Relations enumerable from positive information.
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| Title: | Relations enumerable from positive information. |
|---|---|
| Authors: | Csima, Barbara F1 (AUTHOR), MacLean, Luke2 (AUTHOR), Rossegger, Dino3 (AUTHOR) |
| Source: | Journal of Logic & Computation. Jul2025, Vol. 35 Issue 5, p1-16. 16p. |
| Subjects: | Recursion theory, Mathematical formulas, Relation algebras |
| Abstract: | We study countable structures from the viewpoint of enumeration reducibility. Since enumeration reducibility is based on only positive information, in this setting it is natural to consider structures given by their positive atomic diagram—the computable join of all relations of the structure. Fixing a structure |${\mathcal{A}}$| , a natural class of relations in this setting are the relations |$R$| such that |$R^{\hat{\mathcal{A}}}$| is enumeration reducible to the positive atomic diagram of |$\hat{\mathcal{A}}$| for every |$\hat{\mathcal{A}}\cong{\mathcal{A}}$| – the relatively intrinsically positively enumerable (r.i.p.e.) relations. We show that the r.i.p.e. relations are exactly the relations that are definable by |$\varSigma ^{p}_{1}$| formulas, a subclass of the infinitary |$\varSigma ^{0}_{1}$| formulas. We then introduce a new natural notion of the jump of a structure and study its interaction with other notions of jumps. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 187148426 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Relations enumerable from positive information. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Csima%2C+Barbara+F%22">Csima, Barbara F</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22MacLean%2C+Luke%22">MacLean, Luke</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Rossegger%2C+Dino%22">Rossegger, Dino</searchLink><relatesTo>3</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Jul2025, Vol. 35 Issue 5, p1-16. 16p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Recursion+theory%22">Recursion theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+formulas%22">Mathematical formulas</searchLink><br /><searchLink fieldCode="DE" term="%22Relation+algebras%22">Relation algebras</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We study countable structures from the viewpoint of enumeration reducibility. Since enumeration reducibility is based on only positive information, in this setting it is natural to consider structures given by their positive atomic diagram—the computable join of all relations of the structure. Fixing a structure |${\mathcal{A}}$| , a natural class of relations in this setting are the relations |$R$| such that |$R^{\hat{\mathcal{A}}}$| is enumeration reducible to the positive atomic diagram of |$\hat{\mathcal{A}}$| for every |$\hat{\mathcal{A}}\cong{\mathcal{A}}$| – the relatively intrinsically positively enumerable (r.i.p.e.) relations. We show that the r.i.p.e. relations are exactly the relations that are definable by |$\varSigma ^{p}_{1}$| formulas, a subclass of the infinitary |$\varSigma ^{0}_{1}$| formulas. We then introduce a new natural notion of the jump of a structure and study its interaction with other notions of jumps. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1093/logcom/exaf034 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 16 StartPage: 1 Subjects: – SubjectFull: Recursion theory Type: general – SubjectFull: Mathematical formulas Type: general – SubjectFull: Relation algebras Type: general Titles: – TitleFull: Relations enumerable from positive information. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Csima, Barbara F – PersonEntity: Name: NameFull: MacLean, Luke – PersonEntity: Name: NameFull: Rossegger, Dino IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 07 Text: Jul2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 0955792X Numbering: – Type: volume Value: 35 – Type: issue Value: 5 Titles: – TitleFull: Journal of Logic & Computation Type: main |
| ResultId | 1 |
