A Büchi-Like Theorem for Weighted Tree Automata over Multioperator Monoids

Abstract

A multioperator monoid $\mathcal{A}$ is a commutative monoid with additional operations on its carrier set. A weighted tree automaton over $\mathcal{A}$ is a finite state tree automaton of which each transition is equipped with an operation of $\mathcal{A}$. We define M-expressions over $\mathcal{A}$ in the spirit of formulas of weighted monadic second-order logics and, as our main result, we prove that if $\mathcal{A}$ is absorptive, then the class of tree series recognizable by weighted tree automata over $\mathcal{A}$ coincides with the class of tree series definable by M-expressions over $\mathcal{A}$. This result implies the known fact that for the series over semirings recognizability by weighted tree automata is equivalent with definability in syntactically restricted weighted monadic second-order logic. We prove this implication by providing two purely syntactical transformations, from M-expressions into formulas of syntactically restricted weighted monadic second-order logic, and vice versa.