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On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables

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Abstract

Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non-commuting, then the product of any two aperiodic series is again aperiodic. We also show that if K is idempotent and the matrix monoids over K have a Burnside property (satisfied, e.g. by the tropical semiring), then the aperiodic and the star-free series coincide. This generalizes a classical result of Schützenberger (Inf. Control 4:245–270, 1961) for aperiodic regular languages and subsumes a result of Guaiana et al. (Theor. Comput. Sci. 97:301–311, 1992) on aperiodic trace languages.

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Authors and Affiliations

  1. Institut für Informatik, Universität Leipzig, Augustusplatz 10-11, 04109, Leipzig, Germany

    Manfred Droste

  2. LSV, CNRS UMR 8643 & ENS de Cachan, 61, Av. du Président Wilson, 94235, Cachan Cedex, France

    Paul Gastin

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  1. Manfred Droste
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  2. Paul Gastin
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Correspondence to Manfred Droste.

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This work partly supported by the DAAD-PROCOPE project Temporal and Quantitative Analysis of Distributed Systems.

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Droste, M., Gastin, P. On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables. Theory Comput Syst 42, 608–631 (2008). https://doi.org/10.1007/s00224-007-9064-z

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