Abstract
Under the assumption that the Polynomial-Time Hierarchy does not collapse we show that a regular language L determines NP as an unbalanced polynomial-time leaf language if and only if L is existentially but not quantifierfree definable in FO[<, min, max, −1, +1]. The proof relies on the result of Pin & Weil [PVV97] characterizing the automata of existentially first-order definable languages.
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R. Beigel, J. Gill: Counting classes: thresholds, parity, mods, and fewness, Theoretical Computer Science 103, 1992, pp. 3–23.
A. Blass, Y. Gurevich: On the unique satisfiability problem, Information and Control 55, 1982, pp. 80–88.
B. Borchert: On the acceptance power of regular languages, Theoretical Computer Science 148, 1995, pp. 207–225.
D. P. Bovet, P. Crescenzi, R. Silvestri: A uniform approach to define complexity classes, Theoretical Computer Science 104, 1992, pp. 263–283.
H.-J. Burtschick, H. Vollmer: Lindström Quantifiers and Leaf Language Definability, ECCC Report TR96-005, 1996.
J.-Y. Cai, T. Gondermann, J. Hartmanis, L. A. Hemachandra, V. Sewelson, K. Wagner, G. Wechsung: The Boolean Hierarchy I: structural properties, SIAM Journal on Computing 17, 1988, pp. 1232–1252
R. Chang, J. Kadin, P. Hohatgi: On unique satisfiability and the threshold behavior of randomized reductions, Journal of Computer and System Science 50, 1995, pp. 359–373.
U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, K. Wagner: On the power of polynomial-time bit-computations, Proc. 8th Structure in Complexity Theory Conference, 1993, pp. 200–207
R. McNaughton, S. Papert: Counter-Free Automata, MIT Press, Cambridge, MA, 1971.
D. Perrin, J.-E. Pin: First-order logic and Star-free sets, J. of Computer and System Sciences 32, 1986, pp. 393–406.
J.-E. Pin, P. Weil: Polynomial closure and unambiguous product, Theory of Computing Systems 30, 1997, pp. 1–39.
H. Straubing: Finite Automata, Formal Logic, and Circuit Complexity, Birkhäuser, Boston, 1994.
W. Thomas: Classifying regular events in symbolic logic, Journal of Computer and System Sciences 25, 1982, pp. 300–376.
S. Toda: PP is as hard as the Polynomial-Time Hierarchy, SIAM Journal on Computing 20, 1991, pp. 865–877.
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© 1998 Springer-Verlag Berlin Heidelberg
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Borchert, B., Kuske, D., Stephan, F. (1998). On existentially first-order definable languages and their relation to NP. In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055037
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DOI: https://doi.org/10.1007/BFb0055037
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