An Automaton-based Characterisation of First-Order Logic over Infinite Trees

2025-09-17

AI Summary

This paper investigates the expressive power of first-order logic (FO) over infinite trees and its relationship with branching-time temporal logics such as CTL and CTL*. Addressing FO’s inherent limitations in characterizing path properties on trees, we establish, for the first time, an automata-theoretic semantic characterization of FO over infinite trees. Specifically, we introduce two classes of hesitant tree automata that precisely capture the expressive power of polcCTLp and cCTL*[f], respectively, and prove both are expressively equivalent to FO. Our results reveal that FO over infinite trees can express only those tree path properties that are either safety or co-safety, exposing a fundamental limitation in its ability to describe general branching-time properties. This work establishes a novel connection between FO and tree automata, yields natural normal forms for temporal logics, and provides new technical tools for model checking and logical expressiveness analysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we study First Order Logic (FO) over (unordered) infinite trees and its connection with branching-time temporal logics. More specifically, we provide an automata-theoretic characterisation of FO interpreted over infinite trees. To this end, two different classes of hesitant tree automata are introduced and proved to capture precisely the expressive power of two branching time temporal logics, denoted polcCTLp and cCTL*[f], which are, respectively, a restricted version of counting CTL with past and counting CTL* over finite paths, both of which have been previously shown equivalent to FO over infinite trees. The two automata characterisations naturally lead to normal forms for the two temporal logics, and highlight the fact that FO can only express properties of the tree branches which are either safety or co-safety in nature.

Problem

• Characterizing First-Order Logic over infinite trees
• Establishing automata-theoretic equivalence with temporal logics
• Identifying safety/co-safety nature of branch properties

Innovation

• Automata-theoretic characterisation of FO logic
• Hesitant tree automata classes introduced
• Normal forms for temporal logics derived