# 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}
}
• Published 1 February 2000
• Computer Science
• SIAM J. Comput.
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

• Mathematics
Arch. Math. Log.
• 1997
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

• Mathematics
Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science
• 1997
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

• Mathematics, Computer Science
Bulletin of Symbolic Logic
• 1997
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

• Mathematics
Journal of Symbolic Logic
• 1973
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

• Computer Science
Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science
• 1997
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

### The Classical Decision Problem

• Mathematics
Perspectives in Mathematical Logic
• 1993
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.