Algorithm to determine if a top-down non deterministic tree automaton accepts some tree?

Question

Given a non-deterministic top-down tree automaton, is there an algorithm that can determine if there exists any tree that is accepted by this automaton? if so, what is the most efficient algorithm known?

Thanks.

Answer 1

Yes, you can do it in linear time. Associate to each automaton state $q$ a boolean variable $x_q$; the intended meaning is $x_q$ should be true if there exists any tree that is accepted by the automaton if we start with state $q$ at the root. Then, each transition rule of the automaton corresponds to a Horn clause on these boolean variables. For instance, the transition rule $f(q_1,\dots,q_n) \to q'(f)$ corresponds to the Horn clause $(x_{q_1} \land \cdots \land x_{q_n}) \implies x_{q'}$. Now, solve the associated HornSAT problem using any standard algorithm to find the minimal satisfying assignment to all of these clauses; this can be done in linear time. If you find any final state $q$ of the automaton such that $x_q$ is set to true by this assignment, then the answer is yes, there exists a tree that's accepted by this automaton; otherwise, the answer is no.