Abstract
We study the problem of the satisfiability of guarded formulas in models in which some distinguished binary symbols are interpreted as equivalence relations or as transitive relations. We sharpen the undecidability result for the two-variable guarded fragment with transitive relations by reducing the number of transitive relations to two. We prove that the satisfiability problem for the two-variable guarded fragment with two equivalence relations is 2EXPTIME-complete. We consider the guarded fragment with equivalence relations in guards and show that this variant is easily reducible to the variant with transitive relations in guards. However, in the case of two variables, the version with equivalence relations is easier: NEXPTIME-complete. Finally we show that the decidability results for the guarded fragment with either equivalence relations or transitive relations in guards cannot be generalized to the loosely guarded fragment.
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References
Andréka, H., van Benthem, J., Németi, I.: Modal Languages and Bounded Fragments of Predicate Logic. J. Philos. Logic 27(3), 217–274 (1998)
van Benthem, J.: Dynamic bits and pieces, ILLC Research Report (1997)
Ganzinger, H., Meyer, C., Veanes, M.: The Two-Variable Guarded Fragment with Transitive Relations. In: Proc. of 14-th LICS, pp. 24–34 (1999)
Grädel, E.: On the Restraining Power of Guards. J. Symbolic Logic 64, 1719–1742 (1999)
Grädel, E., Kolaitis, P., Vardi, M.: On the Decision Problem for Two-Variable First Order Logic. Bull. of Symbolic Logic 3(1), 53–96 (1997)
Grädel, E., Otto, M.: On Logics with Two Variables. TCS 224, 77–113 (1999)
Grädel, E., Otto, M., Rosen, E.: Undecidability Results on Two-Variable First-Order Logic. Archive of Mathematical Logic 38, 313–354 (1999)
Grädel, E., Walukiewicz, I.: Guarded Fixpoint Logic. In: Proc. of 14-th LICS, pp. 45–54 (1999)
Kieroński, E.: EXPSPACE-Complete Variant of Guarded Fragment with Transitivity. In: Proc. of 19-th STACS, pp. 608–619 (2002)
Kieroński, E.: The Two-Variable Guarded Fragment with Transitive Guards is 2EXPTIMEHard. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 299–312. Springer, Heidelberg (2003)
Kieroński, E., Otto, M.: Small Substructures and Decidability Issues for First-Order Logic with Two Variables. accepted for LICS 2005 (2005)
Lewis, H.R.: Complexity Results for Classes of Quantificational Formulas. J. Comp. and System Sci. 21, 317–353 (1980)
Otto, M.: Two Variable First-Order Logic Over Ordered Domains. J. Symb. Log. 66 (2001)
Szwast, W., Tendera, L.: On the Decision Problem for the Guarded Fragment with Transitivity. In: Proc. of 16-th LICS, pp. 147–156 (2001)
Szwast, W., Tendera, L.: The Guarded Fragment with Transitive Guards. Annals of Pure and Applied Logic 128, 227–276 (2004)
Tendera, L.: Counting in the two variable guarded logic with transitivity. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 83–96. Springer, Heidelberg (2005)
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Kieroński, E. (2005). Results on the Guarded Fragment with Equivalence or Transitive Relations. In: Ong, L. (eds) Computer Science Logic. CSL 2005. Lecture Notes in Computer Science, vol 3634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538363_22
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DOI: https://doi.org/10.1007/11538363_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28231-0
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