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Two variable first-order logic over ordered domains

Published online by Cambridge University Press:  12 March 2014

Martin Otto
Martin Otto*
Affiliation:
Computer Science Department, University of Wales Swansea, Swansea SA2 8PP, U.K., E-mail: m.otto@swan.ac.uk
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Abstract

The satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations.

It is shown that FO2 over ordered, respectively wellordered. domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO2. namely non-deterministic exponential time.

In contrast FO2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings. wellorderings. or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO2 and computation tree logic CTL.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 685 - 702
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Börger, E., Grädel, E., and Gurevich, Y., The classical decision problem, Springer, 1997.CrossRefGoogle Scholar
[2]Etessami, K., Vardi, M., and Wilke, T., First-order logic with two variables and unary temporal logic, Proceedings of the 12th ieee symposium on logic in compter science, 1997, pp. 228–235.Google Scholar
[3]Grädel, E., Kolaitis, P., , and Vardi, M., On the complexity of the decision problem for two-variable first-order logic, The Bulletin of Symbolic Logic, vol. 3 (1997), pp. 53–69.CrossRefGoogle Scholar
[4]Grädel, E. and Otto, M., On logics with two variables, Theoretical Computer Science, vol. 224 (1999), pp. 73–113.CrossRefGoogle Scholar
[5]Grädel, E., Otto, M., and Rosen, E., Undecidability results on two-variable logics, Archive of Mathematical Logic, vol. 38 (1999), pp. 313–354, A preliminary version appeared in Proceedings of 14th Symposium on Theoretical Aspects of Computer Science 1997, LNCS, vol. 1200, Springer 1997, pp. 249–260.Google Scholar
[6]Gurevich, Y. and Koryakov, I., Remarks on Berger's paper on the domino problem, Siberian Mathematical Journal, vol. 13 (1972), pp. 319–321.CrossRefGoogle Scholar
[7]Immerman, N. and Vardi, M., Model checking and transitive closure logic, Proceedings of the 9th conference on computer-aided verification, 1997, pp. 291–302.CrossRefGoogle Scholar
[8]Mortimer, M., On languages with two variables, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 135–140.CrossRefGoogle Scholar
[9]Scott, D., A decision methodfor validity of sentences in two variables, this Journal, vol. 27 (1962), p. 377.Google Scholar
[10]Vardi, M., Why is modal logic so robustly decidable?, Descriptive complexity and finite models (Princeton, NJ, 1996), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, no. 31, American Mathematical Society, 1997, pp. 149–184.CrossRefGoogle Scholar