# Expressivity of Coalgebraic Modal Logic: The Limits and Beyond

• Lutz Schröder
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3441)

## Abstract

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.

### References

1. 1.
Barr, M.: Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114, 299–315 (1993)
2. 2.
Bartels, F., Sokolova, A., de Vink, E.: A hierarchy of probabilistic system types. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 82. Elsevier, Amsterdam (2003)Google Scholar
3. 3.
Chellas, B.: Modal logic, Cambridge (1980)Google Scholar
4. 4.
Cîrstea, C.: A compositional approach to defining logics for coalgebras. Theoret. Comput. Sci. 327, 45–69 (2004)
5. 5.
Cîrstea, C., Pattinson, D.: Modular construction of modal logics. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 258–275. Springer, Heidelberg (2004)
6. 6.
D’Agostino, G., Visser, A.: Finality regained: A coalgebraic study of Scott-sets and multisets. Arch. Math. Logic 41, 267–298 (2002)
7. 7.
Hansen, H.H., Kupke, C.: A coalgebraic perspective on monotone modal logic. In: Adámek, J., Milius, S. (eds.) Coalgebraic Methods in Computer Science. ENTCS, vol. 106, pp. 121–143. Elsevier, Amsterdam (2004)Google Scholar
8. 8.
Hennessy, M., Milner, R.: Algebraic laws for non-determinism and concurrency. J. ACM 32, 137–161 (1985)
9. 9.
Jacobs, B.: Towards a duality result in the modal logic of coalgebras. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 33. Elsevier, Amsterdam (2000)Google Scholar
10. 10.
Jónnson, B., Tarski, A.: Boolean algebras with operators I. Amer. J. Math. 73, 891–939 (1951)
11. 11.
Klin, B.: A coalgebraic approach to process equivalence and a coinduction principle for traces. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 106, pp. 201–218. Elsevier, Amsterdam (2004)Google Scholar
12. 12.
Kurz, A.: Logics for coalgebras and applications to computer science, Ph.D. thesis, Universität München (2000)Google Scholar
13. 13.
Kurz, A.: Specifying coalgebras with modal logic. Theoret. Comput. Sci. 260, 119–138 (2001)Google Scholar
14. 14.
Kurz, A.: Logics admitting final semantics. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 238–249. Springer, Heidelberg (2002)
15. 15.
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inform. Comput. 94, 1–28 (1991)
16. 16.
Moss, L.: Coalgebraic logic. Ann. Pure Appl. Logic 96, 277–317 (1999)
17. 17.
Mossakowski, T., Schröder, L., Roggenbach, M., Reichel, H.: Algebraic-co-algebraic specification in CoCASL. J. Logic Algebraic Programming (to appear)Google Scholar
18. 18.
Pattinson, D.: Expressivity results in the modal logic of coalgebras, Ph.D. thesis, Universität München (2001)Google Scholar
19. 19.
Pattinson, D.: Semantical principles in the modal logic of coalgebras. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 514–526. Springer, Heidelberg (2001)
20. 20.
Pattinson, D.: Expressive logics for coalgebras via terminal sequence induction. Notre Dame J. Formal Logic 45, 19–33 (2004)
21. 21.
Power, J., Watanabe, H.: An axiomatics for categories of coalgebras. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 11. Elsevier, Amsterdam (2000)Google Scholar
22. 22.
Rößiger, M.: Coalgebras and modal logic. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 33. Elsevier, Amsterdam (2000)Google Scholar
23. 23.
Rothe, J., Tews, H., Jacobs, B.: The Coalgebraic Class Specification Language CCSL. J. Universal Comput. Sci. 7, 175–193 (2001)
24. 24.
Rutten, J.: Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249, 3–80 (2000)
25. 25.
Siekmann, J., Szabo, P.: A noetherian and confluent rewrite system for idempotent semigroups. Semigroup Forum 25, 83–110 (1982)