An Automata Theoretic Characterization of Weighted First-Order Logic

TL;DR

This paper provides an automata-theoretic characterization of the full weighted first-order logic, extending classical algebraic results to the quantitative setting with polynomially ambiguous automata.

Contribution

It offers a new characterization linking extended weighted automata and full weighted first-order logic under polynomial ambiguity constraints.

Findings

Characterizes the fragment of extended weighted automata matching full weighted first-order logic.

Extends classical algebraic characterizations to the weighted, quantitative setting.

Identifies conditions under which automata recognize exactly the logic.

Abstract

Since the 1970s with the work of McNaughton, Papert and Sch\"utzenberger, a regular language is known to be definable in the first-order logic if and only if its syntactic monoid is aperiodic. This algebraic characterisation of a fundamental logical fragment has been extended in the quantitative case by Droste and Gastin, dealing with polynomially ambiguous weighted automata and a restricted fragment of weighted first-order logic. In the quantitative setting, the full weighted first-order logic (without the restriction that Droste and Gastin use, about the quantifier alternation) is more powerful than weighted automata, and extensions of the automata with two-way navigation, and pebbles or nested capabilities have been introduced to deal with it. In this work, we characterise the fragment of these extended weighted automata that recognise exactly the full weighted first-order logic, under the condition that automata are polynomially ambiguous.