Asymptotic Density for Equivalence

Matecki, Grzegorz

doi:10.1016/j.entcs.2005.06.022

Cited by 11 publications

(6 citation statements)

References 6 publications

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“…Some variants involving expressions with other logical connectives have also been considered. Genitrini and Kozik have studied the influence of adding the connectors ∨ and ∧ to implication [7], while Matecki [13] considered the case of the single equivalence connector. For two connectors again, the and/or case has already received much attention -see Lefmann and Savický [12], Chauvin, Flajolet, Gardy and Gittenberger [1], Gardy and Woods [6], Woods [18] and Kozik [11].…”

Section: Introductionmentioning

confidence: 99%

Tautologies over implication with negative literals

Fournier¹,

Gardy

Genitrini

et al. 2010

Mathematical Logic Qtrly

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Section: Introductionmentioning

confidence: 99%

Tautologies over implication with negative literals

Fournier¹,

Gardy

Genitrini

et al. 2010

Mathematical Logic Qtrly

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“…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. For And/Or trees, we refer the reader to Woods's result that the tautologies have asymptotic probability 3/4k, and that almost all of them have the simple form ℓ ∨ · · · ∨l ∨ .…”

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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“…A generalisation which takes commutativity of the connectives into account and maps this property into the model was presented recently in [10]. The question for the probability of a tautology was pursued for instance in [13,14,15,18] for various logical systems. An overview is presented in Gardy's survey [8].…”

Section: Introductionmentioning

confidence: 99%

Probabilities of Boolean Functions given by Random Implicational Formulas

Genitrini

Gittenberger

Kraus

et al. 2012

Electron. J. Combin.

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We study the asymptotic relation between the probability and the complexity of Boolean functions in the implicational fragment which are generated by large random Boolean expressions involving variables and implication, as the number of variables tends to infinity. In contrast to models studied in the literature so far, we consider two expressions to be equal if they differ only in the order of the premises. A precise asymptotic formula is derived for functions of low complexity. Furthermore, we show that this model does not exhibit the Shannon effect.An erratum was added to this paper on Feb 20, 2014.

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Asymptotic Density for Equivalence

Cited by 11 publications

References 6 publications

Tautologies over implication with negative literals

Tautologies over implication with negative literals

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

Probabilities of Boolean Functions given by Random Implicational Formulas

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