Chapter

Automata, Languages, and Programming

Volume 9135 of the series Lecture Notes in Computer Science pp 414-426

Date:

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

  • Joseph SwernofskyAffiliated withIndependent Researcher
  • , Michael WeharAffiliated withUniversity at Buffalo Email author 

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Abstract

We apply a construction of Cook (1971) to show that the intersection non-emptiness 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 \(c_1\) and \(c_2\) such that for every k, intersection non-emptiness for one PDA and k DFA’s is solvable in \(O(n^{c_1 k})\) time, but is not solvable in \(O(n^{c_2 k})\) time. Then, for every k, we reduce intersection non-emptiness for one PDA and \(2^k\) 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 \(c_1\) and \(c_2\) such that for every k, intersection non-emptiness for k tree automata is solvable in \(O(n^{c_1 k})\) time, but is not solvable in \(O(n^{c_2 k})\) time.