A method for the computation of Haar spectral coefficients is described using the OBDD representation of a function. An algebraic relationship between the circuit output probabilities and the Haar spectral coefficients is derived and used. The circuit output probabilities are computed by applying graph algorithms to the OBDDs. A fundamental dependence between the spectral coefficients and n + 1 simple Boolean equivalence relationships is noted.

New symmetries of degree two are introduced, along with spectral techniques for identifying these symmetries. Some applications of these symmetries are discussed, in particular their application to the construction of binary decision diagrams and the implementation of Boolean functions. 1.

Methods based on AND/OR graph representations of Boolean relations provide a promising new way of approaching VLSI CAD design automation problems. AND/OR graphs can represent any Boolean network and they allow for systematic reasoning through the application of the technique of recursive learning. An approach to build and analyze AND/OR graphs that makes use of hashing techniques in a way similar to that for modern Decision Diagram (DD) packages is described. Additionally, the problem of extracting spectral information from AND/OR graphs is also examined. Spectral information can be used for many CAD system tasks including synthesis, verification and test vector generation. It is shown that spectral information may be calculated directly from output probabilities and a method for estimating output probabilities from AND/OR graphs is presented. Experimental results regarding the AND/OR graph package efficiency and the extraction of spectral information are provided. 1 Introdu...

Introduction: Digital logic synthesis is an integral step in modern integrated circuit design. Coupled with the popular usage of synthesis techniques is the use of BDDs, which are data structures that represent Boolean functions [1]. These directed acyclic graphs (DAGs) have generated great interest due to their ability to represent many Boolean functions in a very compact manner and the existence of associated algorithms for their manipulation. Although BDDs have traditionally been viewed as purely behavioral entities, it is well known that they can represent "trees of multiplexers" due to the correspondence between their structure and the Shannon decomposition. With this point of view, a minimized BDD can represent a correspondingly small structural representation of a circuit in the form of a network of multiplexers. The motivation for developing a BDD based synthesis tool based on the structure of the DAG, is given by the fact that many good variable ordering heuristics have

A technique to detect and realize a parity logic function using a linear number of spectral coefficients is presented. Recent advances in computation methods for the spectra of a Boolean function have resulted in the determination of a single coefficient very efficiently. The use of a small set of spectral coefficients offers low cost and attractive alternatives to more traditional digital logic design and analysis techniques when coupled with the use of the new spectral computation approach. The detection and realization of the class of parity functions is an important problem since it generally requires the computation of 2 n Walsh coefficients when spectral methods are used, or, extensive symbolic algebraic manipulation when other methods are used. The method presented in this paper only requires the computation of n + 2 Walsh coefficients and n + 1 non-Walsh coefficients to detect and realize a parity function. 1 Introduction With the recent advent of an efficient method for th...

In this paper we provide polynomial time approximation techniques which allow us to calculate, to arbitrary levels of accuracy and with high probability of success, the spectral coefficients and autocorrelation coefficients of Boolean functions, given that those functions are expressed in either Sum-of-Products or Product-of-Sums form. 1 Introduction The focus of this paper is on the generation of spectral coefficients and autocorrelation coefficients for Boolean functions. Efficient calculation of those coefficients would allow digital logic analysts to draw on the large body of research already effectively employed in the area of signal processing. Utilisation of coefficient-based techniques in areas such as logic testing and synthesis has traditionally been hampered by the computational requirements for coefficient calculation. To reduce the computational demands, we use an approximation technique to estimate the coefficient values. As arbitrary levels of accuracy can still be obta...

This paper presents an algorithm to generate Walsh functions in four different orderings: Hadamard, Harmuth, Paley and strict sequency. By the analysis of the properties and mutual relations among these four orderings, the authors found a unified approach to generate any of the orderings from the primary set of Rademacher functions. By using these properties, the authors developed a programmable Walsh function generator for 64 outputs by both field programmable gate arrays and lookup table cascades to estimate the amount of hardware and performance. Such a programmable Walsh function generator can be used in VLSI testing, CDMA, pattern recognition, as well as image and signal processing.

In this paper we provide polynomial time approximation techniques which allow us to calculate, to arbitrary levels of accuracy and with high probability of success, the spectral coefficients and autocorrelation coefficients of Boolean functions, given that those functions are expressed in either Sum-of-Products or Product-of-Sums form. 1 Introduction The focus of this paper is on the generation of spectral coefficients and autocorrelation coefficients for Boolean functions. Efficient calculation of those coefficients would allow digital logic analysts to draw on the large body of research already effectively employed in the area of signal processing. Utilisation of coefficient-based techniques in areas such as logic testing and synthesis has traditionally been hampered by the computational requirements for coefficient calculation. To reduce the computational demands, we use an approximation technique to estimate the coefficient values. As any specified level of accuracy can still be o...

This paper presents a new methodology for the computation of the Reed-Muller spectral coefficients of a function of any fixed polarity using its OBDD representation. By using past results that allow the computations to be performed using real arithmetic, an efficient technique may be developed in the real domain with the resulting coefficients obtained by using a simple mapping relation to the GF (2) domain. These results mathematically justify the OBDD based approach developed in this work. This result is novel since it relies on the use of OBDDs and the concept of a Boolean function output probability to compute the coefficients. This approach is also very general in that it allows other types of coefficients (such as the Walsh) to be computed as well as the Reed-Muller forms with a single OBDD operation. A small example of this technique is given to illustrate the approach followed by some experimental results. I. Introduction The Reed-Muller (RM) spectrum can be used to provide i...

Goal models allow software engineers to model software system requirements and stake holder objectives at a high level of abstraction. We extend the use of goal models by presenting a probability model for them. The probability model can be used to compute the probability of satisfaction and denial of goals, and may have other applications as well. In our probability model, goal satisfaction and denial is expressed as a function of leaf level tasks being satisfied and denied given that all contribution links are satisfied. We also translate goal models to Bayesian networks that satisfy our probability model. Using Bayesian networks allows for the probability to be calculated efficiently, which we demonstrate in our experimental evaluation. Furthermore, using Bayesian networks allows the research on Bayesian networks to be applied on goal models. 1