Ian Pratt-Hartmann
ABSTRACT
We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a single distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NExpTime-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable.
- C. Chang and H. Keisler. Model Theory. North Holland, Amsterdam, 1973.Google Scholar
- F. Eisenbrand and G. Shmonin. Carathéodory bounds for integer cones. Operations Research Letters, 34(5):564--568, 2006. Google ScholarDigital Library
- E. Grädel, P. Kolaitis, and M. Vardi. On the decision problem for two-variable first-order logic. Bulletin of Symbolic Logic, 3(1):53--69, 1997.Google ScholarCross Ref
- E. Grädel, M. Otto, and E. Rosen. Two-variable logic with counting is decidable. In Logic in Computer Science, pages 306--317. IEEE, 1997. Google ScholarDigital Library
- A. Janiczak. Undecidability of some simple formalized theories. Fundamenta Mathematicae, 40(1):131--139, 1953.Google ScholarCross Ref
- Y. Kazakov. Saturation-based decision procedures for extensions of the guarded fragment. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany, 2006.Google Scholar
- Y. Kazakov, U. Sattler, and E. Zolin. How many legs do I have? Non-simple roles in number restrictions revisited. In Proc. of LPAR, volume 4790 of LNCS, Berlin, 2007. Springer. Google ScholarDigital Library
- E. Kieroński. Results on the guarded fragment with equivalence or transitive relations. In Computer Science Logic, volume 3634 of LNCS, pages 309--324. Springer, 2005. Google ScholarDigital Library
- E. Kieroński and M. Otto. Small substructures and decidability issues for first-order logic with two variables. In Logic in Computer Science, pages 448--457. IEEE, 2005. Google ScholarDigital Library
- E. Kieroński and L. Tendera. On finite satisfiability of two-variable first-order logic with equivalence relations. In Logic in Computer Science. IEEE, 2009. Google ScholarDigital Library
- E. Kieroński, J. Michalyszyn, I. Pratt-Hartmann, and L. Tendera. Two-variable first-order logic with equivalence closure. SIAM Journal on Computing, 43(3):1012--1063, 2014.Google ScholarCross Ref
- M. Mortimer. On languages with two variables. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 21:135--140, 1975.Google ScholarCross Ref
- L. Pacholski, W. Szwast, and L. Tendera. Complexity of two-variable logic with counting. In Logic in Computer Science, pages 318--327. IEEE, 1997. Google ScholarDigital Library
- I. Pratt-Hartmann. Complexity of the two-variable fragment with counting quantifiers. Journal of Logic, Language and Information, 14(3):369--395, 2005. Google ScholarDigital Library
- A. Schrijver. Theory of Linear and Integer Programming. John Wiley & sons, New York, 1998. Google ScholarDigital Library
- D. Scott. A decision method for validity of sentences in two variables. Journal Symbolic Logic, 27:477, 1962.Google Scholar
- W. Szwast and L. Tendera. FO2 with one transitive relation is decidable. In STACS, volume 20 of LIPIcs, pages 317--328. Schloß Dagstuhl - Leibniz-Zentrum für Informatik, 2013.Google Scholar
- L. Tendera. Counting in the two-variable guarded fragment with transitivity. In Proc. of STACS, volume 3404 of Lecture Notes in Computer Science, pages 83--96, Berlin, 2005. Springer. Google ScholarDigital Library
Index Terms
- Logics with counting and equivalence
Recommendations
First-order satisfiability in Gödel logics: An NP-complete fragment
Defined over sets of truth values V which are closed subsets of [0,1] containing both 0 and 1, Godel logics G"V are prominent examples of many-valued logics. We investigate a first-order fragment of G"V extended with @D, that is powerful enough to ...
Deciding the Satisfiability of Propositional Formulas in Finitely-Valued Signed Logics
ISMVL '08: Proceedings of the 38th International Symposium on Multiple Valued LogicSigned logic is a way of expressing the semantics of many-valued connectives and quantifiers in a formalism that is well-suited for automated reasoning. In this paper we consider propositional, finitely-valued formulas in clausal normal form. We show ...
Non-deterministic Semantics in Polynomial Format
The method for automatic theorem proving proposed in [Carnielli, W. A., Polynomial ring calculus for many-valued logics, Proceedings of the 35th International Symposium on Multiple-Valued Logic, IEEE Computer Society. Calgary, Canada (2005), 20-25], ...
Comments