This paper argues that the ideas underlying the renormalization group technique applicable to phase transitions in physical systems are useful in distinguishing computational complexity classes. The paper presents a renormalization group transformation that maps an arbitrary Boolean function of N Boolean variables to one of N − 1 variables and shows that when this transformation is applied repe...

We prove a lower bound of 5n − o(n) for the circuit complexity of an explicit (constructible in deterministic polynomial time) Boolean function , over the basis U2. That is, we obtain a lower bound of 5n−o(n) for the number of {and, or} gates needed to compute a certain Boolean function, over the basis {and, or, not} (where the not gates are not counted). Our proof is based on a new combinatori...

In this article, a relationship between the Walsh-Hadamard spectrum and σ f , the sum-of-squares-modulus indicator (SSMI) of the generalized Boolean function is presented. It is demonstrated for every s-plateaued generalized Boolean function in n variables that σ f = 22n+s. Two constructions on generalized semi-bent Boolean functions are presented. A class of generalized semi-bent functions in ...

Arithmetic Walsh transform(AWT) of Boolean function caught our attention due to their arithmetic analogs of Walsh-Hadamard transform(WHT) recently. We present new results on AWT in this paper. Firstly we characterize the existence of linear structure of Boolean functions in terms of AWT. Secondly we show that the relation between AWT and WHT of a balanced Boolean function with a linear structur...

It is well known that Exclusive Sum-Of-Products (ESOP) expressions for Boolean functions require on average the smallest number of cubes. Thus, a simple complexity measure for a Boolean function is the number of cubes in its simplest ESOP. It will be shown that this structure-oriented measure of the complexity can be improved by a unique complexity measure which is based on the function. Thus, ...

In various elds of computer science, it is often required to represent Boolean functions succinctly and to manipulate them e ciently. An Ordered Binary Decision Diagram (OBDD) [1, 2] is a directed acyclic graph representing a Boolean function. OBDDs have the following properties. 1) For a given variable ordering, OBDDs have a reduced canonical form for each Boolean function. 2) Many practical B...

Because of the algebraic attacks, a high algebraic immunity is now an important criteria for Boolean functions used in stream ciphers. In this paper, by using the relationship between some flats and support of a n variables Boolean function f , we introduce a general method to determine the algebraic immunity of a Boolean function and finally construct some balanced functions with optimum algeb...

We consider the problem of evaluating a Boolean function on PRAMs. We exhibit a Boolean function f : {0, 1} → {0, 1} that can be evaluated in time O(log log n) on a deterministic CROW (Concurrent Read Owner Write) PRAM, but requires time Ω(log n) on EROW (Exclusive Read Owner Write) PRAMs. Our lower bound also holds for randomized Monte Carlo EROW PRAMs. This Boolean function is derived from th...

We exhibit links between pseudo-Boolean optimization, graph theory and logic. We show the equivalence of maximizing a pseudo-Boolean function and finding a maximum weight stable set; symmetrically minimizing a pseudo-Boolean function is shown to be equivalent to solving a weighted satisfiability problem.

In this paper we use the support of a Boolean function to establish some algebraic properties. These properties allow us to obtain the degree of a Boolean function without having to calculate its algebraic normal form. Furthermore, we derive some algorithms and compute the average time to obtain the degree of some Boolean functions from its support.