Abstract
Verification of properties expressed in the two-variable fragment of first-order logic FO2 has been investigated in a number of contexts. The satisfiability problem for FO2 over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO2 is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees FO2 has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO2 gives a 2EXPTIME bound for satisfiability of FO2 over trees. This work contains a comprehensive analysis of the complexity of FO2 on trees, and on the size and depth of models. We show that the exact complexity varies according to the vocabulary used, the presence or absence of a schema, and the encoding of labels on trees. We also look at a natural restriction of FO2, its guarded version, GF2. Our results depend on an analysis of types in models of FO2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO2 sentences over finite trees.
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References
Andréka, H., van Benthem, J., Németi, I.: Modal languages and bounded fragments of predicate logic. J. Phil. Logic 27, 217–274 (1998)
Benaim, S., Benedikt, M., Lenhardt, R., Worrell, J.: Controlling the depth, size, and number of subtrees in two variable logic over trees. CoRR abs/1304.6925 (2013)
Benedikt, M., Fan, W., Geerts, F.: XPath satisfiability in the presence of DTDs. J. ACM 55(2), 8:1–8:79 (2008)
Benedikt, M., Koch, C.: XPath Leashed. ACM Comput. Surv. 41(1), 3:1–3:54 (2008)
Benedikt, M., Lenhardt, R., Worrell, J.: Verification of two-variable logic revisited. In: QEST, pp. 114–123. IEEE (2012)
Bojańczyk, M., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data trees and XML reasoning. J. ACM 56(3) (2009)
Charatonik, W., Kieroński, E., Mazowiecki, F.: Satisfiability of the two-variable fragment of first-order logic over trees. CoRR abs/1304.7204 (2013)
Charatonik, W., Witkowski, P.: Two-variable logic with counting and trees. In: LICS. IEEE (to appear, 2013)
Etessami, K., Vardi, M.Y., Wilke, T.: First-order logic with two variables and unary temporal logic. Inf. Comput. 179(2), 279–295 (2002)
Figueira, D.: Satisfiability for two-variable logic with two successor relations on finite linear orders. CoRR abs/1204.2495 (2012)
Grädel, E., Kolaitis, P.G., Vardi, M.Y.: On the decision problem for two-variable first-order logic. Bull. Symb. Logic 3(1), 53–69 (1997)
Kieroński, E.: EXPSPACE-complete variant of guarded fragment with transitivity. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 608–619. Springer, Heidelberg (2002)
Kieroński, E.: Results on the guarded fragment with equivalence or transitive relations. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 309–324. Springer, Heidelberg (2005)
Marx, M.: XPath with conditional axis relations. In: Bertino, E., Christodoulakis, S., Plexousakis, D., Christophides, V., Koubarakis, M., Böhm, K. (eds.) EDBT 2004. LNCS, vol. 2992, pp. 477–494. Springer, Heidelberg (2004)
Marxand, M., de Rijke, M.: Semantic characterization of navigational XPath. In: TDM. CTIT Workshop Proceedings Series, pp. 73–79 (2004)
Szwast, W., Tendera, L.: FO2 with one transitive relation is decidable. In: STACS. LIPIcs, vol. 20, pp. 317–328, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)
Stockmeyer, L.J.: The Complexity of Decision Problems in Automata Theory and Logic. PhD thesis, Massachusetts Institute of Technology (1974)
Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Springer (1997)
Weis, P.: Expressiveness and Succinctness of First-Order Logic on Finite Words. PhD thesis, University of Massachusetts (2011)
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Benaim, S. et al. (2013). Complexity of Two-Variable Logic on Finite Trees. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39212-2_10
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DOI: https://doi.org/10.1007/978-3-642-39212-2_10
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