We say that a reversible boolean function on n bits has alternation depth d if it can be written as the sequential composition of d reversible boolean functions, each of which acts only on the top n − 1 bits or on the bottom n− 1 bits. We show that every reversible boolean function of n > 4 bits has alternation depth 9.

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...

Decomposition of any Boolean Function BFn of n binary inputs into an optimal inverter coupled network of Symmetric Boolean functions SFk (k ≤ n) is described. Each SF component is implemented by Threshold Logic Cells, forming a complete and compact T -Cell Library. Optimal phase assignment of input polarities maximizes local symmetries. Rank spectrum is a new BFn description independent of inpu...

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...

Although there are plenty of AutoCAD books available for architects, interior designers, and engineers, it has been difficult for the author to find an appropriate textbook for students who major in interior design while teaching AutoCAD, especially 3-D AutoCAD at the sophomore level. Most existing 3-D AutoCAD textbooks are more engineering oriented. Industrial machine components such as bolts ...

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, ...

We give an adaptive algorithm that tests whether an unknown Boolean function f : {0, 1}n →{0, 1} is unate (i.e. every variable of f is either non-decreasing or non-increasing) or -far from unate with one-sided error and ̃ O(n/ ) many queries. This improves on the best adaptive O(n/ )-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri [1] when 1/ n. Combined with...

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...