Natural Deduction System in Paraconsistent Setting: Proof Search for PCont

Bolotov, Alexander; Shangin, Vasilyi

doi:10.1515/jisys-2011-0021

Cited by 13 publications

(13 citation statements)

References 6 publications

(5 reference statements)

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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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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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“…Now we observe that our algorithm NPComp ALG is a modification of the proof search algorithm for classical ND [7], [8] preserving its style, and most of the procedures. Considering our classical proof search technique, it is possible to show that any ndF proof can be transformed into the proof by the classical ND calculus.…”

Section: Correctness and Efficiencymentioning

confidence: 99%

“…This logic was also given natural deduction formulation and algorithmic proof search [7], based on the same generic foundation as PComp which meets our criterion (iii). Having in mind that the algorithmic proof search has the core goal-directed procedure adapted in both cases it also meets our adaptivity element in our requirement (iii).…”

Section: Introductionmentioning

confidence: 99%

Tackling Incomplete System Specifications Using Natural Deduction in the Paracomplete Setting

Bolotov

Shangin

2014

2014 IEEE 38th Annual Computer Software and Applications Conference

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Abstract-In many modern computer applications the significance of specification based verification is well accepted. However, when we deal with such complex processes as the integration of heterogeneous systems, parts of specification may be not known. Therefore it is important to have techniques that are able to cope with such incomplete information. An adequate formal set up is given by so called paracomplete logics, where, contrary to the classical framework, for some statements we do not have evidence to conclude if they are true or false. As a consequence, for example, the law of excluded middle is not valid. In this paper we justify how the automated proof search technique for paracomplete logic PComp can be efficiently applied to the reasoning about systems with incomplete information. Note that for many researchers, one of the core features of natural deduction, the opportunity to introduce arbitrary formulae as assumptions, has been a point of great scepticism regarding the very possibility of the automation of the proof search. Here, not only we show the contrary, but we also turned the assumptions management into an advantage showing the applicability of the proposed technique to assume-guarantee reasoning.

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“…In this paper, we will deal with the correspondence analysis for the extensions of Kleene's strong three-valued logic K 3 [25,24] which was presented by Tamminga [55]. Based upon Tamminga's paper, we develop an original sound and complete proofsearching technique in the spirit of [10,36] for the natural deduction systems for the binary extensions of K 3 . 1 Natural deduction successfully overcame skepticism concerning its appropriateness to automated theorem-proving [13].…”

Section: Introductionmentioning

confidence: 99%

Automated Proof-searching for Strong Kleene Logic and its Binary Extensions via Correspondence Analysis

Petrukhin

Shangin

2018

LLP

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Natural Deduction System in Paraconsistent Setting: Proof Search for PCont

Cited by 13 publications

References 6 publications

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

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

Tackling Incomplete System Specifications Using Natural Deduction in the Paracomplete Setting

Automated Proof-searching for Strong Kleene Logic and its Binary Extensions via Correspondence Analysis

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