Abstract
We carefully reexamine a construction of Karakostas, Lipton, and Viglas (2003) to show that the intersection non-emptiness problem for DFA’s (deterministic finite automata) characterizes the complexity class NL. In particular, if restricted to a binary work tape alphabet, then there exist constants c 1 and c 2 such that for every k intersection non-emptiness for k DFA’s is solvable in c 1 k log(n) space, but is not solvable in c 2 k log(n) space. We optimize the construction to show that for an arbitrary number of DFA’s intersection non-emptiness is not solvable in \(o(\frac{n}{\log(n)\log(\log(n))})\) space. Furthermore, if there exists a function f(k) = o(k) such that for every k intersection non-emptiness for k DFA’s is solvable in n f(k) time, then P ≠ NL. If there does not exist a constant c such that for every k intersection non-emptiness for k DFA’s is solvable in n c time, then P does not contain any space complexity class larger than NL.
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References
Blondin, M., Krebs, A., McKenzie, P.: The complexity of intersecting finite automata having few final states. In: Computational Complexity, CC (to appear, 2014)
Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press, New York (2008)
Jones, N.D., Lien, Y.E., Laaser, W.T.: New problems complete for nondeterministic log space. Mathematical Systems Theory 10 (1976)
Karakostas, G., Lipton, R.J., Viglas, A.: On the complexity of intersecting finite state automata and NL versus NP. Theoretical Computer Science 302, 257–274 (2003)
Kozen, D.: Lower bounds for natural proof systems. In: Proc. 18th Symp. on the Foundations of Computer Science, pp. 254–266 (1977)
Lange, K.-J., Rossmanith, P.: The emptiness problem for intersections of regular languages. In: Havel, I.M., Koubek, V. (eds.) MFCS 1992. LNCS, vol. 629, pp. 346–354. Springer, Heidelberg (1992)
Lipton, R.J.: On the intersection of finite automata. Gödel’s Lost Letter and P=NP (August 2009)
Lipton, R.J., Regan, K.W.: The power of guessing. Gödel’s Lost Letter and P=NP (November 2012)
Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal (1959)
Rampersad, N., Shallit, J.: Detecting patterns in finite regular and context-free languages. Information Processing Letters 110 (2010)
Veanes, M.: On computational complexity of basic decision problems of finite tree automata. UPMAIL Technical Report 133 (1997)
Wehar, M.: Intersection emptiness for finite automata. Honors thesis, Carnegie Mellon University (2012)
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Wehar, M. (2014). Hardness Results for Intersection Non-Emptiness. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43951-7_30
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DOI: https://doi.org/10.1007/978-3-662-43951-7_30
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