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A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

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Algebra, Meaning, and Computation

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4060))

Abstract

This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten’s description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a so-called bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic structure of regular expressions and languages. This culminates in a reformulated proof via finality of Kozen’s completeness result. It yields a complete axiomatisation of observational equivalence (bisimilarity) on regular expressions. We suggest that this situation is paradigmatic for (theoretical) computer science as the study of “generated behaviour”.

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References

  1. Arbib, M.A., Manes, E.G.: Foundations of system theory: Decomposable systems. Automatica 10, 285–302 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arbib, M.A., Manes, E.G.: Algebraic Approaches to Program Semantics. In: Texts and Monogr. Comp. Sci., Springer, Heidelberg (1986)

    Google Scholar 

  3. de Bakker, J.W., Vink, E.: Control Flow Semantics. MIT Press, Cambridge (1996)

    MATH  Google Scholar 

  4. Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Berlin (1985), Revised and corrected version available from www.cwru.edu/artsci/math/wells/pub/ttt.html

    MATH  Google Scholar 

  5. Bartels, F.: On generalised coinduction and probabilistic specification formats. Distributive laws in coalgebraic modelling. PhD thesis, Free Univ. Amsterdam (2004)

    Google Scholar 

  6. Beck, J.: Distributive laws. In: Eckman, B. (ed.) Seminar on Triples and Categorical Homolgy Theory. Lect. Notes Math., vol. 80, pp. 119–140. Springer, Berlin (1969)

    Chapter  Google Scholar 

  7. Bloom, B., Istrail, S., Meyer, A.R.: Bisimulation can’t be traced. Journ. ACM 42(1), 232–268 (1988)

    Article  MathSciNet  Google Scholar 

  8. Brzozowski, J.A.: Derivatives of regular expressions. Journ. ACM 11(4), 481–494 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  9. Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)

    MATH  Google Scholar 

  10. Fokkink, W.: On the completeness of the equations for the Kleene star in bisimulation. In: Nivat, M., Wirsing, M. (eds.) AMAST 1996. LNCS, vol. 1101, pp. 180–194. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  11. Goguen, J.A.: Minimal realization of machines in closed categories. Bull. Amer. Math. Soc. 78(5), 777–783 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goguen, J.A.: Realization is universal. Math. Syst. Theor. 6(4), 359–374 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goguen, J.A.: Discrete-time machines in closed monoidal categories. I. Journ. Comp. Syst. Sci. 10, 1–43 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  14. Groote, J.F., Vaandrager, F.: Structured operational semantics and bisimulation as a congruence. Inf. Comp. 100(2), 202–260 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Inf. Comp. 145, 107–152 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jacobs, B.: Objects and classes, co-algebraically. In: Freitag, B., Jones, C.B., Lengauer, C., Schek, H.-J. (eds.) Object-Orientation with Parallelism and Persistence, pp. 83–103. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

  17. Jacobs, B.: Exercises in coalgebraic specification. In: Blackhouse, R., Crole, R.L., Gibbons, J. (eds.) Algebraic and Coalgebraic Methods in the Mathematics of Program Construction. LNCS, vol. 2297, pp. 237–280. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  18. Jacobs, B.: Distributive laws for the coinductive solution of recursive equations. Inf. & Comp. 204(4), 561–587 (2006), Earlier version in number 106 in Elect. Notes in Theor. Comp. Sci.

    Article  MATH  MathSciNet  Google Scholar 

  19. Johnstone, P.T.: Adjoint lifting theorems for categories of algebras. Bull. London Math. Soc. 7, 294–297 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kick, M.: Bialgebraic modelling of timed processes. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 525–536. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  21. Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies. Annals of Mathematics Studies, vol. 34, pp. 3–41. Princeton University Press, Princeton (1956)

    Google Scholar 

  22. Koushik, S., Rosu, G.: Generating optimal monitors for extended regular expressions. In: Runtime Verification (RV 2003). Elect. Notes in Theor. Comp. Sci., vol. 89(2), Elsevier, Amsterdam (2003)

    Google Scholar 

  23. Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Inf. Comp. 110(2), 366–390 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kozen, D.: Myhill-nerode relations on automatic systems and the completeness of Kleene algebra. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 27–38. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  25. Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)

    MATH  Google Scholar 

  26. Lenisa, M., Power, J., Watanabe, H.: Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads. In: Reichel, H. (ed.) Coalgebraic Methods in Computer Science. Elect. Notes in Theor. Comp. Sci. vol. 33, Elsevier, Amsterdam (2000)

    Google Scholar 

  27. Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theor. Comp. Sci. 54(2-3), 267–276 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  28. Perrin, D.: Finite automata. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 1–55. Elsevier/MIT Press (1990)

    Google Scholar 

  29. Reichel, H.: An approach to object semantics based on terminal co-algebras. Math. Struct. Comp. Sci. 5, 129–152 (1995)

    MATH  MathSciNet  Google Scholar 

  30. Rutten, J.: Automata and coinduction (an exercise in coalgebra). In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  31. Rutten, J.: Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theor. Comp. Sci. 308, 1–53 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rutten, J., Turi, D.: Initial algebra and final coalgebra semantics for concurrency. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) REX 1993. LNCS, vol. 803, pp. 530–582. Springer, Heidelberg (1994)

    Google Scholar 

  33. Rutten, J.J.M.M.: Automata, power series, and coinduction: Taking input derivatives seriously (extended abstract). In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 645–654. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  34. Turi, D.: Functorial operational semantics and its denotational dual. PhD thesis, Free Univ. Amsterdam (1996)

    Google Scholar 

  35. Turi, D., Plotkin, G.: Towards a mathematical operational semantics. In: Logic in Computer Science, pp. 280–291. IEEE, Computer Science Press (1997)

    Google Scholar 

  36. Uustalu, T., Vene, V., Pardo, A.: Recursion schemes from comonads. Nordic Journ. Comput. 8(3), 366–390 (2001)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Computing and Information Sciences, Radboud University Nijmegen, P.O. Box 9010, 6500 GL, Nijmegen, The Netherlands

    Bart Jacobs

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  1. Bart Jacobs
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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology (JAIST),  

    Kokichi Futatsugi

  2. LIX, Projet INRIA TypiCal, École Polytechnique and CNRS, 91400, Palaiseau, France

    Jean-Pierre Jouannaud

  3. CS Dept., University of Illinois at Urbana-Champaign, USA

    José Meseguer

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Jacobs, B. (2006). A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages. In: Futatsugi, K., Jouannaud, JP., Meseguer, J. (eds) Algebra, Meaning, and Computation. Lecture Notes in Computer Science, vol 4060. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780274_20

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  • DOI: https://doi.org/10.1007/11780274_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35462-8

  • Online ISBN: 978-3-540-35464-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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