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

Manfred Droste, Paul Gastin
2007 Theory of Computing Systems  
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
more » ... 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 (1961) for aperiodic regular languages and subsumes a result of Guaiana, Restivo and Salemi (1992) on aperiodic trace languages.
doi:10.1007/s00224-007-9064-z fatcat:v3hrgjung5bc3i5z7gtwouv2xm