## Two-Variable Logic with Counting is Decidable (1996)

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Citations: | 58 - 3 self |

### BibTeX

@MISC{Grädel96two-variablelogic,

author = {Erich Grädel and Martin Otto and Eric Rosen},

title = {Two-Variable Logic with Counting is Decidable},

year = {1996}

}

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### Abstract

We prove that the satisfiability problem for C² is decidable. C² is first-order logic with only two variables in the presence of arbitrary counting quantifiers 9 ?m , m ? 1. It considerably extends L², plain first-order with only two variables, which is known to be decidable by a result of Mortimer. Unlike L², C² does not have the finite model property. As C² extends L² by expressive means for counting, significant applications arise from the fact that C² embeds corresponding counting extensions of modal logics.

### Citations

264 |
The classical decision problems
- Börger, Grädel, et al.
- 1997
(Show Context)
Citation Context ...ecision problem of mathematical logic, i.e., the problem of classifying fragments of first-order logic, according to whether their satisfiability and finite satisfiability problems are decidable (see =-=[2]-=-). These are the main results of the paper: Theorem 1.2. sat(C 2 ) is decidable. Theorem 1.3. fin-sat(C 2 ) is decidable. In the present exposition we provide a full proof for the decidability of sat(... |

57 | Describing graphs: A first-order approach to graph canonization
- Lander
- 1990
(Show Context)
Citation Context ...itted in C 2 without increasing its expressive power. We point out one particularly interesting piece of evidence for the expressive power of C 2 , which far exceeds that of L 2 . Immerman and Lander =-=[6]-=- show that the C 2 -theory of a finite graph (which is actually axiomatized by a single sentence of C 2 ) exactly characterises the stable colouring of that graph. By results of Babai, Erd"os, and Se... |

49 |
S.M.: Random graph isomorphism
- Babai, Erdős, et al.
- 1980
(Show Context)
Citation Context ... that the C 2 -theory of a finite graph (which is actually axiomatized by a single sentence of C 2 ) exactly characterises the stable colouring of that graph. By results of Babai, Erd"os, and Selkow =-=[1]-=- it follows that almost all finite graphs are characterized up to isomorphism by their C 2 -theory -- almost all in the sense of asymptotic probabilities: the proportion of graphs with vertices 0; : :... |

47 | On the decision problem for twovariable first-order logic
- Grädel, Kolaitis, et al.
- 1997
(Show Context)
Citation Context ...ry satisfiable L 2 -sentence has a model of double exponential size, so there is a more direct argument for decidability. Mortimer's proof and the quality of this bound were recently improved upon in =-=[3]-=-, where the complexity of sat(L 2 ) was also determined precisely: sat(L 2 ) is complete for nondeterministic exponential time. Some of the important applications of this result arise in the context o... |

32 | L.: Canonical labelling of graphs in linear average time - Babai, Kučera - 1979 |

27 |
Bounded Variable Logics and Counting — A Study in Finite Models
- Otto
- 1997
(Show Context)
Citation Context ...es 0; : : : ; n \Gammas1 having this property tends to 1 as n goes to infinity. For more information on the expressive power and model-theoretic properties of C 2 (and extensions thereof) we refer to =-=[8, 9]-=-. That C 2 does not have the finite model property is witnessed by 8x9 =1 yExy ^ 8x9 61 yEyx ^ 9x8y:Eyx; which asserts that E is the graph of an injective function from the universe to itself which fa... |

26 | Undecidability results on twovariable logics - Grädel, Otto, et al. - 1997 |

23 | On the Decision Problem for Two-Variable First-Order Logic - Kolaitis, Vardi |

21 |
Complexity of two-variable logic with counting
- Pacholski, Szwast, et al.
- 1997
(Show Context)
Citation Context ...for description logics with counting restrictions and modal logics with graded modalities. Remark. An upper complexity bound for sat(C 2 ) has recently been announced by Pacholski, Szwast and Tendera =-=[10]-=-. 2. Notation and basic definitions Let o/ be some finite vocabulary of unary and binary relation symbols and constants. It is not difficult to see that in two-variable logics (even in the presence of... |

18 | The Classical Decision Problem - örger, ädel, et al. - 1997 |

10 | On languages with two variables, Zeitschr. f. math. Logik u. Grundlagen d - Mortimer - 1975 |

6 |
On languages with two variables. Zeitschr.f.math.Logik u.Grundlagen d.Math
- Mortimer
- 1975
(Show Context)
Citation Context ...ulae involving no variables apart from x and y under Boolean operations and 9x, 9y. Throughout this paper we restrict attention to finite vocabularies consisting of relation symbols and constants. By =-=[7]-=-, L 2 has the finite model property, i.e. every satisfiable sentence has a finite model. Consequently, the satisfiability problem for L 2 is decidable. Standard terminology uses the following: ffl sat... |

4 | Ptime canonization for two variables with counting
- Otto
- 1995
(Show Context)
Citation Context ...es 0; : : : ; n \Gammas1 having this property tends to 1 as n goes to infinity. For more information on the expressive power and model-theoretic properties of C 2 (and extensions thereof) we refer to =-=[8, 9]-=-. That C 2 does not have the finite model property is witnessed by 8x9 =1 yExy ^ 8x9 61 yEyx ^ 9x8y:Eyx; which asserts that E is the graph of an injective function from the universe to itself which fa... |

2 |
Undecidability results on two-variable logics, to appear in Archive of Mathematical Logic. A preliminary version has appeared
- adel, Otto, et al.
- 1997
(Show Context)
Citation Context ...ional dynamic logic PDL, computation tree logic CTL, or propositional _-calculus L_ (which feature expressive means that make them useful as process logics) are known to be decidable, it was shown in =-=[4]-=- that several corresponding extensions of L 2 are no longer decidable. Unlike FO, L 2 is not closed under assertions that there are at least m elements satisfying some property for m ? 2. It is theref... |