Reversible Weighted Automata over Finite Rings and Monoids with Commuting Idempotents
Peter Kostolányi, Andrej Ravinger
TL;DR
This work characterizes languages realized by reversible weighted automata over nontrivial locally finite commutative rings, proving that the support of any realised series is a rational language whose syntactic monoid has commuting idempotents ($M_L \in \mathbf{ECom}$). For the two-element field $\mathbb{F}_2$, reversible weighted automata capture exactly the Boolean closure of the usual reversible languages over the Boolean semiring, and this extends to general rings $R$ via $\mathrm{RevL}(R) = \mathrm{RevL}(\mathbb{F}_2)$. The authors provide an effective decision procedure for reversibility of rational series over effective locally finite commutative rings by reducing membership in $\mathbf{ECom}$ to the automaton's transition monoid. Collectively, these results yield a new automata-theoretic characterization of the pseudovariety $\mathbf{ECom}$ and clarify the interplay between reversibility, Boolean closure, and algebraic language theory.
Abstract
Reversible weighted automata are introduced and considered in a specific setting where the weights are taken from a nontrivial locally finite commutative ring such as a finite field. It is shown that the supports of series realised by such automata are precisely the rational languages such that the idempotents in their syntactic monoids commute. In particular, this is true for reversible weighted automata over the finite field $\mathbb{F}_2$, where the realised series can be directly identified with such languages. A new automata-theoretic characterisation is thus obtained for the variety of rational languages corresponding to the pseudovariety of finite monoids $\mathbf{ECom}$, which also forms the Boolean closure of the reversible languages in the sense of J.-É. Pin. The problem of determining whether a rational series over a locally finite commutative ring can be realised by a reversible weighted automaton is decidable as a consequence.