# Aperiodic Weighted Automata and Weighted First-Order Logic

## TL;DR

This paper extends the classical equivalence between first-order definable and aperiodic languages to a quantitative setting using weighted automata and weighted first-order logic, covering various weight structures.

## Contribution

It introduces a weighted first-order logic and proves its expressive equivalence with aperiodic polynomially ambiguous weighted automata, generalizing classical results.

## Key findings

- Weighted first-order logic matches aperiodic polynomially ambiguous weighted automata.
- Equivalence also holds for weighted sublogics and unambiguous automata.
- Results apply to general weight structures, including semirings and lattices.

## Abstract

By fundamental results of Sch\"utzenberger, 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.