On Closure under Complementation of Equational Tree Automata for Theories Extending AC

We study the problem of closure under complementation of languages accepted by one-way and two-way tree automata modulo equational theories. We deal with the equational theories of commutative monoids ( ), idempotent commutative monoids ( ), Abelian groups ( ), and the theories of exclusive-or ( ), generalized exclusive-or ( ), and distributive minus symbol ( ). While the one-way automata for all these theories are known to be closed under intersection, the situation is strikingly different for complementation. We show that one-way and automata are closed under complementation, but one-way , , and automata are not. The same results hold for the two-way automata, except for the theory , as the two-way automata modulo all these theories except are known to be as expressive as the one-way automata. The question of closure under intersection and complementation of two-way automata is open.