Abstract
It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and wellstudied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to FO2[<] and FO 2[<; Suc], the latter of which includes the binary successor relation in addition to the linear ordering on string positions.
For both languages, our structuretheorems showexactly whatis expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m ≤ n. Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages. The question whether there was such a hierarchy had been completely open. As another consequence of our structural results, we show that satisfiability for FO2[<], which is NEXP-complete in general, becomes NP-complete once we only consider alphabets of a bounded size.
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References
Adler, M., Immerman, N.: An n! lower bound on formula size. ACM Transactions on Computational Logic 4(3), 296–314 (2003)
Brzozowski, J., Knast, R.: The dot-depth hierarchy of star-free languages is infinite. Journal of Computer and System Science 16, 37–55 (1978)
Etessami, K., Vardi, M.Y., Wilke, T.: First-order logic with two variables and unary temporal logic. In: IEEE Symposium on Logic in Computer Science (1997)
Etessami, K., Vardi, M.Y., Wilke, T.: First-order logic with two variables and unary temporal logic. Information and Computation 179(2), 279–295 (2002)
Grohe, M., Schweikardt, N.: The succinctness of first-order logic on linear orders. Logical Methods in Computer Science 1, 1–6 (2005)
Immerman, N.: Descriptive complexity. Springer, Heidelberg (1999)
Immerman, N., Kozen, D.: Definability with bounded number of bound variables. Information and Computation 83(2), 121–139 (1989)
Kamp, J.A.: Tense logic and the theory of linear order. PhD thesis, University of California, Los Angeles (1968)
Karchmer, M., Wigderson, A.: Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal of Discrete Mathematics 3(2), 255–265 (1990)
McNaughton, R., Papert, S.A.: Counter-free automata. MIT Press, Cambridge (1971)
Pin, J.-E., Weil, P.: Polynomial closure and unambiguous product. Theory of Computing Systems 30, 1–39 (1997)
Schützenberger, M.P.: Sur le produit de concatenation non ambigu. Semigroup Forum 13, 47–75 (1976)
Schwentick, T., Thérien, D., Vollmer, H.: Partially-ordered two-way automata: a new characterization of DA. In Developments in Language Theory (2001)
Straubing, H., Thérien, D.: Weakly iterated block products. In: 5th Latin American Theoretical Informatics Conference (2002)
Tesson, P., Thérien, D.: Diamonds are forever: the variety DA. In Semigroups, Algorithms, Automata and Languages (2001)
Tesson, P., Thérien, D.: Algebra meets logic: the case of regular languages. Logical Methods in Computer Science 3, 1–4 (2007)
Thérien, D., Wilke, T.: Over words, two variables are as powerful as one quantifier alternation. In: ACM Symposium on Theory of Computing (1998)
Thomas, W.: Classifying regular events in symbolic logic. Journal of Computer and System Science 25, 360–376 (1982)
Thomas, W.: An application of the Ehrenfeucht-Fraïssé game in formal language theory. Mémoires de la S.M.F 16, 11–21 (1984)
Weis, P., and Immerman, N.: Structure theorem and strict alternation hierarchy for FO2 on words (2007), http://www.cs.umass.edu/~immerman/pub/FO2_on_words.pdf
Wilke, T.: Personal communication (2007)
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Weis, P., Immerman, N. (2007). Structure Theorem and Strict Alternation Hierarchy for FO2 on Words. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_27
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DOI: https://doi.org/10.1007/978-3-540-74915-8_27
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