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Causality in Bounded Petri Nets is MSO Definable

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Logic, Language, Information, and Computation (WoLLIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9803))

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Abstract

In this work we show that the causal behaviour of any bounded Petri net is definable in monadic second order (MSO) logic. Our proof relies in a definability vs recognizability result for DAGs whose edges and vertices can be covered by a constant number of paths. Our notion of recognizability is defined in terms of saturated slice automata, a formalism for the specification of infinite families of graphs. We show that a family \(\mathfrak {G}\) of k-coverable DAGs is recognizable by a saturated slice automaton if and only if \(\mathfrak {G}\) is definable in monadic second order logic. This result generalizes Büchi’s theorem from the context of strings, to the context of k-coverable DAGs.

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Notes

  1. 1.

    Proposals of proofs of Courcelle’s conjecture for graphs of constant pathwidth and constant treewidth have been provided in [20] and [23] respectively. Nevertheless, both proposed proofs contained substantial gaps, and Courcelle’s conjecture is regarded to be open in both cases [8].

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Acknowledgements

The author gratefully acknowledges financial support from the European Research Council, ERC grant agreement 339691, within the context of the project Feasibility, Logic and Randomness (FEALORA).

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Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Zitná 25, 115 67, Praha 1, Czech Republic

    Mateus de Oliveira Oliveira

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  1. Mateus de Oliveira Oliveira
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Correspondence to Mateus de Oliveira Oliveira .

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Editors and Affiliations

  1. Dept of Mathematics & Statistics, Univ of Helsinki Dept of Mathematics & Statistics, Helsinki, Finland

    Jouko Väänänen

  2. Department of Mathematics and Statistics, University of Helsinki , Helsinki, Finland

    Åsa Hirvonen

  3. Centro de Informática , Recife, Pernambuco, Brazil

    Ruy de Queiroz

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de Oliveira Oliveira, M. (2016). Causality in Bounded Petri Nets is MSO Definable. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_13

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  • DOI: https://doi.org/10.1007/978-3-662-52921-8_13

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