On the Number of Fanout-Free Functions and Unate Cascade Functions

Sasao,; Kinoshita,

doi:10.1109/tc.1979.1675227

Cited by 18 publications

(17 citation statements)

References 21 publications

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“…Bender and Butler [2] and Sasao and Kinoshita [18] independently found the number of unate cascade functions, among other fanout-free functions. As an immediate corollary of Theorem 2.8, we therefore know the number of NCFs in n variables, for a given value of n. We use the recursive formula found by Sasao and Kinoshita [18], in the following corollary.…”

Section: Theorem 28-a Boolean Function Is Nested Canalyzing If and Omentioning

confidence: 99%

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Nested canalyzing, unate cascade, and polynomial functions

Jarrah¹,

Raposa²,

Laubenbacher³

2007

Physica D: Nonlinear Phenomena

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This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.

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Section: Theorem 28-a Boolean Function Is Nested Canalyzing If and Omentioning

confidence: 99%

“…As an immediate corollary of Theorem 2.8, we therefore know the number of NCFs in n variables, for a given value of n. We use the recursive formula found by Sasao and Kinoshita [18], in the following corollary. …”

Section: Theorem 28-a Boolean Function Is Nested Canalyzing If and Omentioning

confidence: 99%

Nested canalyzing, unate cascade, and polynomial functions

Jarrah¹,

Raposa²,

Laubenbacher³

2007

Physica D: Nonlinear Phenomena

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“…Definition 1 [21]. f is a unate cascade function if f can be represented as where x Ã i is either x i or " x x i and } i is either the OR (_) or AND (^) operator.…”

Section: Unate Cascade Functionsmentioning

confidence: 99%

Average Path Length of Binary Decision Diagrams

Butler

Sasao

Matsuura

2005

IEEE Trans. Comput.

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Abstract-The traditional problem in binary decision diagrams (BDDs) has been to minimize the number of nodes since this reduces the memory needed to store the BDD. Recently, a new problem has emerged: minimizing the average path length (APL). APL is a measure of the time needed to evaluate the function by applying a sequence of variable values. It is of special significance when BDDs are used in simulation and design verification. A main result of this paper is that the APL for benchmark functions is typically much smaller than for random functions. That is, for the set of all functions, we show that the average APL is close to the maximum path length, whereas benchmark functions show a remarkably small APL. Surprisingly, however, typical functions do not achieve the absolute maximum APL. We show that the parity functions are unique in having that distinction. We show that the APL of a BDD can vary considerably with variable ordering. We derive the APL for various functions, including the AND, OR, threshold, Achilles' heel, and certain arithmetic functions. We show that the unate cascade functions uniquely achieve the absolute minimum APL.

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“…The number of nested canalizing functions is B(n, n). A recurrence for this was independently derived in the 1970s by engineers studying unate cascade functions [BB78,SK79], and then a closed formula was found by mathematicians studying NCFs [LAM + 13]. The quantity B(4, k) was recently computed in [RDC14].…”

Section: Enumeration Of K-canalizing Functionsmentioning

confidence: 99%

Stratification and enumeration of Boolean functions by canalizing depth

Macauley

2016

Physica D: Nonlinear Phenomena

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ABSTRACT. Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively "picked off", until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.

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On the Number of Fanout-Free Functions and Unate Cascade Functions

Cited by 18 publications

References 21 publications

Nested canalyzing, unate cascade, and polynomial functions

Nested canalyzing, unate cascade, and polynomial functions

Average Path Length of Binary Decision Diagrams

Stratification and enumeration of Boolean functions by canalizing depth

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