Strongly Complete Logics for Coalgebras

Coalgebras for a functor model different types of transition systems in a uni- form way. This paper focuses on a uniform account of finitary logics for set-based coalge- bras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts. Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor. Part II investigates algebras for a functor over ind-completions and extends the theorem of Jonsson and Tarski on canonical extensions of Boolean algebras with operators to this setting. Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T. Based on Part II we prove the logic to be strongly complete under a reasonable condition on T.

[1]  J. Adámek,et al.  Locally presentable and accessible categories , 1994 .

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

[3]  Martin Rö,et al.  From modal logic to terminal coalgebras , 2001, Theor. Comput. Sci..

[4]  Bartek Klin,et al.  Coalgebraic Modal Logic Beyond Sets , 2007, MFPS.

[5]  Law Fw FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963 .

[6]  Alexander Kurz,et al.  Specifying Coalgebras with Modal Logic , 1998, CMCS.

[7]  Samson Abramsky,et al.  Domain Theory in Logical Form , 1991, LICS.

[8]  Yde Venema,et al.  Algebras and coalgebras , 2007, Handbook of Modal Logic.

[9]  Alexander Kurz,et al.  On universal algebra over nominal sets , 2010, Math. Struct. Comput. Sci..

[10]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[11]  Alexander Kurz,et al.  Equational Coalgebraic Logic , 2009, MFPS.

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

[13]  P. Gabriel,et al.  Lokal α-präsentierbare Kategorien , 1971 .

[14]  Peter Aczel,et al.  Non-well-founded sets , 1988, CSLI lecture notes series.

[15]  Marcello M. Bonsangue,et al.  Presenting Functors by Operations and Equations , 2006, FoSSaCS.

[16]  Dirk Pattinson,et al.  Strong Completeness of Coalgebraic Modal Logics , 2009, STACS.

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

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

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

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

[21]  S. Maclane,et al.  Categorical Algebra , 2007 .

[22]  Lutz Schröder,et al.  A finite model construction for coalgebraic modal logic , 2006, J. Log. Algebraic Methods Program..

[23]  James Worrell,et al.  On the final sequence of a finitary set functor , 2005, Theor. Comput. Sci..

[24]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[25]  F. W. Lawvere,et al.  FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

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

[27]  Erwin Engeler,et al.  Languages with expressions of infinite length , 1966 .

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

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

[30]  Jiří Adámek,et al.  On the duality between varieties and algebraic theories , 2003 .

[31]  Tadeusz Litak,et al.  An algebraic approach to incompleteness in modal logic , 2005 .

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

[33]  Jiří Adámek,et al.  On sifted colimits and generalized varieties. , 2001 .

[34]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

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

[36]  R. Goldblatt Metamathematics of modal logic , 1974, Bulletin of the Australian Mathematical Society.

[37]  Jean-Louis Loday,et al.  Generalized bialgebras and triples of operads , 2006, Astérisque.

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

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

[40]  M. Stone The theory of representations for Boolean algebras , 1936 .

[41]  Andrew M. Pitts,et al.  Nominal Equational Logic , 2007, Electron. Notes Theor. Comput. Sci..

[42]  Marcello M. Bonsangue,et al.  Duality for Logics of Transition Systems , 2005, FoSSaCS.

[43]  Murdoch James Gabbay,et al.  Nominal (Universal) Algebra: Equational Logic with Names and Binding , 2009, J. Log. Comput..

[44]  Jiri Rosicky,et al.  A characterization of locally D-presentable categories , 2004 .

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

[46]  Yde Venema,et al.  Stone Coalgebras , 2004, CMCS.

[47]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[48]  F. E. J. Linton,et al.  An outline of functorial semantics , 1969 .

[49]  Peter Aczel,et al.  A Final Coalgebra Theorem , 1989, Category Theory and Computer Science.

[50]  Alexander Kurz,et al.  Algebraic Semantics for Coalgebraic Logics , 2004, CMCS.

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

[52]  Alexander Kurz,et al.  The Goldblatt-Thomason Theorem for Coalgebras , 2007, CALCO.

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

[54]  J Ad Amek,et al.  On Sifted Colimits and Generalized Varieties , 1999 .

[55]  Alexander Kurz,et al.  Presenting functors on many-sorted varieties and applications , 2010, Inf. Comput..

[56]  Samson Abramsky,et al.  A Cook's Tour of the Finitary Non-Well-Founded Sets , 2011, We Will Show Them!.