Natural Deduction Calculus for Linear-Time Temporal Logic

Bolotov, Alexander; Basukoski, Artie; Grigoriev, Oleg; Shangin, Vasilyi

doi:10.1007/11853886_7

Cited by 10 publications

(21 citation statements)

References 7 publications

(10 reference statements)

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“…Moreover, it makes our systems amenable to extensions to other logics as we have begun investigating towards LTL and to the branching-time logics CTL and CTL * . To that end, we plan to capitalize on the labeled ND systems for LTL given in [4,12], which both make use of a specific rule for induction.…”

Section: Discussionmentioning

confidence: 99%

“…4 Note that the presentation of the system could be simplified by introducing a unique symbol for falsum (say ), shared by the labeled and the relational sub-systems. In that case, we would not need the rules uf 1 and uf 2, while the rules for falsum elimination RAA ⊥ and RAA ∅ could be replaced by the following rule, where with −ϕ we denote the negation of a generic formula (labeled or relational):…”

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confidence: 99%

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Labeled Natural Deduction Systems for a Family of Tense Logics

Vigano

Volpe

2008

2008 15th International Symposium on Temporal Representation and Reasoning

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We give labeled natural deduction systems for a family of tense logics extending the basic linear tense logic Kl . We prove that our systems are sound and complete with respect to the usual Kripke semantics, and that they possess a number of useful normalization properties (in particular, derivations reduce to a normal form that enjoys a subformula property). We also discuss how to extend our systems to capture richer logics like (fragments of) LTL.

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Section: Discussionmentioning

confidence: 99%

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confidence: 99%

Labeled Natural Deduction Systems for a Family of Tense Logics

Vigano

Volpe

2008

2008 15th International Symposium on Temporal Representation and Reasoning

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“…In the future, we point out studying consequences of the precise definition with an application to complexity problems [7]. Last, not least, we look forward to formulating proof searching procedures for these ND systems in the fashion of [3,4].…”

Section: Final Remarksmentioning

confidence: 99%

A Precise Definition of an Inference (by the Example of Natural Deduction Systems for Logics $I_{\langle \alpha,\beta \rangle}$

Shangin¹

2017

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In the paper, we reconsider a precise definition of a natural deduction inference given by V. Smirnov. In refining the definition, we argue that all the other indirect rules of inference in a system can be considered as special cases of the implication introduction rule in a sense that if one of those rules can be applied then the implication introduction rule can be applied, either, but not vice versa. As an example, we use logics I ⟨α,β⟩ , α, β ∈ {0, 1, 2, 3, . . . ω}, such that I ⟨0,0⟩ is propositional classical logic, presented by V. Popov. He uses these logics, in particular, a Hilbertstyle calculus HI ⟨α,β⟩ , α, β ∈ {0, 1, 2, 3, . . . ω}, for each logic in question, in order to construct examples of effects of Glivenko theorem's generalization. Here we, first, propose a subordinated natural deduction system N I ⟨α,β⟩ , α, β ∈ {0, 1, 2, 3, . . . ω}, for each logic in question, with a precise definition of a N I ⟨α,β⟩ -inference. Moreover, we, comparatively, analyze precise and traditional definitions. Second, we prove that, for each α, β ∈ {0, 1, 2, 3, . . . ω}, a Hilbert-style calculus HI ⟨α,β⟩ and a natural deduction system N I ⟨α,β⟩ are equipollent, that is, a formula A is provable in HI ⟨α,β⟩ iff A is provable in N I ⟨α,β⟩ .

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“…In [3] it was further extended to capture propositional linear-time temporal logic PLTL and in [6] the ND system was proposed for the computation tree logic CTL.…”

Section: Introductionmentioning

confidence: 99%

Natural Deduction System in Paraconsistent Setting: Proof Search for PCont

Bolotov

Shangin²

2012

Journal of Intelligent Systems

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This paper continues a systematic approach to build natural deduction calculi and corresponding proof procedures for non-classical logics. Our attention is now paid to the framework of paraconsistent logics. These logics are used, in particular, for reasoning about systems where paradoxes do not lead to the 'deductive explosion', i.e., where formulae of the type 'A follows from false', for any A, are not valid. We formulate the natural deduction system for the logic PCont, explain its main concepts, define a proof searching technique and illustrate it by examples. The presentation is accompanied by demonstrating the correctness of these developments.

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Natural Deduction Calculus for Linear-Time Temporal Logic

Cited by 10 publications

References 7 publications

Labeled Natural Deduction Systems for a Family of Tense Logics

Labeled Natural Deduction Systems for a Family of Tense Logics

A Precise Definition of an Inference (by the Example of Natural Deduction Systems for Logics $I_{\langle \alpha,\beta \rangle}$

Natural Deduction System in Paraconsistent Setting: Proof Search for PCont

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