meyer,et al.  Graded Modalities in Epistemic Logic , 1992, LFCS.

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

[14]  Falk Bartels,et al.  Generalised coinduction , 2003, Mathematical Structures in Computer Science.

[15]  Leandro Chaves Rêgo,et al.  Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic , 2006, IJCAI.

[16]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

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

[18]  Jana Koehler,et al.  Modal Logics, Description Logics and Arithmetic Reasoning , 1999, Artif. Intell..

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

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

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

[22]  Eric Pacuit,et al.  Majority Logic , 2004, KR.

[23]  Marc Pauly,et al.  On the role of language in social choice theory , 2008, Synthese.

[24]  Moshe Y. Vardi.  Why is Modal Logic So Robustly Decidable? , 1996, Descriptive Complexity and Finite Models.

[25]  Richard E. Ladner,et al.  The Computational Complexity of Provability in Systems of Modal Propositional Logic , 1977, SIAM J. Comput..

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

[27]  Martin Rößiger,et al.  Coalgebras and Modal Logic , 2000, CMCS.

[28]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

[29]  Yde Venema.  Automata and fixed point logic: A coalgebraic perspective , 2006, Inf. Comput..

[30]  Alexander Kurz.  Specifying coalgebras with modal logic , 2001, Theor. Comput. Sci..

[31]  Alexander Kurz,et al.  Ultrafilter Extensions for Coalgebras , 2005, CALCO.

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

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

[34]  Dirk Pattinson,et al.  Rank-1 Modal Logics Are Coalgebraic , 2007, STACS.

[35]  Ronald Fagin,et al.  Reasoning about knowledge and probability , 1988, JACM.

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

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

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

[39]  Francesco Caro.  Graded modalities, II (canonical models) , 1988, Stud Logica.

[40]  Gordon D. Plotkin,et al.  Towards a mathematical operational semantics , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

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

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

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

[44]  Dirk Pattinson,et al.  Semantical Principles in the Modal Logic of Coalgebras , 2001, STACS.

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

[46]  Joseph Y. Halpern,et al.  Decision procedures and expressiveness in the temporal logic of branching time , 1982, STOC '82.