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The Depth of All Boolean Functions

1977
*
SIAM journal on computing (Print)
*

It is shown that every

doi:10.1137/0206026
fatcat:hdyhsusfljavjfjoelyrrqnmda
*Boolean**function**of*n argumentshasacircuitofdepthn+]overthe basis {fl f : {0,.| }2 * {0'1 }i. l. ... Lntroduction ipira showed in If] that for any k > On there is a r'.uber N(k) sucli that if n > N(k) then any n argument*Boolean**function*has a circuit*of*denth n + log2lo82 .lo8rn. ... Schemes ' Our present constnuctions, and*all*pr:evious ones for minimizing*depth*that we know*of*, have*the*property*of*being "uniform" for aj-'l*functions**of*n arguments. ...##
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Stratification and enumeration of Boolean functions by canalizing depth

2016
*
Physica D : Non-linear phenomena
*

This generalizes recent work on

doi:10.1016/j.physd.2015.09.016
fatcat:j7iy667s5ndsvlwex5e6cywwfu
*the*algebraic structure*of*nested canalizing*functions*, and it yields a stratification*of**all**Boolean**functions*by their canalizing*depth*. ... Many*of*these networks use canalizing*Boolean**functions*, which has led to increased interest in*the*study*of*these*functions*. ... We will also study*the*impact*of**the*stratification*of**all**Boolean**functions*on random*Boolean*networks (RBNs). ...##
###
Stratification and enumeration of Boolean functions by canalizing depth
[article]

2015
*
arXiv
*
pre-print

This generalizes recent work on

arXiv:1504.07591v1
fatcat:ktit5t73angy7cydgjmk72minm
*the*algebraic structure*of*nested canalizing*functions*, and it yields a stratification*of**all**Boolean**functions*by their canalizing*depth*. ... Many*of*these networks use canalizing*Boolean**functions*, which has led to increased interest in*the*study*of*these*functions*. ... This gave us a stratification*of**the*set*of*n-variable*Boolean**functions*by canalizing*depth*. ...##
###
New Bounds for Energy Complexity of Boolean Functions
[article]

2020
*
arXiv
*
pre-print

For a

arXiv:1808.07199v2
fatcat:ddq3tbnu3zdaxn24cnn3glpjm4
*Boolean**function*f:{0,1}^n →{0,1} computed by a circuit C over a finite basis ℬ,*the*energy complexity*of*C (denoted by _(C)) is*the*maximum over*all*inputs {0,1}^n*the*numbers*of*gates*of**the*circuit ... to*Boolean*formulas, we show (F) = Ω (√(L(F))-*depth*(F) ) where L(F) is*the*size and*depth*(F) is*the**depth**of*a formula F. ... Acknowledgments*The*authors would like to thank*the*anonymous reviewers for their constructive comments. ...##
###
New Bounds for Energy Complexity of Boolean Functions
[chapter]

2018
*
Lecture Notes in Computer Science
*

For a

doi:10.1007/978-3-319-94776-1_61
fatcat:5qjhgjwgi5ahfet5gez3256sai
*Boolean**function*f : {0, 1} n → {0, 1} computed by a circuit C over a finite basis B,*the*energy complexity*of*C (denoted by EC B (C)) is*the*maximum over*all*inputs {0, 1} n*the*numbers*of*gates ... Energy Complexity*of*a*Boolean**function*over a finite basis B denoted by EC B ( f ) def * ... Acknowledgments*The*authors would like to thank*the*anonymous reviewers for their constructive comments. ...##
###
Limiting Negations in Bounded-Depth Circuits: An Extension of Markov's Theorem
[chapter]

2003
*
Lecture Notes in Computer Science
*

Then, we present tight upper bounds on

doi:10.1007/978-3-540-24587-2_13
fatcat:bw5evgz6rzcifmartr4l6g5oby
*the*number*of*negations for computing an arbitrary collection*of**Boolean**functions*. ... From a theorem*of*Markov,*the*minimum number*of*negation gates in a circuit sufficient to compute any collection*of**Boolean**functions*on n variable is ℓ = ⌈log(n + 1)⌉. ... From (4), for 2 ≤ ℓ ≤ D each*of*g ℓ j 's is*the*negation*of*a monotone*Boolean**function**of**all*h i (x) for 1 ≤ i ≤ m and*all*g k j (x)'s for 1 ≤ k ≤ ℓ − 1, where each g k j (x) is computed by some gate ...##
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The Dynamics of Canalizing Boolean Networks
[article]

2019
*
arXiv
*
pre-print

Motivated by these observations, we conduct mathematical study

arXiv:1902.00056v3
fatcat:gjcxbwaw3ng5zjf3mtqth4cjty
*of**the*attractor structure*of*a random*Boolean*network*of*canalizing*depth*one (i.e.,*the*smallest positive*depth*). ... For every positive integer ℓ, we give an explicit formula for*the*limit*of**the*expected number*of*attractors*of*length ℓ in an n-state random*Boolean*network as n goes to infinity. ... Data Availability Statement Python/sage code and*the*results*of*simulations used to support*the*findings*of*this study have been deposited at https://github.com/MathTauAthogen/Canalizing-*Depth*-Dynamics ...##
###
The Dynamics of Canalizing Boolean Networks

2020
*
Complexity
*

Motivated by these observations, we conduct mathematical study

doi:10.1155/2020/3687961
fatcat:na5p2dkworhhlojpthenppiq6i
*of**the*attractor structure*of*a random*Boolean*network*of*canalizing*depth*one (i.e.,*the*smallest positive*depth*). ... For every positive integer ℓ, we give an explicit formula for*the*limit*of**the*expected number*of*attractors*of*length ℓ in an n-state random*Boolean*network as n goes to infinity. ... EP, GP, and WQ are grateful to*the*New York Math Circle, where their collaboration started. ...##
###
A Multi-start Heuristic for Multiplicative Depth Minimization of Boolean Circuits
[chapter]

2018
*
Lecture Notes in Computer Science
*

In this work we propose a multi-start heuristic which aims at minimizing

doi:10.1007/978-3-319-78825-8_23
fatcat:i5frwy445rhxtiz34bifwnowwu
*the*multiplicative*depth**of**boolean*circuits. ...*The*multiplicative*depth*objective is encountered in*the*field*of*homomorphic encryption where ciphertext size depends on*the*number*of*consecutive multiplications. ... In order to decrease*the*overall multiplicative*depth**of*a*boolean*circuit by one,*all**the*parallel critical paths*of*this circuit must be rewritten. ...##
###
Enumerating Optimal Quantum Circuits using Spectral Classification

2020
*
2020 IEEE International Symposium on Circuits and Systems (ISCAS)
*

We show that any

doi:10.1109/iscas45731.2020.9180792
fatcat:bdntbnsckreb5nei6pu2t54ydy
*Boolean**function*can be derived from*the*implementation*of*its class representative without increasing any*of**the*stated cost*functions*. ...*The*database contains three circuits for each spectral-equivalent class representative, which are respectively optimized for*the*T -count,*the*T -*depth*, and*the*number*of*qubits. ... Acknowledgments This research was supported by*the*Swiss National Science Foundation (200021-169084 MAJesty). ...##
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Non-cancellative Boolean circuits: A generalization of monotone boolean circuits

2000
*
Theoretical Computer Science
*

We show that in

doi:10.1016/s0304-3975(98)00170-4
fatcat:auuq7iuaqrexrmvcuhh3undfiu
*the*absence*of*cancellation,*Boolean*circuits require super-polynomial size to compute*the*determinant interpreted over GF(2). This non-monotone*Boolean**function*is known to be in P. ... In*the*spirit*of*monotone complexity classes, we deÿne complexity classes based on non-cancellative*Boolean*circuits. ... in details in*the*proofs*of*Lemmas 2.1 and 2.2. ...##
###
Polynomial Threshold Functions, $AC^0 $ Functions, and Spectral Norms

1992
*
SIAM journal on computing (Print)
*

This result complements

doi:10.1137/0221003
fatcat:7b2ukvg6nfaarp4ktcdvhvurie
*the*results*of*[12]. (iii) A lower bound*of*Q(nJ'o'Y1og(n)) on*the*size*of*a*depth*-2 circuit*of*MAJORITY gates that computes an ACo*function*. ... Toronto, Toronto, Canada M5S 1A4. an R(npo'ylog(n)) lower bound on*the*number*of*terms needed to compute exactly a*Boolean*ACO*function*as a sign*function**of*a polynomial. ... Acknowledgement: We thank Noga Alon for his useful observations that greatly improved this paper and to Chaim Gotsman and Sunny Siu for their comments on an early draft*of*this paper. ...##
###
Non-cancellative Boolean circuits: A generalization of monotone Boolean circuits
[chapter]

1996
*
Lecture Notes in Computer Science
*

This non-monotone

doi:10.1007/3-540-62034-6_58
fatcat:qzufwzk55zhl5ildtnvozeudzm
*Boolean**function*is known to be in P. In*the*spirit*of*monotone complexity classes, we de ne complexity classes based on non-cancellative*Boolean*circuits. ... We s h o w that in*the*absence*of*cancellation,*Boolean*circuits require super-polynomial size to compute*the*determinant i n terpreted over GF(2). ... in details in*the*proofs*of*Lemmas 2.1 and 2.2. ...##
###
Optimal decision trees and one-time-only branching programs for symmetric Boolean functions

1984
*
Information and Control
*

It has turned out to be very hard to prove good lower bounds on

doi:10.1016/s0019-9958(84)80031-5
fatcat:bifskep5u5cbzewy34qxtigtly
*the*combinational complexity or*the**depth**of*explicitly defined*Boolean**functions*. ... Combinational complexity and*depth*are*the*most important complexity measures for*Boolean**functions*. ... For*all*symmetric*Boolean**functions*,*all**functions*whose prime implicants have length n, and*all**functions*whose prime clauses have length n*the*size*of*optimal branching programs*of*minimum*depth*equals ...##
###
Upper bounds for the formula size of symmetric Boolean functions

2014
*
Russian Mathematics (Izvestiya VUZ. Matematika)
*

We prove that

doi:10.3103/s1066369x14050041
fatcat:66igea26azgbjb3uxkotsevuxq
*the*complexity*of**the*implementation*of**the*counting*function**of*n*Boolean*variables by binary formulas is at most n 3.03 , and it is at most n 4.47 for DeMorgan formulas. ...*The*following bounds are proved for*the*formula size*of*any symmetric*Boolean**function**of*n variables: n 3.04 for binary formulas and n 4.48 for DeMorgan ones. ... INTRODUCTION In this paper we study*the*complexity*of**the*implementation*of*symmetric*Boolean**functions*by formulas over*the*basis B 2*of**all*binary*Boolean**functions*and over*the*standard basis B 0 = ...
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