Complexity Of Boolean Functions Research Articles

On parity decision trees for Fourier-sparse Boolean functions

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows: Let \(f : \mathbb {F}_2^n \rightarrow \lbrace -1, 1\rbrace\) be a Boolean function with Fourier support 𝒮 and Fourier sparsity k .

Trap spaces of Boolean networks are conflict-free siphons of their Petri net encoding

Boolean network modeling of gene regulation but also of post-transcriptomic systems has proven over the years that it can bring powerful analyses and corresponding insight to the many cases where precise biological data is not sufficiently available to build a detailed quantitative model. Besides simulation, the analysis of such models is mostly based on attractor computation, since those correspond roughly to observable biological phenotypes. The recent use of trap spaces made a real breakthrough in that field allowing to consider medium-sized models that used to be out of reach. However, with the continuing increase in model size and complexity of Boolean update functions, the state-of-the-art computation of minimal trap spaces based on prime implicants shows its limits due to the difficulty of the prime-implicant computation.In this article we explore and prove for the first time a connection between trap spaces of a general Boolean network and siphons of its Petri net encoding. Besides important theoretical applications in studying properties of trap spaces, the connection enables us to propose an alternative approach to compute minimal trap spaces, and hence complex attractors, of a general Boolean network. It replaces the need for prime implicants by a completely different technique, namely the enumeration of maximal siphons in the Petri net encoding of the original model. We then demonstrate its efficiency and compare it to the state-of-the-art methods on a large collection of real-world and randomly generated models.

Multiplicative Complexity of XOR Based Regular Functions

XOR-AND Graphs (XAGs) are an enrichment of the classical AND-Inverter Graphs (AIGs) with XOR nodes. In particular, XAGs are networks composed by ANDs, XORs, and inverters. Besides several emerging technologies applications, XAGs are often exploited in cryptography-related applications based on the multiplicative complexity of a Boolean function. The multiplicative complexity of a function is the minimum number of AND gates (i.e., multiplications) that are sufficient to represent the function over the basis {AND, XOR, NOT}. In fact, the minimization of the number of AND gates is important for high-level cryptography protocols such as secure multiparty computation, where processing AND gates is more expensive than processing XOR gates. Moreover, it is an indicator of the degree of vulnerability of the circuit, as a small number of AND gates corresponds to a high vulnerability to algebraic attacks. In this paper we study the multiplicative complexity of Boolean functions characterized by two particular regularities, called autosymmetry and D-reducibility. Moreover, we exploit these regularities for decreasing the number of AND nodes in XAGs. The experimental results validate the proposed approaches.

Certificate complexity of elementary symmetric Boolean functions

Boolean functions have important applications in information technology and computer science. Certificate complexity is an important combinatorial measure of Boolean function complexity. In this paper, we first introduce many concise and efficient notations then we study the elementary symmetric Boolean functions and obtain their certificate complexities when their degrees are odd or powers of 2. We show that both upper and lower bounds of certificate complexities can be attained.

Low-Latency Boolean Functions and Bijective S-boxes

In this paper, we study the gate depth complexity of (vectorial) Boolean functions in the basis of {NAND, NOR, INV} as a new metric, called latency complexity, to mathematically measure the latency of Boolean functions. We present efficient algorithms to find all Boolean functions with low-latency complexity, or to determine the latency complexity of the (vectorial) Boolean functions, and to find all the circuits with the minimum latency complexity for a given Boolean function. Then, we present another algorithm to build bijective S-boxes with low-latency complexity which with respect to the computation cost, this algorithm overcomes the previous methods of building S-boxes.As a result, for latency complexity 3, we present n-bit S-boxes of 3 ≤ n ≤ 8 with linearity 2n−1 and uniformity 2n−2 (except for 5-bit S-boxes for whose the minimum achievable uniformity is 6). Besides, for latency complexity 4, we present several n-bit S-boxes of 5 ≤ n < 8 with linearity 2n−2 and uniformity 2n−4.

On the complexity of Boolean functions in some classes of hypercontact schemes

В классах $\operatorname{РМ}_F^{(n)}$ расширенных матриц над кольцами полиномов с идемпотентными переменными определены подклассы (гиперконтактных схем): $\operatorname{ГС}_F^{(n)}$ (над произвольным полем $F$) и $\operatorname{ГС}_Z^{(n)}$ (над кольцом целых чисел), - алгебраически расширяющие класс матриц инциденций контактных схем ($\operatorname{КС}^{(n)}$) и реализующие произвольные булевы функции $n$ переменных со сложностью менее $3\sqrt{2}\cdot2^{n/2}$ контактов. Такой же порядок нижней оценки получен для соответствующей функции Шеннона в классе $\operatorname{ГС}_{F_q}^{(n)}$ над произвольным конечным полем $F_q$. Для матриц класса $\operatorname{ГС}_Z^{(n)}$ найдена физическая интерпретация в виде матриц инциденций-зацеплений контактно-трансформаторных схем.

On the average-case complexity of Boolean functions under binomial distribution on their domains

Abstract Given a binomial probability distribution on the n-dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n-place Boolean functions is established.

Implementation complexity of Boolean functions with a small number of ones

Abstract We consider the class F n, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class F n, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis { x & y , x ‾ } $ \{x\&y,\overline x\} $ (or in the basis { x ∨ y , x ‾ } ) $ \{x\vee y,\overline x\}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.

Asymptotics for the Complexity of Boolean Functions with Small Number of Ones

Рассматривается класс булевых функций $F_{n,k}$, состоящий из всех тех функций от $n$ переменных, каждая из которых обращается в единицу ровно на $k$ наборах значений переменных. При малых $k$, например, при $k<\ln n$, для каждой функции из $F_{n,k}$ найдена асимптотика для сложности реализации этой функции схемой из функциональных элементов в неизбыточном базисе, содержащем $x\to y$ и $\overline{x\to y}$. Библиография: 7 названий.

Asymptotics for the Complexity of Boolean Functions with Small Number of Ones

Asymptotics for the Complexity of Boolean Functions with Small Number of Ones

New bounds for energy complexity of Boolean functions

For a Boolean function f:{0,1}n→{0,1} computed by a Boolean circuit C over a finite basis B, the energy complexity of C (denoted by ECB(C)) is the maximum over all inputs {0,1}n of the number gates of the circuit C (excluding the inputs) that output a one. Energy Complexity of a Boolean function over a finite basis B denoted by ECB(f)=defminC⁡ECB(C) where C is a Boolean circuit over B computing f.We study the case when B={∧2,∨2,¬}, the standard Boolean basis. It is known that any Boolean function can be computed by a circuit (with potentially large size) with an energy of at most 3n(1+ϵ(n)) for a small ϵ(n) (which we observe is improvable to 3n−1). We show several new results and connections between energy complexity and other well-studied parameters of Boolean functions.•For all Boolean functions f, EC(f)≤O(DT(f)3) where DT(f) is the optimal decision tree depth of f.•We define a parameter positive sensitivity (denoted by psens), a quantity that is smaller than sensitivity (Cook et al. 1986, [3]) and defined in a similar way, and show that for any Boolean circuit C computing a Boolean function f, EC(C)≥psens(f)/3.•For a monotone function f, we show that EC(f)=Ω(KW+(f)) where KW+(f) is the cost of monotone Karchmer-Wigderson game of f.•Restricting the above notion of energy complexity to Boolean formulas, we show EC(F)=Ω(L(F)−Depth(F)) where L(F) is the size and Depth(F) is the depth of a formula F.

Revisiting Deutsch-Jozsa algorithm

The Deutsch-Jozsa algorithm is essentially faster than any possible deterministic classical algorithm for solving a promise problem that is in fact a symmetric partial Boolean function, named as the Deutsch-Jozsa problem. The Deutsch-Jozsa problem can be equivalently described as a partial function DJn0:{0,1}n→{0,1} defined as: DJn0(x)=1 for |x|=n/2, DJn0(x)=0 for |x|=0,n, and it is undefined for the remaining cases, where n is even, and |x| is the Hamming weight of x. The Deutsch-Jozsa algorithm needs only one query to compute DJn0 but the classical deterministic algorithm requires n2+1 queries to compute it in the worse case.We present all symmetric partial Boolean functions with degree 1 and 2; We prove the exact quantum query complexity of all symmetric partial Boolean functions with degree 1 and 2. We prove Deutsch-Jozsa algorithm can compute any symmetric partial Boolean function f with exact quantum 1-query complexity.

The generalized complexity of linear Boolean functions

AbstractWe study generalized (in terms of bases) complexity of implementation of linear Boolean functions by Boolean circuits in arbitrary functionally complete bases; the complexity of a circuit is defined as the number of gates. Let L*(n) be the minimal number of gates sufficient for implementation of an arbitrary linear Boolean function of n variables in an arbitrary functionally complete basis. We show that L*(0) = L*(1) = 3 and L*(n) = 7(n – 1) for any natural n ≥ 2.

Implementation complexity of Boolean functions with a small number of ones

Рассматривается класс булевых функций $F_{n,k}$, состоящий из всех тех функций от $n$ переменных, каждая из которых обращается в единицу ровно на $k$ наборах значений переменных. При небольших $k$ класс $F_{n,k}$ разбивается на подклассы, и для каждого подкласса находится асимптотика функции Шеннона для сложности реализации функций из этого подкласса схемами из функциональных элементов в базисе $\{x&y,\overline x\}$ (или в базисе $\{x\vee y,\overline x\}$); весами элементов базисов при этом могут быть любые строго положительные числа.

Upper Bounds on the Multiplicative Complexity of Symmetric Boolean Functions.

A special metric of interest about Boolean functions is multiplicative complexity (MC): the minimum number of AND gates sufficient to implement a function with a Boolean circuit over the basis {XOR, AND, NOT}. In this paper we study the MC of symmetric Boolean functions, whose output is invariant upon reordering of the input variables. Based on the Hamming weight method from Muller and Preparata (1975), we introduce new techniques that yield circuits with fewer AND gates than upper bounded by Boyar et al. in 2000 and by Boyar and Peralta in 2008. We generate circuits for all such functions with up to 25 variables. As a special focus, we report concrete upper bounds for the MC of elementary symmetric functions and counting functions with up to n = 25 input variables. In particular, this allows us to answer two questions posed in 2008: both the elementary symmetric and the counting functions have MC 6. Furthermore, we show upper bounds for the maximum MC in the class of n-variable symmetric Boolean functions, for each n up to 132.

Quadratically Tight Relations for Randomized Query Complexity

In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions R(f). The certificate complexity C(f) is such a complexity measure for the zero-error randomized query complexity R0(f): C(f) ≤R0(f) ≤C(f)2. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity EC(f), which is also a quadratically tight bound on R0(f): EC(f) ≤R0(f) = O(EC(f)2). For R(f), we prove that EC2/3 ≤R(f). We then prove that EC(f) ≤C(f) ≤EC(f)2 and show that there is a quadratic separation between the two, thus EC(f) gives a tighter upper bound for R0(f). The measure is also related to the fractional certificate complexity FC(f) as follows: FC(f) ≤EC(f) = O(FC(f)3/2). This also connects to an open question by Aaronson whether FC(f) is a quadratically tight bound for R0(f), as EC(f) is in fact a relaxation of FC(f). In the second part of the work, we investigate whether the corruption bound corr𝜖(f) quadratically approximates R(f). By Yao’s theorem, it is enough to prove that the square of the corruption bound upper bounds the distributed query complexity \(\mathsf {D}^{\mu }_{\epsilon }(f)\) for all input distributions μ. Here, we show that this statement holds for input distributions in which the various bits of the input are distributed independently. This is a natural and interesting subclass of distributions, and is also in the spirit of the input distributions studied in communication complexity in which the inputs to the two communicating parties are statistically independent. Our result also improves upon a result of Harsha et al. (2016), who proved a similar weaker statement. We also note that a similar statement in the communication complexity is open.

Complexity of Boolean functions' representations in classes of extended pair-generated operator forms

Complexity of Boolean functions' representations in classes of extended pair-generated operator forms

The difficulty of Monte Carlo approximation of multivariate monotone functions

We study the L1-approximation of d-variate monotone functions based on information from n function evaluations. It is known that this problem suffers from the curse of dimensionality in the deterministic setting, that is, the number n(ε,d) of function evaluations needed in order to approximate an unknown monotone function within a given error threshold ε grows at least exponentially in d. In the randomized setting (Monte Carlo setting) the complexity n(ε,d) grows exponentially in d (modulo logarithmic terms) only. An algorithm exhibiting this complexity is presented. The problem remains difficult as best methods known are deterministic if ε is comparably small, namely ε⪯1∕d. This inherent difficulty is confirmed by lower complexity bounds which reveal a joint (ε,d)-dependence and from which we deduce that the problem is not weakly tractable. The lower bound proof also has implications on the complexity of learning Boolean monotone functions.

Exact Value of the Nonmonotone Complexity of Boolean Functions

We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function f (the inversion complexity of the function f) is equal to ⌈log2(d(f) + 1)⌉, where d(f) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis B of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function f is equal to ⌈log2(d(f)/D(B) +1)⌉, where D(B) = max d(ω); the maximum is taken over all nonmonotone functions ω of the basis B.

Complexity of the Realization of a Linear Boolean Function in the Class of π-Schemes

Using Khrapchenko’s method, we obtain the exact lower bound of 40 for the complexity in the class of π-schemes of a linear Boolean function depending substantially on 6 variables. We give a simplified proof of several lower bounds for the complexity of linear Boolean functions which are previously obtained on the basis of the same method.