Combining Equational Tree Automata over AC and ACI Theories

In this paper, we study combining equational tree automata in two different senses: (1) whether decidability results about equational tree automata over disjoint theories E1 and E2 imply similar decidability results in the combined theory E1 ∪ E2; (2) checking emptiness of a language obtained from the Boolean combination of regular equational tree languages. We present a negative result for the first problem. Specifically, we show that the intersection-emptiness problem for tree automata over a theory containing at least one AC symbol, one ACI symbol, and 4 constants is undecidable despite being decidable if either the AC or ACI symbol is removed. Our result shows that decidability of intersectionemptiness is a non-modular property even for the union of disjoint theories. Our second contribution is to show a decidability result which implies the decidability of two open problems: (1) If idempotence is treated as a rule f(x, x)→ x rather than an equation f(x, x) = x, is it decidable whether an AC tree automata accepts an idempotent normal form? (2) If E contains a single ACI symbol and arbitrary free symbols, is emptiness decidable for a Boolean combination of regular E-tree languages?