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

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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.

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Notes

  1. Here \(X+Y\) denotes the disjoint union of two sets X, Y and 1 stands for the singleton set \(\{\bullet \}\).

  2. \(\#_a(w)\) stands for the number of occurrences of the letter a in the word w.

  3. An object 1 is final, whenever there exists, for each object X, a unique arrow \(f:X\rightarrow 1\).

  4. A semiring is locally finite if each finitely generated subsemiring is finite.

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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.

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  1. Universität Duisburg-Essen, Duisburg, Germany

    Barbara König & Sebastian Küpper

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  1. Barbara König
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  2. Sebastian Küpper
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Correspondence to Sebastian Küpper.

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

Additional information

Communicated by M. Droste, Z. Esik, K. Larsen.

This article is based on our previous work in König and Küpper (2014); however, it is extended by fully working out all proofs and gives a more detailed description on the instantiation of the algorithm for weighted automata. Research was partially supported by DFG Project BEMEGA.

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König, B., Küpper, S. A generalized partition refinement algorithm, instantiated to language equivalence checking for weighted automata. Soft Comput 22, 1103–1120 (2018). https://doi.org/10.1007/s00500-016-2363-z

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