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Logics with definitional reflection rules.

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Title: Logics with definitional reflection rules.
Authors: NORIHIRO KAMIDE1 drnkamide08@kpd.biglobe.ne.jp
Source: Journal of Logic & Computation. Jul2017, Vol. 27 Issue 5, p1523-1548. 26p.
Subjects: Logic programming languages, Mathematics theorems, Mathematical formulas, Set theory, Sequent calculus
Abstract: Definitional reflection rules (DRRs) provide a proof-theoretic framework for dealing with a set of clauses. An infinite version of definitional reflection logic (DRL), which has some infinite-premise DRRs, is introduced according to Gentzen-type sequent calculus for classical propositional logic. A finite version of DRL is obtained from the infinite version of DRL by replacing the infinite-premise DRRs with finite ones. A theorem for embedding the infinite version into infinitary propositional logic is proved, and a theorem for embedding the finite version into classical propositional logic is shown. The cut-elimination theorems for Gentzen-type sequent calculi for these versions are obtained using these embedding theorems. The finite version is shown to be decidable. Some similar results for the infinite and finite versions of generalized definitional reflection logic (GDRL) which has generalized definitional reflection rules (GDRRs) are also obtained. Some paraconsistent and temporal extensions of the above-mentioned classical versions of DRL and GDRL are also investigated. [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.)
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  Data: <searchLink fieldCode="DE" term="%22Logic+programming+languages%22">Logic programming languages</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+theorems%22">Mathematics theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+formulas%22">Mathematical formulas</searchLink><br /><searchLink fieldCode="DE" term="%22Set+theory%22">Set theory</searchLink><br /><searchLink fieldCode="DE" term="%22Sequent+calculus%22">Sequent calculus</searchLink>
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  Data: Definitional reflection rules (DRRs) provide a proof-theoretic framework for dealing with a set of clauses. An infinite version of definitional reflection logic (DRL), which has some infinite-premise DRRs, is introduced according to Gentzen-type sequent calculus for classical propositional logic. A finite version of DRL is obtained from the infinite version of DRL by replacing the infinite-premise DRRs with finite ones. A theorem for embedding the infinite version into infinitary propositional logic is proved, and a theorem for embedding the finite version into classical propositional logic is shown. The cut-elimination theorems for Gentzen-type sequent calculi for these versions are obtained using these embedding theorems. The finite version is shown to be decidable. Some similar results for the infinite and finite versions of generalized definitional reflection logic (GDRL) which has generalized definitional reflection rules (GDRRs) are also obtained. Some paraconsistent and temporal extensions of the above-mentioned classical versions of DRL and GDRL are also investigated. [ABSTRACT FROM AUTHOR]
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  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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        Value: 10.1093/logcom/exw013
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      – Code: eng
        Text: English
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        PageCount: 26
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    Subjects:
      – SubjectFull: Logic programming languages
        Type: general
      – SubjectFull: Mathematics theorems
        Type: general
      – SubjectFull: Mathematical formulas
        Type: general
      – SubjectFull: Set theory
        Type: general
      – SubjectFull: Sequent calculus
        Type: general
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      – TitleFull: Logics with definitional reflection rules.
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              Text: Jul2017
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