Many-Sorted Coalgebraic Modal Logic: a Model-theoretic Study

This paper gives a semantical underpinning for a many- sorted modal logic associated with certain dynamical systems, like tran- sition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial func- tors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many- sorted Boolean Algebras with Operators, combining standard (categor- ical) models of modal logic and of many-sorted predicate logic. In this setting we will see Lindenbaum MBAO models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as nal objects. They will be used to (re)prove completeness results, and Hennessey{Milner style characterisation results for the modal logic, rst established by Roiger. Mathematics Subject Classication. 03G05, 03G30, 06E25.

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

[2]  Hans-Jörg Schek,et al.  Object Orientation with Parallelism and Persistence , 1996 .

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

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

[5]  Yde Venema,et al.  Points, Lines and Diamonds: A two-sorted Modal Logic for Projective Planes , 1999, J. Log. Comput..

[6]  B. Jacobs,et al.  A tutorial on (co)algebras and (co)induction , 1997 .

[7]  Robert Goldblatt,et al.  What is the coalgebraic analogue of Birkhoff's variety theorem? , 2001, Theor. Comput. Sci..

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

[9]  Horst Reichel,et al.  An approach to object semantics based on terminal co-algebras , 1995, Mathematical Structures in Computer Science.

[10]  Bart Jacobs,et al.  Objects and Classes, Co-Algebraically , 1995, Object Orientation with Parallelism and Persistence.

[11]  Jan Rutten,et al.  On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces , 1998, Mathematical Structures in Computer Science.

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

[13]  Alexandru Baltag,et al.  A Logic for Coalgebraic Simulation , 2000, CMCS.

[14]  Bart Jacobs,et al.  The temporal logic of coalgebras via Galois algebras , 2002, Mathematical Structures in Computer Science.

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

[16]  E. J. Lemmon,et al.  Algebraic semantics for modal logics II , 1966, Journal of Symbolic Logic.

[17]  Marcelo P Fiore,et al.  Topology via Logic , 1999 .

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

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  Bart Jacobs,et al.  Reasonong about Classess in Object-Oriented Languages: Logical Models and Tools , 1998, ESOP.

[21]  Bart Jacobs,et al.  Structural Induction and Coinduction in a Fibrational Setting , 1998, Inf. Comput..

[22]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[23]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[24]  E. J. Lemmon,et al.  Algebraic semantics for modal logics I , 1966, Journal of Symbolic Logic (JSL).

[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]  Robert Goldblatt.  Duality for some categories of coalgebras , 2001 .

[27]  Martin Rößiger.  From modal logic to terminal coalgebras , 2001 .

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

[29]  M. Stone.  Applications of the theory of Boolean rings to general topology , 1937 .

[30]  Martin Rößiger.  Languages for Coalgebras on Datafunctors , 1999, CMCS.

[31]  R. Goldblatt.  Mathematics of modality , 1993 .

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

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