# Aperiodic Weighted Automata and Weighted First-Order Logic

arXiv: https://arxiv.org/abs/1902.08149 · HTML: https://mlsanity.com/paper/1902.08149/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2019.

## Built on by (most cited first, 1 of 1)

- [Probability Distributions Computed by Autoregressive Transformers](https://mlsanity.com/paper/2510.27118/) (2025)

## Abstract (arXiv)

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.