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 (ACU), idempotent commutative monoids (ACUI), Abelian groups (ACUM), and the theories of exclusive-or (ACUX), generalized exclusive-or (ACUX n ), and distributive minus symbol (ACUD). 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 ACU and ACUD automata are closed under complementation, but one-way ACUX, ACUX n , ACUM and ACUI automata are not. The same results hold for the two-way automata, except for the theory ACUI, as the two-way automata modulo all these theories except ACUI are known to be as expressive as the one-way automata. The question of closure under intersection and complementation of two-way ACUI automata is open.

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