A proof-theoretic investigation of a logic of positions

Baratella, Stefano; Masini, Andrea

doi:10.1016/s0168-0072(03)00021-6

Cited by 14 publications

(11 citation statements)

References 13 publications

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“…The suggested interpretation of the structural connectives justifies a number of 'display postulates' (dp) (we omit outer brackets) 3 :…”

Section: A Display Calculus For Ib[l]mentioning

confidence: 90%

“…In [16], the strong normalization theorem for a typed λ-calculus for the {→, ∧, X, G}-fragment of IB[l] is shown, but the (strong) normalization theorem for the full system is not discussed. There are a lot of natural deduction systems and typed-λ-calculi for LTL and its neighbors, and a survey of such systems is also given in [3,9,16]. Note that our systems somewhat resemble Baratella's and Masini's system PNJ for an intuitionistic LTL which is called a logic of positions [3].…”

Section: Natural Deductionmentioning

confidence: 94%

“…Natural deduction proof systems and typed λ-calculi for bounded intuitionistic linear-time temporal logics are surveyed in [16]. See also [3].…”

Section: Definition 1 (Ib[l]mentioning

confidence: 99%

“…The display sequent calculus δIB [l] consists of the above sequent rules together with the following right and left introduction rules 4 : 3 Note that the following display postulates are derivable:…”

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confidence: 99%

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Combining linear-time temporal logic with constructiveness and paraconsistency

Kamide

Wansing

2010

Journal of Applied Logic

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It is known that linear-time temporal logic (LTL), which is an extension of classical logic, is useful for expressing temporal reasoning as investigated in computer science. In this paper, two constructive and bounded versions of LTL, which are extensions of intuitionistic logic or Nelson's paraconsistent logic, are introduced as Gentzen-type sequent calculi. These logics, IB[l] and PB [l], are intended to provide a useful theoretical basis for representing not only temporal (linear-time), but also constructive, and paraconsistent (inconsistency-tolerant) reasoning. The time domain of the proposed logics is bounded by a fixed positive integer. Despite the restriction on the time domain, the logics can derive almost all the typical temporal axioms of LTL. As a merit of bounding time, faithful embeddings into intuitionistic logic and Nelson's paraconsistent logic are shown for IB [l] and PB[l], respectively. Completeness (with respect to Kripke semantics), cut-elimination, normalization (with respect to natural deduction), and decidability theorems for the newly defined logics are proved as the main results of this paper. Moreover, we present sound and complete display calculi for IB[l] and PB [l]. ] it has been emphasized that intuitionistic linear-time logic (ILTL) admits an elegant characterization of safety and liveness properties. The system ILTL, however, has been presented only in an algebraic setting. The present paper is the first semantical and proof-theoretical study of bounded constructive linear-time temporal logics containing either intuitionistic or strong negation.

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“…The suggested interpretation of the structural connectives justifies a number of 'display postulates' (dp) (we omit outer brackets) 3 :…”

Section: A Display Calculus For Ib[l]mentioning

confidence: 90%

Section: Natural Deductionmentioning

confidence: 94%

“…Natural deduction proof systems and typed λ-calculi for bounded intuitionistic linear-time temporal logics are surveyed in [16]. See also [3].…”

Section: Definition 1 (Ib[l]mentioning

confidence: 99%

mentioning

confidence: 99%

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Combining linear-time temporal logic with constructiveness and paraconsistency

Kamide

Wansing

2010

Journal of Applied Logic

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“…Both K and K4 are complete with respect to the class of models where the accessibility relation is not always defined. This means that the correspondence between positions and nodes could be undefined at some position, a situation reminiscent of the case of first order logic with undefined terms 2 . In fact, we will treat this case with an existence predicate for positions, a tool introduced by D. Scott in the late seventies [27] to deal with empty domains, and therefore with partially defined terms.…”

Section: Partial Logicsmentioning

confidence: 99%

From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

Martini

Masini

Zorzi

2021

ACM Trans. Comput. Logic

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We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules.

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Temporal Gödel‐Gentzen and Girard translations

Kamide

2013

Mathematical Logic Qtrly

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A theorem for embedding a first‐order linear‐time temporal logic LTL into its intuitionistic counterpart ILTL is proved using Baratella‐Masini's temporal extension of the Gödel‐Gentzen negative translation of classical logic into intuitionistic logic. A substructural counterpart LLTL of ILTL is introduced, and a theorem for embedding ILTL into LLTL is proved using a temporal extension of the Girard translation of intuitionistic logic into intuitionistic linear logic. These embedding theorems are proved syntactically based on Gentzen‐type sequent calculi.

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A proof-theoretic investigation of a logic of positions

Cited by 14 publications

References 13 publications

Combining linear-time temporal logic with constructiveness and paraconsistency

Combining linear-time temporal logic with constructiveness and paraconsistency

From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

Temporal Gödel‐Gentzen and Girard translations

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