2010
DOI: 10.1002/malq.200810053
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Tautologies over implication with negative literals

Abstract: We consider logical expressions built on the single binary connector of implication and a finite number of literals (Boolean variables and their negations). We prove that asymptotically, when the number of variables becomes large, all tautologies have the following simple structure: either a premise equal to the goal, or two premises which are opposite literals.

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Cited by 6 publications
(12 citation statements)
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References 17 publications
(35 reference statements)
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“…As for the classical implication model, we can prove that the sprouting tree distribution pn,k tends to δTrue when n tends to infinity. In this new labelling model, there are two kinds of simple tautologies: simple tautologies of first kind, defined in the same way as in the classic labelling model (see Definition ), and simple tautologies of second kind, which we now define: Definition (()). A tautology of second kind is a Boolean expression in which two nice premises are labelled respectively with a variable and its negation (see Fig .…”
Section: Tautologies In the Implication Modelmentioning
confidence: 99%
See 2 more Smart Citations
“…As for the classical implication model, we can prove that the sprouting tree distribution pn,k tends to δTrue when n tends to infinity. In this new labelling model, there are two kinds of simple tautologies: simple tautologies of first kind, defined in the same way as in the classic labelling model (see Definition ), and simple tautologies of second kind, which we now define: Definition (()). A tautology of second kind is a Boolean expression in which two nice premises are labelled respectively with a variable and its negation (see Fig .…”
Section: Tautologies In the Implication Modelmentioning
confidence: 99%
“…Fournier et al () have shown that, both in the Catalan and in the Galton‐Watson models with positive and negative literals, all the tautologies are simple tautologies of either first or second kind, asymptotically when k tends to infinity. We prove that this is not the case in the sprouting tree model.…”
Section: Tautologies In the Implication Modelmentioning
confidence: 99%
See 1 more Smart Citation
“…One motivation for considering this system came from its relation to intuitionistic logic, which was explored in [35,21]. It was shown in [12] that asymptotically almost every tautology of classical logic is intuitionistic considered in [13]. See also [25,20] for the expressions built on the single equivalence connector.…”
Section: Introductionmentioning
confidence: 99%
“…It has been shown that in both the Catalan trees and in the Galton-Watson model, all the tautologies are simple tautologies of either first or second kind, asymptotically when k tends to infinity [FGGZ10]. We show that it is not the case in the growing tree model:…”
Section: Proof Of Theoremmentioning
confidence: 68%