Tautologies over implication with negative literals

Fournier, Hervé; Gardy, Danièle; Genitrini, Antoine; Zaionc, Marek

doi:10.1002/malq.200810053

Cited by 6 publications

(12 citation statements)

References 17 publications

(35 reference statements)

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“…As for the classical implication model, we can prove that the sprouting tree distribution

p_{n, k}

tends to

δ_{T r u e}

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%

“…Simple tautologies have also been studied under a variation of the implication model, where negative literals are allowed [10]. As for the classical implication model, we can prove that the sprouting tree distribution p n,k tends to δ True when n tends to infinity.…”

Section: Allowing Positive and Negative Literalsmentioning

confidence: 99%

See 2 more Smart Citations

A sprouting tree model for random boolean functions

Chauvin

Gardy

Mailler

2014

Random Struct. Alg.

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We define a new probability distribution for Boolean functions of k variables. Consider the random Binary Search Tree of size n, and label its internal nodes by connectives and its leaves by variables or their negations. This random process defines a random Boolean expression which represents a random Boolean function. Finally, let n tend to infinity: the asymptotic distribution on Boolean functions exists; we call it the sprouting tree distribution. We study this model and compare it with two previously‐known distributions induced by two other random trees: the Catalan tree and the Galton‐Watson tree. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 635–662, 2015

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“…As for the classical implication model, we can prove that the sprouting tree distribution

p_{n, k}

tends to

δ_{T r u e}

Section: Tautologies In the Implication Modelmentioning

confidence: 99%

Section: Tautologies In the Implication Modelmentioning

confidence: 99%

See 1 more Smart Citation

A sprouting tree model for random boolean functions

Chauvin

Gardy

Mailler

2014

Random Struct. Alg.

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“…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%

The fraction of large random trees representing a given Boolean function in implicational logic

Fournier¹,

Gardy²,

Genitrini³

et al. 2011

Random Struct Algorithms

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We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f . We also prove that most expressions computing any given function in this system are "simple", in a sense that we make precise. The probability of all read-once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process.

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“…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%

The growing tree distribution on Boolean functions.

Chauvin¹,

Gardy²,

Mailler³

2011

2011 Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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We define a probability distribution over the set of Boolean functions of k variables induced by the tree representation of Boolean expressions. The law we are interested in is inspired by the growth model of Binary Search Trees: we call it the growing tree law. We study it over different logical systems and compare the results we obtain to already known distributions induced by the tree representation: Catalan trees, Galton-Watson trees and balanced trees.

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Tautologies over implication with negative literals

Cited by 6 publications

References 17 publications

A sprouting tree model for random boolean functions

A sprouting tree model for random boolean functions

The fraction of large random trees representing a given Boolean function in implicational logic

The growing tree distribution on Boolean functions.

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