A Coalgebraic Perspective on Minimization and Determinization
Abstract
Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this paper, we use the coalgebraic view on systems to derive, in a uniform way, abstract procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization. First, we show that for coalgebras in categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. Then, we consider coalgebras in categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. We will apply our theory to the following examples: conditional transition systems and (non-deterministic) automata.
Keywords
Transition System Label Transition System Unique Morphism Input Alphabet Concrete CategoryReferences
- 1.Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolin. 15, 589–602 (1974)MATHGoogle Scholar
- 2.Adámek, J., Bonchi, F., Hülsbusch, M., König, B., Milius, S., Silva, A.: A coalgebraic perspective on minimization and determinization (extended version), http://alexandrasilva.org/files/fossacs12-extended.pdf
- 3.Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories – The Joy of Cats. Wiley (1990)Google Scholar
- 4.Adámek, J., Koubek, V.: On the greatest fixed point of a set functor. TCS 150, 57–75 (1995)MATHCrossRefGoogle Scholar
- 5.Adámek, J., Milius, S., Moss, L.S., Sousa, L.: Well-pointed coalgebras. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, Springer, Heidelberg (2012)Google Scholar
- 6.Boreale, M.: Weighted Bisimulation in Linear Algebraic Form. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 163–177. Springer, Heidelberg (2009)CrossRefGoogle Scholar
- 7.Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. Mathematical Theory of Automata 12(6), 529–561 (1962)Google Scholar
- 8.Brzozowski, J., Tamm, H.: Theory of Átomata. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 105–116. Springer, Heidelberg (2011)CrossRefGoogle Scholar
- 9.Ferrari, G.L., Montanari, U., Tuosto, E.: Coalgebraic minimization of HD-automata for the pi-calculus using polymorphic types. TCS 331(2-3), 325–365 (2005)MathSciNetMATHCrossRefGoogle Scholar
- 10.Fitting, M.: Bisimulations and boolean vectors. In: Advances in Modal Logic, vol. 4, pp. 1–29. World Scientific Publishing (2002)Google Scholar
- 11.Gumm, H.P.: From T-Coalgebras to Filter Structures and Transition Systems. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 194–212. Springer, Heidelberg (2005)CrossRefGoogle Scholar
- 12.Gumm, H.P.: On minimal coalgebras. Applied Categorical Structures 16, 313–332 (2008)MathSciNetMATHCrossRefGoogle Scholar
- 13.Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace semantics via coinduction. LMCS 3(4:11), 1–36 (2007)MathSciNetGoogle Scholar
- 14.Hennessy, M., Lin, H.: Symbolic bisimulations. TCS 138(2), 353–389 (1995)MathSciNetMATHCrossRefGoogle Scholar
- 15.Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Information and Computation 145, 107–152 (1998)MathSciNetMATHCrossRefGoogle Scholar
- 16.Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 3rd edn. Wesley (2006)Google Scholar
- 17.Kurz, A.: Logics for Coalgebras and Applications to Computer Science. PhD thesis, Ludwigs-Maximilians-Universität München (2000)Google Scholar
- 18.Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)MATHGoogle Scholar
- 19.Mulry, P.S.: Lifting Theorems for Kleisli Categories. In: Main, M.G., Melton, A.C., Mislove, M.W., Schmidt, D., Brookes, S.D. (eds.) MFPS 1993. LNCS, vol. 802, pp. 304–319. Springer, Heidelberg (1994)CrossRefGoogle Scholar
- 20.Panangaden, P.: Duality in probabilistic automata Slides (May 19, 2011), http://www.cs.mcgill.ca/~prakash/Talks/duality_talk.pdf
- 21.Power, J., Turi, D.: A coalgebraic foundation for linear time semantics. In: Proc. of CTCS 1999. ENTCS, vol. 29, pp. 259–274 (1999)Google Scholar
- 22.Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. TCS 249, 3–80 (2000)MathSciNetMATHCrossRefGoogle Scholar
- 23.Staton, S.: Relating coalgebraic notions of bisimulation. LMCS 7(1) (2011)Google Scholar
- 24.Worrell, J.: On the final sequence of a finitary set functor. TCS 338(1-3), 184–199 (2005)MathSciNetMATHCrossRefGoogle Scholar