DS-1995-03:
Marx, Maarten
(1995)
*Algebraic Relativization and Arrow Logic.*
Doctoral thesis, University of Amsterdam.

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## Abstract

We investigate several weakened versions of first--order logic, and of

the logic of binary relations, as provided by representable relation

algebras. The most important reason to weaken these two well-known and

often-used logics is their complexity: the theory of both systems is

undecidable. These logics are not only used in areas where this

complexity is needed, as in mathematics, but also in other disciplines

(computer science, linguistics) which deal with simpler (i.e.,

decidable) questions. For this reason it is desirable to develop new

versions of these logics, which are close to the original with respect

to expressive power and semantics, but behave better computationally.

We change first--order logic such that, given a model $M=<D,I>$ the set

of assignments for $M$ is just a subset of ${^\omega D}$. We show

decidability of such a weakened first--order logic, using filtration. We

also investigate Craig interpolation and Beth definability of these logics.

J.~van Benthem baptized the weakened version of the logic of binary

relations ``arrow--logic''. This is a modal logic, interpreted on a

set of arrows, with modalities for composition and inverse of arrows,

and a (constant) modality denoting that the source and target of an

arrow are the same. In this work, we identify arrows with the pair

$<${\em source, target}$>$. We investigate the

complete spectrum of arrow--logics in which the domain of the models

is a binary relation, satisfying a combination of the conditions

$\{$ reflexivity, symmetry, transitivity, Cartesian product $\}$.

We systematically treat decidability, finite axiomatizability, Craig

interpolation and Beth definability. Our results can be summarized in

one sentence: an arrow--logic has one or more of these positive

properties if and only if the domains of the models are not

necessarily transitive relations.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-1995-03 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 1995 |

Subjects: | Computation Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/1976 |

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