Complexity Results for First-Order Two-Variable Logic with Counting

@article{Pacholski2000ComplexityRF,
  title={Complexity Results for First-Order Two-Variable Logic with Counting},
  author={Leszek M. Pacholski and Wieslaw Szwast and Lidia Tendera},
  journal={SIAM J. Comput.},
  year={2000},
  volume={29},
  pages={1083-1117}
}
Let $C^2_p$ denote the class of first-order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) $i$" for $i\leq p$, and let $C^2$ be the union of $C^2_p$ taken over all integers $p$. We prove that the satisfiability problem for $C^2_1$ sentences is NEXPTIME-complete. This strengthens the results by [E. Gradel, Ph. Kolaitis, and M. Vardi, Bull. Symbolic Logic, 3 (1997), pp. 53--69], who showed that the satisfiability problem for the first-order… 

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References

SHOWING 1-10 OF 41 REFERENCES

Undecidability results on two-variable logics

It is shown that going beyond L2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, such as transitive closure or monadic fixed-point operations.

Complexity of two-variable logic with counting

It is proved that the problem of satisfiability of sentences of C/sub 1//sup 2/ is NEXPTIME-complete, which easily implies that the satisfiability problem for C/sup2/ is in non-deterministic, doubly exponential time.

On the Decision Problem for Two-Variable First-Order Logic

Improve Mortimer's bound by one exponential and show that every satisfiable FO2-sentence has a model whose size is at most exponential in the size of the sentence, establishing that the satisfiability problem for FO2 is NEXPTIME-complete.

Decidability of second-order theories and automata on infinite trees

Introduction. In this paper we solve the decision problem of a certain secondorder mathematical theory and apply it to obtain a large number of decidability results. The method of solution involves

Decidability of second-order theories and automata on infinite trees.

Introduction. In this paper we solve the decision problem of a certain secondorder mathematical theory and apply it to obtain a large number of decidability results. The method of solution involves

The decision problem for standard classes

It is shown that, for any theory T, the decision problem for any class of prenex T -sentences specified by restrictions reduces to that for the standard classes, and there are finitely many standard classes such that any undecidable standard class contains one of K 1, …, K n.

The decision problem for formulas with a small number of atomic subformulas

This paper considers classes of quantificational formulas specified by restrictions on the number of atomic subformulas appearing in a formula, and shows the undecidability of the class of those formulas containing five atomic sub formulas and with prefixes of the form ∀∃∀…∀.

Two-variable logic with counting is decidable

We prove that the satisfiability and the finite satisfiability problems for C/sup 2/ are decidable. C/sup 2/ is first-order logic with only two variables in the presence of arbitrary counting

Complexity Results for Classes of Quantificational Formulas

The Classical Decision Problem

The Undecidable Standard Classes for Pure Predicate Logic, a Treatise on the Transformation of the Classical Decision Problem, and some Results and Open Problems are presented.