Automaton-based Characterisations of First Order Logic over Infinite Trees

2026-04-29

AI Summary

This work addresses the long-standing lack of a clear automata-theoretic characterization for first-order logic (FO) over unordered infinite trees. By introducing two classes of hesitant tree automata, it establishes precise correspondences with the branching-time temporal logics PolPCTL and CTLsf, thereby providing the first unified automata-based semantic characterization of FO’s expressive power on infinite trees. The main contributions include proving the exact equivalence FO ≡ PolPCTL ∩ CTLsf, proposing a natural normal form called PolCTLs, and uncovering a fundamental limitation of FO: along every path, it can express only safety or co-safety properties. This result delineates the structural boundaries of FO’s expressiveness in the setting of infinite unordered trees.

Abstract

We study the expressive power of First-Order Logic (\FO) over (unordered) infinite trees, with the aim of identifying robust characterisations in terms of branching-time specification formalisms. While such correspondences are well understood in the linear-time setting, the branching-time case presents well-known structural challenges. To this end, we introduce two classes of hesitant tree automata and show that they capture precisely the expressive power of two branching-time temporal logics, namely \PolPCTL and \CTLsf, both of which have been previously shown to be equivalent to \FO over infinite trees. These results provide uniform automata-theoretic characterisations and yield a natural normal form for the latter in terms of a new fragment of \CTLs called \PolCTLs. As a consequence, we identify a fundamental limitation of \FO in this setting: along each branch, it can express only properties that are either safety or co-safety, thereby revealing a sharp expressive boundary for first-order definability over infinite trees.

Problem

• First-Order Logic
• infinite trees
• branching-time
• expressive power
• automata-theoretic characterisation

Innovation

• hesitant tree automata
• first-order logic
• branching-time temporal logic
• PolCTLs
• expressive boundary