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

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.