On the Complexity of Intersecting Regular, Context-Free, and Tree Languages

We apply a construction of Cook (1971) to show that the intersection nonemptiness problem for one PDA (pushdown automaton) and a finite list of DFA’s (deterministic finite automata) characterizes the complexity class P. In particular, we show that there exist constants c1 and c2 such that for every k, intersection non-emptiness for one PDA and k DFA’s is solvable in O(nc1k) time, but is not solvable in O(nc2k) time. Then, for every k, we reduce intersection non-emptiness for one PDA and 2k DFA’s to non-emptiness for multi-stack pushdown automata with k-phase switches to obtain a tight time complexity lower bound. Further, we revisit a construction of Veanes (1997) to show that the intersection non-emptiness problem for tree automata also characterizes the complexity class P. We show that there exist constants c1 and c2 such that for every k, intersection non-emptiness for k tree automata is solvable in O(nc1k) time, but is not solvable in O(nc2k) time.