## Equality and Monodic First-Order Temporal Logic (2002)

Venue: | Studia Logica |

Citations: | 15 - 7 self |

### BibTeX

@ARTICLE{Degtyarev02equalityand,

author = {Anatoli Degtyarev and Michael Fisher and Alexei Lisitsa},

title = {Equality and Monodic First-Order Temporal Logic},

journal = {Studia Logica},

year = {2002},

volume = {72},

pages = {200--2}

}

### Years of Citing Articles

### OpenURL

### Abstract

It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e. the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.

### Citations

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Citation Context ...nd TL \ TL 2 1 , together with their intersection TL \ TL mo 1 \ TL 2 1 , become not even partially decidable, i.e. not recursively enumerable. Our proof is based on the argument that Minsky machines =-=[Min67]-=- can be simulated by formulae of TL mo 1 \ TL 2 1 . As to the guarded fragment TL \ TL G 1 the question of its decidability/enumerability remains open. 2 Minsky machines The (two-counter) Minsky machi... |

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Citation Context ...onodic and guarded fragment with equality is open as before. Related papers dealing with undecidable guarded fragments of non-temporal firstorder logic with added transitive relations are [GMV99] and =-=[Gra99]-=-. In [Gra99] it is shown that the three-variable guarded fragment equipped with two transitive binary relations is not recursively enumerable, while in [GMV99] the authors have shown that the two-vari... |

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Citation Context ...for the logic 1 , or equivalently, the set of valid formulae of the logic is not recursively enumerable. Recently, the interesting monodic fragment of first-order temporal logic has been investigated =-=[HWZ00]-=-, which has a quite transparent (and intuitive) syntactic definition and a finite inference system [WZ01]. Moreover many important subfragments of the monodic fragment turn out to be decidable [HWZ00,... |

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Citation Context ...agments does not destroy their decidability [BGG97]. The proof of incompleteness of the monodic fragment with equality given in [WZ01] was based on the reduction to Craig's [Cra50] and Trakhtenbrot's =-=[Tra5-=-0] result about incompleteness of the set of all first-order formulae valid in all finite interpretations. Roughly speaking, a formula ^ 0 of first-order temporal logic with equality was presented cha... |

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Citation Context ...ecently, the interesting monodic fragment of first-order temporal logic has been investigated [HWZ00], which has a quite transparent (and intuitive) syntactic definition and a finite inference system =-=[WZ01]-=-. Moreover many important subfragments of the monodic fragment turn out to be decidable [HWZ00, WZ01]. Unfortunately, all the positive properties of the monodic fragment concerning completeness and de... |

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Citation Context ...riable fragment with equality is not recursively enumerable. Let us note that, in classical first-order logic, adding equality to monadic or two-variable fragments does not destroy their decidability =-=[BGG97]-=-. The proof of incompleteness of the monodic fragment with equality given in [WZ01] was based on the reduction to Craig's [Cra50] and Trakhtenbrot's [Tra50] result about incompleteness of the set of a... |

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Citation Context ...ity of the monodic and guarded fragment with equality is open as before. Related papers dealing with undecidable guarded fragments of non-temporal firstorder logic with added transitive relations are =-=[GMV99]-=- and [Gra99]. In [Gra99] it is shown that the three-variable guarded fragment equipped with two transitive binary relations is not recursively enumerable, while in [GMV99] the authors have shown that ... |

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1 |
with respect to validity in every finite nonempty domain, of first-order functional calculus
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(Show Context)
Citation Context ... monadic or two-variable fragments does not destroy their decidability [BGG97]. The proof of incompleteness of the monodic fragment with equality given in [WZ01] was based on the reduction to Craig's =-=[Cra5-=-0] and Trakhtenbrot's [Tra50] result about incompleteness of the set of all first-order formulae valid in all finite interpretations. Roughly speaking, a formula ^ 0 of first-order temporal logic with... |

1 | Incompleteness of first-order logic with until - Szalas, Holenderski - 1988 |

1 |
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Citation Context ... As distances between time points are finite, the domain of any model must be finite as well. This formula is obtained from the formula without predicate symbols but with a flexible variable given in =-=[Sza95]-=- after replacing the flexible variable by a (flexible) monadic predicate symbol. 2 Another interesting and important monodic fragment, for which decidability without equality was proved in [HWZ00], is... |

1 |
Sentences true in all constructive models
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Citation Context ...raints related to '. In our case a very simple temporal formula took the place of ' due to immediate simulation of Minsky machines. Taking into account further results on Trakhtenbrot's theorem (e.g. =-=[Vua60]-=-) it seems to be possible to restrict ' such that it would contain, besides monadic predicates, only one binary predicate. In such a way it would be possible to extend the proof of incompleteness of m... |