Set Automata and Limits of Decidability of Two-Variable Logic on Data Words

2026-05-09

AI Summary

This study investigates the decidability boundary of two-variable logic with data values extended by protected regular binary predicates. By constructing a set automaton model and establishing its expressive equivalence with ordered quasi-regular set automata, the work leverages algebraic language theory, multi-counter automata, and model-checking techniques to identify an ordered quasi-regular subclass for which the emptiness problem is decidable. The main contribution lies in precisely characterizing the class of idempotent monoids that render the satisfiability problem of the extended logic decidable. Specifically, it is shown that decidability holds when the underlying monoid is bi-ideal linearly ordered—such as in the case of linear bands—thereby delineating a sharp semantic condition under which the enriched logical framework remains tractable.

Abstract

We extend the two-variable logic on data words with guarded regular binary predicates of the form $\widetilde{L}(x,y)$ that is true if positions $x$ and $y$ are in the same class and the factor strictly between $x$ and $y$ is in the regular language $L$. We characterise the class of monoids for which the extension of the two-variable logic with guarded predicates recognised by the monoid is decidable, namely the class of idempotent monoids whose two-sided ideals are linearly ordered. For this, we introduce an automata formalism, set automata, that is equivalent to the class automata of Bojańczyk and Lasota and thus has an undecidable emptiness problem. We identify a subclass of set automata called ordered quasi-normal set automata that has a decidable emptiness problem by reduction to the emptiness problem of ordered multicounter automata. We show that the two-variable logic extended with guarded regular predicates recognised by a semigroup $S$ is expressively equivalent to a quasi-normal set automaton with the semigroup of transformations $S$. In particular, if $S$ is a linear band monoid then the resulting automaton is ordered, and the decidability result follows.

Problem

• two-variable logic
• data words
• guarded predicates
• decidability
• monoids

Innovation

• set automata
• two-variable logic
• data words
• decidability
• ordered multicounter automata