Aperiodic Weighted Automata and Weighted First-Order Logic

Abstract

By fundamental results of Schützenberger, McNaughton and Papert from the 1970s, the classes of first-order definable and aperiodic languages coincide. Here, we extend this equivalence to a quantitative setting. For this, weighted automata form a general and widely studied model. We define a suitable notion of a weighted first-order logic. Then we show that this weighted first-order logic and aperiodic polynomially ambiguous weighted automata have the same expressive power. Moreover, we obtain such equivalence results for suitable weighted sublogics and finitely ambiguous or unambiguous aperiodic weighted automata. Our results hold for general weight structures, including all semirings, average computations of costs, bounded lattices, and others.

ℤ-polyregular functions

“…For this class of functions the aperiodic variant ZSF has been shown to be decidable [12]. Beware that several non-equivalent notions of aperiodicity coexist for rational series [40,19,12], and that we only refer to the one applying to rational series of polynomial growths introduced in [12]. This approach through rational series has several limitations.…”