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The Depth of All Boolean Functions

W. F. McColl, M. S. Patterson
1977 SIAM journal on computing (Print)  
It is shown that every 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.  ... 
doi:10.1137/0206026 fatcat:hdyhsusfljavjfjoelyrrqnmda

Stratification and enumeration of Boolean functions by canalizing depth

Qijun He, Matthew Macauley
2016 Physica D : Non-linear phenomena  
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.  ...  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).  ... 
doi:10.1016/j.physd.2015.09.016 fatcat:j7iy667s5ndsvlwex5e6cywwfu

Stratification and enumeration of Boolean functions by canalizing depth [article]

Qijun He, Matthew Macauley
2015 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1504.07591v1 fatcat:ktit5t73angy7cydgjmk72minm

New Bounds for Energy Complexity of Boolean Functions [article]

Krishnamoorthy Dinesh, Samir Otiv, Jayalal Sarma
2020 arXiv   pre-print
For a 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.  ... 
arXiv:1808.07199v2 fatcat:ddq3tbnu3zdaxn24cnn3glpjm4

New Bounds for Energy Complexity of Boolean Functions [chapter]

Krishnamoorthy Dinesh, Samir Otiv, Jayalal Sarma
2018 Lecture Notes in Computer Science  
For a 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.  ... 
doi:10.1007/978-3-319-94776-1_61 fatcat:5qjhgjwgi5ahfet5gez3256sai

Limiting Negations in Bounded-Depth Circuits: An Extension of Markov's Theorem [chapter]

Shao Chin Sung, Keisuke Tanaka
2003 Lecture Notes in Computer Science  
Then, we present tight upper bounds on 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  ... 
doi:10.1007/978-3-540-24587-2_13 fatcat:bw5evgz6rzcifmartr4l6g5oby

The Dynamics of Canalizing Boolean Networks [article]

Elijah Paul, Gleb Pogudin, William Qin, Reinhard Laubenbacher
2019 arXiv   pre-print
Motivated by these observations, we conduct mathematical study 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  ... 
arXiv:1902.00056v3 fatcat:gjcxbwaw3ng5zjf3mtqth4cjty

The Dynamics of Canalizing Boolean Networks

Elijah Paul, Gleb Pogudin, William Qin, Reinhard Laubenbacher
2020 Complexity  
Motivated by these observations, we conduct mathematical study 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.  ... 
doi:10.1155/2020/3687961 fatcat:na5p2dkworhhlojpthenppiq6i

A Multi-start Heuristic for Multiplicative Depth Minimization of Boolean Circuits [chapter]

Sergiu Carpov, Pascal Aubry, Renaud Sirdey
2018 Lecture Notes in Computer Science  
In this work we propose a multi-start heuristic which aims at minimizing 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.  ... 
doi:10.1007/978-3-319-78825-8_23 fatcat:i5frwy445rhxtiz34bifwnowwu

Enumerating Optimal Quantum Circuits using Spectral Classification

Giulia Meuli, Mathias Soeken, Martin Roetteler, Giovanni De Micheli
2020 2020 IEEE International Symposium on Circuits and Systems (ISCAS)  
We show that any 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).  ... 
doi:10.1109/iscas45731.2020.9180792 fatcat:bdntbnsckreb5nei6pu2t54ydy

Non-cancellative Boolean circuits: A generalization of monotone boolean circuits

Rimli Sengupta, H. Venkateswaran
2000 Theoretical Computer Science  
We show that in 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.  ... 
doi:10.1016/s0304-3975(98)00170-4 fatcat:auuq7iuaqrexrmvcuhh3undfiu

Polynomial Threshold Functions, $AC^0 $ Functions, and Spectral Norms

Jehoshua Bruck, Roman Smolensky
1992 SIAM journal on computing (Print)  
This result complements 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.  ... 
doi:10.1137/0221003 fatcat:7b2ukvg6nfaarp4ktcdvhvurie

Non-cancellative Boolean circuits: A generalization of monotone Boolean circuits [chapter]

Rimli Sengupta, H. Venkateswaran
1996 Lecture Notes in Computer Science  
This non-monotone 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.  ... 
doi:10.1007/3-540-62034-6_58 fatcat:qzufwzk55zhl5ildtnvozeudzm

Optimal decision trees and one-time-only branching programs for symmetric Boolean functions

Ingo Wegener
1984 Information and Control  
It has turned out to be very hard to prove good lower bounds on 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  ... 
doi:10.1016/s0019-9958(84)80031-5 fatcat:bifskep5u5cbzewy34qxtigtly

Upper bounds for the formula size of symmetric Boolean functions

I. S. Sergeev
2014 Russian Mathematics (Izvestiya VUZ. Matematika)  
We prove that 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 =  ... 
doi:10.3103/s1066369x14050041 fatcat:66igea26azgbjb3uxkotsevuxq
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