Random boolean formulas representing any boolean function with asymptotically equal probability

Savický, Petr

doi:10.1007/bfb0017174

Cited by 11 publications

(31 citation statements)

References 2 publications

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“…The Fourier transform was used by Razborov [Raz88] to derive his results on Ramsey graphs, as well as by Savický [Sav90]. The Fourier transform plays a role in many of our results, but it needs to be adapted in various ways to suit different cases.…”

Section: Definitionsmentioning

confidence: 99%

“…To do this we use Savický's [Sav90] argument, which uses induction on the weight of w together with the recurrence…”

Section: Proofmentioning

confidence: 99%

“…Second, by extending Savický's [Sav90] analysis, via "Restriction Lemmas", we characterize growth processes that use monotone connectives, and show that our technique is applicable to growth processes that use other connectives as well. Additionally, we obtain explicit bounds on the convergence rates of several growth processes, including the growth process studied by Savický (1990). …”

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confidence: 99%

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The Boolean functions computed by random Boolean formulas or how to grow the right function

Brodsky

Pippenger

2005

Random Struct Algorithms

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Among their many uses, growth processes (probabilistic amplification), were used for constructing reliable networks from unreliable components, and deriving complexity bounds of various classes of functions. Hence, determining the initial conditions for such processes is an important and challenging problem. In this paper we characterize growth processes by their initial conditions and derive conditions under which results such as Valiant's[Val84] hold. First, we completely characterize growth processes that use linear connectives. Second, by extending Savický's [Sav90] analysis, via "Restriction Lemmas", we characterize growth processes that use monotone connectives, and show that our technique is applicable to growth processes that use other connectives as well. Additionally, we obtain explicit bounds on the convergence rates of several growth processes, including the growth process studied by Savický (1990).

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Section: Definitionsmentioning

confidence: 99%

“…To do this we use Savický's [Sav90] argument, which uses induction on the weight of w together with the recurrence…”

Section: Proofmentioning

confidence: 99%

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confidence: 99%

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The Boolean functions computed by random Boolean formulas or how to grow the right function

Brodsky

Pippenger

2005

Random Struct Algorithms

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“…Then Boppana [2] and Gupta and Mahajan [16] improved Valiant's result for majority; Boppana went on to prove that iteration by a single, well-chosen connector gives a distribution concentrated on one of the threshold functions. Savický [30] showed that iterating a nonlinear and monotone connector leads to the uniform distribution on the set of all Boolean functions. Brodsky and Pippenger [3] presented a systematic study of different classes of connectors and of the distributions induced on Boolean functions; these distributions are either uniform on subsets of Boolean functions, or concentrated on a single function.…”

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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“…Then Boppana [18] and Gupta-Mahajan [19] improved Valiant's result for majority; Boppana went on to prove that iteration by a single, well-chosen connector gives a uniform distribution on the set of threshold functions. Savický [20] showed that iterating a nonlinear balanced connector leads to the uniform distribution on the set of all boolean functions. Finally, Brodsky and Pippenger [21] present a systematic study of different classes of connectors and of the distributions induced on boolean functions; these distributions are either uniform on subsets of boolean functions, or concentrated on a single function.…”

Section: Introductionmentioning

confidence: 99%

Complexity and Limiting Ratio of Boolean Functions over Implication

Fournier

Gardy

Genitrini

et al.

Lecture Notes in Computer Science

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Abstract. 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. We then 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.

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Random boolean formulas representing any boolean function with asymptotically equal probability

Cited by 11 publications

References 2 publications

The Boolean functions computed by random Boolean formulas or how to grow the right function

The Boolean functions computed by random Boolean formulas or how to grow the right function

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

Complexity and Limiting Ratio of Boolean Functions over Implication

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