The Logic of Exact Covers: Completeness and Uniform Interpolation

We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank-1 have the interpolation property, and that in fact interpolation is uniform if the logics just have finitely many modal operators. As immediate applications, we obtain previously unknown interpolation theorems for a range of modal logics, containing probabilistic and graded modal logic, alternating temporal logic and some variants of conditional logic. Technically, this is achieved by translating to and from a new (coalgebraic) logic introduced in this paper, the logic of exact covers. It is interpreted over coalgebrasfor an endofunctor on the category of sets that also directly determines the syntax. Apart from closure under bisimulation quantifiers (and hence interpolation), we also provide a complete tableaux calculus and establish both the Hennessy-Milner and the small model property for this logic.

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

[2]  Yde Venema,et al.  Completeness of the finitary Moss logic , 2008, Advances in Modal Logic.

[3]  Dirk Pattinson,et al.  Cut elimination in coalgebraic logics , 2010, Inf. Comput..

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

[5]  Lawrence S. Moss.  A Note on Expressive Coalgebraic Logics for Finitary Set Functors , 2010, J. Log. Comput..

[6]  Tim French,et al.  Bisimulation quantifiers for modal logics , 2006 .

[7]  A. Visser.  Uniform interpolation and layered bisimulation , 1996 .

[8]  Alessandra Palmigiano,et al.  Proof systems for the coalgebraic cover modality , 2008, Advances in Modal Logic.

[9]  Grigoris Antoniou,et al.  Nonmonotonic reasoning , 1997 .

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

[11]  Andrew M. Pitts,et al.  On an interpretation of second order quantification in first order intuitionistic propositional logic , 1992, Journal of Symbolic Logic.

[12]  Dirk Pattinson,et al.  Cut Elimination for Shallow Modal Logics , 2011, TABLEAUX.

[13]  Yde Venema,et al.  Lax Extensions of Coalgebra Functors , 2012, CMCS.

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

[15]  Silvio Ghilardi,et al.  Unification in intuitionistic logic , 1999, Journal of Symbolic Logic.

[16]  Yde Venema.  A Modal Distributive Law (abstract) , 2007, WoLLIC.

[17]  Marta Bílková,et al.  Uniform Interpolation and Propositional Quantifiers in Modal Logics , 2007, Stud Logica.

[18]  John P. Burgess,et al.  Quick completeness proofs for some logics of conditionals , 1981, Notre Dame J. Formal Log..

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

[20]  Silvio Ghilardi,et al.  An Algebraic Theory of Normal Forms , 1995, Ann. Pure Appl. Log..

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

[22]  Dirk Pattinson,et al.  Coalgebraic semantics of modal logics: An overview , 2011, Theor. Comput. Sci..

[23]  J. Adámek,et al.  Automata and Algebras in Categories , 1990 .

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

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

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

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

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

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

[30]  Václav Koubek,et al.  The colimits in the generalized algebraic categories , 1972 .

[31]  Giovanna D'Agostino.  Interpolation in non-classical logics , 2008, Synthese.

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

[33]  Yde Venema,et al.  Coalgebraic Automata Theory: Basic Results , 2008, Log. Methods Comput. Sci..

[34]  Yde Venema,et al.  Uniform Interpolation for Monotone Modal Logic , 2010, Advances in Modal Logic.

[35]  Sarit Kraus,et al.  Nonmonotonic Reasoning, Preferential Models and Cumulative Logics , 1990, Artif. Intell..

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

[37]  Helle Hvid Hansen,et al.  Neighbourhood Structures: Bisimilarity and Basic Model Theory , 2009, Log. Methods Comput. Sci..

[38]  Dirk Pattinson,et al.  On Modal Logics of Linear Inequalities , 2010, Advances in Modal Logic.

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

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

[41]  Silvio Ghilardi,et al.  Undefinability of propositional quantifiers in the modal system S4 , 1995, Stud Logica.

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