On ground tree transformations and congruences induced by tree automata

Abstract

For a tree automaton A over a ranked alphabet Σ, we study the ground tree transformation π(A) induced by A and the restriction θ(A) of the congruence ↔A∗ to terms over Σ. We define a congruence relation ρ⊆A×A on A, called the determiner of A, and the quotient tree automaton A/ρ. We show the following results. It is decidable if θ(A)=π(A). If A is deterministic, then θ(A)=π(A). The determiner ρ of A can be effectively constructed, A/ρ is deterministic, and θ(A)=θ(A/ρ). For a connected tree automaton A, π(A)=π(A/ρ) if and only if π(A)=π(B) for some deterministic tree automaton B if and only if θ(A)=π(A).