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

Abstract

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 (1961) for aperiodic regular languages and subsumes a result of Guaiana, Restivo and Salemi (1992) on aperiodic trace languages.

Weighted automata and weighted logics with discounting

“…Proposition 4 (Cf. [13]). The class of all recognizable step functions over A and K is closed under sum, scalar product, and Hadamard products.…”

Weighted Automata and Weighted Logics

“…As a consequence of this and of corresponding decidability results given in the full version of [6] for recognizable series over locally finite semirings, we immediately obtain: …”

“…In fact, more generally, any distributive lattice (L, ∨, ∧, 0, 1) with smallest element 0 and largest element 1 is a locally finite semiring, cf. [6] for further basic properties. Examples of infinite but locally finite fields are provided by the algebraic closures of the finite fields Z/pZ for any prime p. We can show (see [7]): Theorem 6.1.…”

“…As is well-known, the first-order definable languages are precisely the starfree languages which in turn coincide with the the aperiodic ones [23,18]. Aperiodic and starfree formal power series were introduced and investigated in [6]. Recall that a monoid M is said to be aperiodic, if there exists some m ≥ 0 such that x m = x m+1 for all x ∈ M .…”

“…Second, if the semiring K is locally finite, we obtain that the semantics of all sentences of our full weighted MSO-logic are representable by weighted automata. Locally finite semirings were investigated in [6]; they form a large class of semirings including e.g. all finite semirings, the max-min-semiring employed for capacity problems of networks, and all Boolean algebras.…”