FO^2 with one transitive relation is decidable

Authors Wieslaw Szwast, Lidia Tendera

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Wieslaw Szwast
Lidia Tendera

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Wieslaw Szwast and Lidia Tendera. FO^2 with one transitive relation is decidable. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 317-328, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


We show that the satisfiability problem for the two-variable first-order logic, FO^2, over transitive structures when only one relation is required to be transitive, is decidable. The result is optimal, as FO^2 over structures with two transitive relations, or with one transitive and one equivalence relation, are known to be undecidable, so in fact, our result completes the classification of FO^2-logics over transitive structures with respect to decidability. We show that the satisfiability problem is in 2-NExpTime. Decidability of the finite satisfiability problem remains open.
  • classical decision problem
  • two-variable first-order logic
  • decidability
  • computational complexity


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