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

Authors:
Manfred Droste;Paul Gastin
Affiliations:
Universität Leipzig, Institut für Informatik, Augustusplatz 10-11, 04109, Leipzig, Germany;CNRS UMR 8643 & ENS de Cachan, LSV, 61, Av. du Préésident Wilson, 94235, Cachan Cedex, France
Venue:
Theory of Computing Systems
Year:
2008

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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.