title: A proof of Bel’tyukov – Lipshitz theorem by quasi-quantifier elimination. I. Definitions and GCD-lemma
metadata.name: dc.contributor.author
metadata.value: Starchak, Mikhail R.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2021-10-07T12:18:49Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2021-10-07T12:18:49Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2021-09
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Starchak M.R. A proof of Bel’tyukov – Lipshitz theorem by quasi-quantifier elimination. I. Definitions and GCD-lemma. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8 (66), issue 3, pp. 455–466.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2021.307
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/33261
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: This paper is the first part of a new proof of decidability of the existential theory of the structure hZ; 0, 1,+,−,≤, |i, where | corresponds to the binary divisibility relation. The decidability was proved independently in 1976 by A. P. Bel’tyukov and L. Lipshitz. In 1977, V. I.Mart’yanov proved an equivalent result by considering the ternary GCD predicate instead of divisibility (the predicates are interchangeable with respect to existential definability). Generalizing in some sense the notion of quantifier elimination (QE) algorithm, we construct a quasi-QE algorithm to prove decidability of the positive existential theory of the structure hZ>0; 1, {a·}a∈Z>0 ,GCDi. We reduce to the decision problem for this theory the decision problem for the existential theory of the structure hZ; 0, 1,+,−,≤,GCDi. A quasi-QE algorithm, which performs this reduction, will be constructed in the second part of the proof. Transformations of formulas are based on a generalization of the Chinese remainder theorem to systems of the form GCD(ai, bi +x) = di for every i ∈ [1..m], where ai, bi, di are some integers such that ai 6= 0, di > 0.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 8 (66); Issue 3
metadata.lang: -
metadata.name: dc.subject
metadata.value: quantifier elimination
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: existential theory
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: divisibility
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: decidability
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Chinese remainder theorem
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: A proof of Bel’tyukov – Lipshitz theorem by quasi-quantifier elimination. I. Definitions and GCD-lemma
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 455-466.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/33261/1/455-466.pdf
resources.format: Adobe PDF
resources.size: 358,2 kB