We study first-order logic with two variables FO2
and establish a small substructure property. Similar
to the small model property for FO2 we obtain an exponential
size bound on embedded substructures, relative to a fixed surrounding
structure that may be infinite. We apply this technique to analyse
the satisfiability problem for FO2 under constraints that require
several binary relations to be interpreted as equivalence relations.
With a single equivalence relation, FO2 has the finite model property
and is complete for non-deterministic exponential time, just as for plain
FO2. With two equivalence relations, FO2 does not have the finite
model property, but is shown to be decidable via a construction of
regular models that admit finite descriptions even though they may
necessarily be infinite. For three or more equivalence relations,
FO2 is undecidable.
H. Andréka, I. Németi, and J. van Benthem Modal languages and bounded fragments of predicate logic, Journal of Philosophical Logic, vol. 27(1998), no. 3, pp. 217–274.
F. Baader, D. Calvanese, D. McGuinness, D. Nardi, and P. Patel-Schneider (editors) The description logic handbook, Cambridge University Press, Cambridge,2003.
R. Berger The undecidability of the domino problem, Memoirs of the American Mathematical Society, vol. 66(1966), p. 72.
Mathematical Reviews (MathSciNet): MR216954
M. Bojańczyk, C. David, A. Muscholl, T. Schwentick, and L. Segoufin Two-variable logic on words with data, LICS, Proceedings of the 21st IEEE Symposium on Logic in Computer Science,2006, pp. 7–16.
E. Börger, E. Grädel, and Y. Gurevich The classical decision problem, Perspectives in Mathematical Logic, Springer,1997.
A. Borgida On the relative expressiveness of description logics and predicate logics, Artificial Intelligence, vol. 82(1996), pp. 353–367.
C. David, L. Libkin, and T. Tan On the satisfiability of two-variable logic over data words, LPAR, Logic for Programming, Artificial Intelligence, and Reasoning, Lecture Notes in Computer Science, vol. 6397,2010, pp. 248–262.
F. Eisenbrand and G. Shmonin Carathéodory bounds for integer cones, Operations Research Letters, vol. 34(2006), no. 5, pp. 564–568.
H. Ganzinger, C. Meyer, and M. Veanes The two-variable guarded fragment with transitive relations, LICS, Proceedings of the 14th Symposium on Logic in Computer Science, IEEE Computer Society, Los Alamitos, CA,1999, pp. 24–34.
W. Goldfarb The unsolvability of the Gödel class with identity, Journal of Symbolic Logic, vol. 49(1984), pp. 1237–1252.
Mathematical Reviews (MathSciNet): MR771790
E. Grädel On the restraining power of guards, Journal of Symbolic Logic, vol. 64(1999), pp. 1719–1742.
E. Grädel, P. Kolaitis, and M. Vardi On the decision problem for two-variable first-order logic, Bulletin of Symbolic Logic, vol. 3(1997), pp. 53–69.
E. Grädel and M. Otto On logics with two variables, Theoretical Computer Science, vol. 224(1999), pp. 73–113.
E. Grädel, M. Otto, and E. Rosen Two-variable logic with counting is decidable, LICS, Proceedings of the 12th IEEE Symposium on Logic in Computer Science,1997, pp. 306–317.
E. Grädel, M. Otto, and E. Rosen Undecidability results on two-variable logics, Archive for Mathematical Logic, vol. 38(1999), pp. 313–354.
Y. Gurevich and I. Koryakov Remarks on Berger's paper on the domino problem, Siberian Mathematical Journal, vol. 13(1972), pp. 319–321.
N. Immerman and E. Lander Describing graphs: a first-order approach to graph canonization, Complexity theory retrospective (A. Selman, editor), Springer,1990, pp. 59–81.
Y. Kazakov Saturation-based decision procedures for extensions of the guarded fragment, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Germany,2006.
E. Kieroński Results on the guarded fragment with equivalence or transitive relations, CSL, Proceedings of Computer Science Logic, Lecture Notes in Computer Science, vol. 3634, Springer,2005, pp. 309–324.
–––– On the complexity of the two-variable guarded fragment with transitive guards, Information and Computation, vol. 204(2006), pp. 1663–1703.
E. Kieroński, J. Michaliszyn, I. Pratt-Hartmann, and L. Tendera Two-variable first-order logic with equivalence closure, in preparation.
E. Kieroński and M. Otto Small substructures and decidability issues for first-order logic with two variables, Proceedings of the 20th IEEE Symposium on Logic in Computer Science,LISC 2005, pp. 448–457.
E. Kieroński and L. Tendera On finite satisfiability of the guarded fragment with equivalence or transitive guards, LPAR, Logic for Programming, Artificial Intelligence, and Reasoning, Lecture Notes in Computer Science, vol. 4790,2007, pp. 318–322.
–––– On finite satisfiability of two-variable first order logic with equivalence relations., LICS, Proceedings of the 23rd IEEE Symposium on Logic in Computer Science,2009, pp. 123–132.
A. Manuel Two variables and two successors, Proceedings of the 35th International Symposium on Mathematical foundations of Computer Science, Lecture Notes in Computer Science, vol. 6281, Springer,MFCS 2010, pp. 513–524.
M. Mortimer On languages with two variables, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21(1975), pp. 135–140.
Mathematical Reviews (MathSciNet): MR396245
M. Niewerth and T. Schwentick Two-variable logic and key constraints on data words, ICDT, Proceedings of the 14th International Conference on Database Theory,2011, pp. 138–149.
M. Otto Two variable first-order logic over ordered domains, Journal of Symbolic Logic, vol. 66(2001), pp. 685–702.
L. Pacholski, W. Szwast, and L. Tendera Complexity of two-variable logic with counting, LICS, Proceedings of the 12th IEEE Symposium on Logic in Computer Science,1997, pp. 318–327.
T. Schwentick and T. Zeume Two-variable logic with two order relations, CSL, Proceedings of Computer Science Logic, Lecture Notes in Computer Science, vol. 6247,2010, pp. 499–513.
D. Scott A decision method for validity of sentences in two variables, Journal of Symbolic Logic, vol. 27(1962), p. 377.
W. Szwast and L. Tendera On the decision problem for the guarded fragment with transitivity, LICS, Proceedings of the 16th IEEE Symposium on Logic in Computer Science,2001, pp. 147–156.