Tree Automata for Non-linear Arithmetic

Abstract

Tree automata modulo associativity and commutativity axioms, called AC tree automata, accept trees by iterating the transition modulo equational reasoning. The class of languages accepted by monotone AC tree automata is known to include the solution set of the inequality $x \times y \geqslant z$, which implies that the class properly includes the AC closure of regular tree languages. In the paper, we characterize more precisely the expressiveness of monotone AC tree automata, based on the observation that, in addition to polynomials, a class of exponential constraints (called monotone exponential Diophantine inequalities) can be expressed by monotone AC tree automata with a minimal signature. Moreover, we show that a class of arithmetic logic consisting of monotone exponential Diophantine inequalities is definable by monotone AC tree automata. The results presented in the paper are obtained by applying our novel tree automata technique, called linearly bounded projection.