Advertisement

Soft Computing

, Volume 22, Issue 4, pp 1103–1120 | Cite as

A generalized partition refinement algorithm, instantiated to language equivalence checking for weighted automata

  • Barbara König
  • Sebastian Küpper
Focus
First Online:

Abstract

We present a generic algorithm, generalizing partition refinement, for deciding behavioural equivalences for various types of transition systems. In order to achieve this generality, we work with coalgebra, which offers a general framework for modelling different types of state-based systems. The underlying idea of the algorithm is to work on the so-called final chain and to factor out redundant information. If the algorithm terminates, the result of the construction is a representative of the given coalgebra that is not necessarily unique and that allows to precisely answer questions about behavioural equivalence. We instantiate the algorithm to the particularly interesting case of weighted automata over semirings in order to obtain a procedure for checking language equivalence for a large number of semirings. We use fuzzy automata with weights from an l-monoid as a case study.

This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We would like to thank Alexandra Silva, Filippo Bonchi, Jacques Sakarovitch and Marcello Bonsangue for several interesting discussions on this topic. Furthermore, we are very grateful to Manfred Droste, Zoltán Ésik and the anonymous reviewers for their helpful comments on this paper. In addition, we would like to thank Christina Mika for her work on the implementation.

Compliance with ethical standards

Conflicts of interest

The authors confirm that there are no known conflicts of interest associated with this publication.

References

  1. Adámek J, Rosický J (1994) Locally presentable and accessible categories, volume 189 of London Mathematical Society Lecture Note Series. Cambridge University Press, CambridgeGoogle Scholar
  2. Adámek J, Koubek V (1995) On the greatest fixed point of a set functor. Theor Comput Sci 150:57–75MathSciNetCrossRefzbMATHGoogle Scholar
  3. Adámek J, Herrlich H, Strecker GE (1990) Abstract and concrete categories—the joy of cats. Wiley, New YorkzbMATHGoogle Scholar
  4. Adámek J, Bonchi F, Hülsbusch M, König B, Milius S, Silva A (2012) A coalgebraic perspective on minimization and determinization. In: Proceedings of FOSSACS ’12, LNCS/ARCoSS 7213. Springer, pp 58–73Google Scholar
  5. Almagor S, Boker U, Kupferman O (2011) What’s decidable about weighted automata? In: Proceedings of ATVA ’11, LNCS 6996. Springer, pp 482–491Google Scholar
  6. Baier C (1996) Polynomial time algorithms for testing probabilistic bisimulation and simulation. In: Proceedings of CAV ’96, LNCS 1102. Springer, pp 50–61Google Scholar
  7. Béal M-P, Lombardy S, Sakarovitch J (2005) On the equivalence of \(\mathbb{Z}\)-automata. In: Proceedings of ICALP ’05, LNCS 3580. Springer, pp 397–409Google Scholar
  8. Béal M-P, Lombardy S, Sakarovitch J (2006) Conjugacy and equivalence of weighted automata and functional transducers. In: Proceedings of CSR ’06, LNCS 3967. Springer, pp 58–69Google Scholar
  9. Berstel J, Reutenauer C (1988) Rational series and their languages. Springer, BerlinCrossRefzbMATHGoogle Scholar
  10. Bloom SL, Ésik Z (1993) Iteration theories: the equational logic of iterative processes. In: EATCS monographs on theoretical computer science. SpringerGoogle Scholar
  11. Bloom SL, Ésik Z (2009) Axiomatizing rational power series over natural numbers. Inf Comput 207(7):793–811MathSciNetCrossRefzbMATHGoogle Scholar
  12. Blyth TS, Janowitz MF (1972) Residuation theory. Pergamon Press, OxfordzbMATHGoogle Scholar
  13. Bonchi F, Petrisan D, Pous D, Rot J (2014) Coinduction up to in a fibrational setting. CoRR abs/1401.6675Google Scholar
  14. Bonsangue M, Milius S, Silva A (2013) Sound and complete axiomatizations of coalgebraic language equivalence. ACM Trans Comput Log 14(1):7. doi: 10.1145/2422085.2422092
  15. Boreale M (2009) Weighted bisimulation in linear algebraic form. In: Proceedings of CONCUR ’09, LNCS 5710. Springer, pp 163–177Google Scholar
  16. Cuninghame-Green RA (1979) Minimax algebra. Lecture notes in economics and mathematical systems. SpringerGoogle Scholar
  17. Droste M, Kuske D (2013) Weighted automata. In: Pin J-E (ed) Automata: from mathematics to applications. European Mathematical Society, Switzerland (to appear)Google Scholar
  18. Droste M, Kuich W, Vogler H (eds) (2009) Handbook of weighted automata. Springer, BerlinzbMATHGoogle Scholar
  19. Eisner J (2003) Simpler and more general minimization for weighted finite-state automata. In: Proceedings of HLT-NAACL ’03. Association for Computational Linguistics, vol 1, pp 64–71Google Scholar
  20. Ésik Z (2011) Multi-linear iterative k-\(\varSigma \)-semialgebras. ENTCS 276:159–170MathSciNetzbMATHGoogle Scholar
  21. Ésik Z, Kuich W (2001) A generalization of Kozen’s axiomatization of the equational theory of the regular sets. In: Words, semigroups, and transductions—Festschrift in Honor of Gabriel Thierrin, pp 99–114Google Scholar
  22. Ésik Z, Maletti A (2010) Simulation vs. equivalence. In: Proceedings of FCS ’10. CSREA Press, pp 119–124Google Scholar
  23. Ésik Z, Kuich W (2013) Free inductive k-semialgebras. J Log Algebr Program 82(3–4):111–122MathSciNetCrossRefzbMATHGoogle Scholar
  24. Ferrari G, Montanari U, Tuosto E (2005) Coalgebraic minimization of HD-automata for the \(\pi \)-calculus using polymorphic types. Theor Comput Sci 331(2–3):325–365MathSciNetCrossRefzbMATHGoogle Scholar
  25. Flouret M, Laugerotte É (1997) Noncommutative minimization algorithms. Inf Process Lett 64(3):123–126MathSciNetCrossRefzbMATHGoogle Scholar
  26. Hasuo I, Jacobs B, Sokolova A (2007) Generic trace semantics via coinduction. Log Methods Comput Sci 3(4:11):1–36MathSciNetzbMATHGoogle Scholar
  27. Hopcroft JE, Ullman J (1979) Introduction to automata theory, languages and computation. Addison Wesley, ReadingzbMATHGoogle Scholar
  28. Kiefer S, Murawski AS., Ouaknine J, Wachter B, Worrell J (2011) Language equivalence for probabilistic automata. In: Proceedings of CAV ’11, LNCS 6806. Springer, pp 526–540Google Scholar
  29. König B, Küpper S (2014) Generic partition refinement algorithms for coalgebras and an instantiation to weighted automata. In: Proceedings of TCS ’14, IFIP AICT, LNCS 8705. Springer, pp. 311–325Google Scholar
  30. Krob D (1991) Expressions rationnelles sur un anneau. In: Topics in invariant theory: Séminaire d’Algèbre P. Dubreil et M.-P. Malliavin 1989–1990 (40ème Année). Springer, pp 215–243Google Scholar
  31. Krob D (1994) The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Int J Algebra Comput 4(3):405–425MathSciNetCrossRefzbMATHGoogle Scholar
  32. Larsen KG, Skou A (1989) Bisimulation through probabilistic testing (preliminary report). In: Proceedings of POPL ’89. ACM, pp 344–352Google Scholar
  33. Mika C (2015) Ein generisches Werkzeug für Sprachäquivalenz bei gewichteten Automaten. Master’s thesis, Universität Duisburg-EssenGoogle Scholar
  34. Mohri M (1997) Finite-state transducers in language and speech processing. Comput Linguist 23:269–311MathSciNetGoogle Scholar
  35. Mohri M (2009) Weighted automata algorithms. In: Droste M, Kuich W, Vogler H (eds) Handbook of weighted automata. Springer, Berlin, pp 213–254CrossRefGoogle Scholar
  36. Morisaki M, Sakai K (1980) A complete axiom system for rational sets with multiplicity. Theor Comput Sci 11(1):79–92MathSciNetCrossRefzbMATHGoogle Scholar
  37. Rahonis G (2009) Fuzzy languages. In: Droste M, Kuich W, Vogler H (eds) Handbook of weighted automata. Springer, Berlin, pp 481–517CrossRefGoogle Scholar
  38. Rot J, Bonchi F, Bonsangue M, Pous D, Rutten J, Silva A (2015) Enhanced coalgebraic bisimulation. Math Struct Comput Sci. doi: 10.1017/S0960129515000523
  39. Rutten JJMM (2000) Universal coalgebra: a theory of systems. Theor Comput Sci 249:3–80MathSciNetCrossRefzbMATHGoogle Scholar
  40. Sakarovitch J (2009) Rational and recognisable power series. In: Droste M, Kuich W, Vogler H (eds) Handbook of weighted automata. Springer, Berlin, pp 105–174Google Scholar
  41. Schützenberger M-P (1961) On the definition of a family of automata. Inf Control 4(2–3):245–270MathSciNetCrossRefzbMATHGoogle Scholar
  42. Staton S (2009) Relating coalgebraic notions of bisimulation. In: Proceedings of CALCO ’09, LNCS 5728. Springer, pp 191–205Google Scholar
  43. Urabe N, Hasuo I (2014) Generic forward and backward simulations III: quantitative simulations by matrices. In: Proceedings of CONCUR ’14, LNCS/ARCoSS 8704. Springer, pp 451–466Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Universität Duisburg-EssenDuisburgGermany

Personalised recommendations