Many-sorted coalgebraic modal logic : a model-theoretic study

Jacobs, Bart

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 1, pp. 31-59.

This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, 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 (categorical) 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 final objects. They will be used to (re)prove completeness results, and Hennessey-Milner style characterisation results for the modal logic, first established by Rößiger.

[1] A. Baltag, A logic for coalgebraic simulation, edited by H. Reichel, Coalgebraic Methods in Computer Science. Elsevier, Amsterdam, Electon. Notes Theor. Comput. Sci. 33 (2000). | MR | Zbl

[2] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Math. Textbooks. Cambridge Univ. Press (1990). | MR | Zbl

[3] R. Goldblatt, Mathematics of Modality. Stanford, CSLI Lecture Notes 43 (1993). | MR | Zbl

[4] R. Goldblatt, Duality for some categories of coalgebras. Algebra Universalis (to appear). | MR | Zbl

[5] R. Goldblatt, What is the coalgebraic analogue of Birkhoff's variety theorem? Theoret. Comput. Sci. (to appear). | MR | Zbl

[6] M. Hennessy and R. Milner, On observing non-determinism and concurrency, edited by J.W. de Bakker and J. van Leeuwen, Mathematical Foundations of Computer Science. Springer, Berlin, Lecture Notes in Comput. Sci. 85 (1980) 299-309. | MR | Zbl

[7] U. Hensel, M. Huisman, B. Jacobs, and H. Tews, Reasoning about classes in object-oriented languages: Logical models and tools, edited by Ch. Hankin, European Symposium on Programming. Springer, Berlin, Lecture Notes in Comput. Sci. 1381 (1998) 105-121.

[8] C. Hermida and B. Jacobs, Structural induction and coinduction in a fibrational setting. Inform. and Comput. 145 (1998) 107-152. | MR | Zbl

[9] B. Jacobs, Objects and classes, co-algebraically, edited by B. Freitag, C.B. Jones, C. Lengauer and H.-J. Schek, Object-Orientation with Parallelism and Persistence. Kluwer Acad. Publ. (1996) 83-103.

[10] B. Jacobs, Categorical Logic and Type Theory. North Holland, Amsterdam (1999). | MR | Zbl

[11] B. Jacobs, The temporal logic of coalgebras via Galois algebras, Technical Report CSI-R9906. Comput. Sci. Inst. Univ. of Nijmegen, Math. Structures Comput. Sci. (to appear). | MR | Zbl

[12] B. Jacobs, Towards a duality result in coalgebraic modal logic, edited by H. Reichel, Coalgebraic Methods in Computer Science. Elsevier, Amsterdam, Electon. Notes Theor. Comput. Sci. 33 (2000). | MR | Zbl

[13] B. Jacobs and J. Rutten, A tutorial on (co)algebras and (co)induction. EATCS Bull. 62 (1997) 222-259. | Zbl

[14] B. Jacobs, J. Van Den Berg, M. Huisman, M. Van Berkum, U. Hensel and H. Tews, Reasoning about classes in Java (preliminary report), in Object-Oriented Programming, Systems, Languages and Applications (OOPSLA). ACM Press (1998) 329-340.

[15] P.T. Johnstone, Stone Spaces. Cambridge Univ. Press, Cambridge Stud. Adv. Math. 3 (1982). | MR | Zbl

[16] B. Jónsson and A. Tarski, Boolean algebras with operators I. Amer. J. Math. 73 (1951) 891-939. | MR | Zbl

[17] A. Kurz, Specifying coalgebras with modal logic. Theoret. Comput. Sci. (to appear). | MR | Zbl

[18] F.W. Lawvere, Functorial semantics. Proc. Nat. Acad. Sci. U.S.A. 50 (1963) 869-872. | MR | Zbl

[19] E.J. Lemmon, Algebraic semantics for modal logics II. J. Symbolic Logic 31 (1966) 191-218. | MR | Zbl

[20] L.S. Moss, Coalgebraic logic. Ann. Pure Appl. Logic 96 (1999) 277-317; Erratum in Ann. Pure Appl. Logic 99 (1999) 241-259. | MR | Zbl

[21] H. Reichel, An approach to object semantics based on terminal co-algebras. Math. Structures Comput. Sci. 5 (1995) 129-152. | MR | Zbl

[22] M. Rößiger, Languages for coalgebras on datafunctors, edited by B. Jacobs and J. Rutten, Coalgebraic Methods in Computer Science. Elsevier, Amsterdam, Electron. Notes Theor. Comput. Sci. 19 (1999). | MR | Zbl

[23] M. Rößiger, Coalgebras and modal logic, edited by H. Reichel, Coalgebraic Methods in Computer Science. Elsevier, Amsterdam, Electron. Notes Theor. Comput. Sci. 33 (2000). | MR | Zbl

[24] M. Rößiger, From modal logic to terminal coalgebras. Theoret. Comput. Sci. (to appear). | MR | Zbl

[25] J. Rothe, H. Tews and B. Jacobs, The coalgebraic class specification language CCSL. J. Universal Comput. Sci. 7 (2001). | MR | Zbl

[26] J. Rutten, Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249 (2000) 3-80. | MR | Zbl

[27] M.H. Stone, The theory of representations for Boolean algebra. Trans. Amer. Math. Soc. 40 (1936) 37-111. | MR | Zbl

[28] M.H. Stone, Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41 (1937) 375-481. | JFM | MR

[29] D. Turi and J. Rutten, On the foundations of final semantics: non-standard sets, metric spaces and partial orders. Math. Structures Comput. Sci. 8 (1998) 481-540. | MR | Zbl

[30] Y. Venema, Points, lines and diamonds: a two-sorted modal logic for projective planes. J. Logic Comput. 9 (1999) 601-621. | MR | Zbl

[31] S. Vickers, Topology Via Logic. Cambridge Univ. Press, Tracts Theor. Comput. Sci. 5 (1989). | MR | Zbl