Effective computer methods for the calculation of Rademacher-Walsh spectrum for completely and incompletely specified Boolean functions

Abstract

A theory has been developed to calculate the Rademacher-Walsh transform from a cube a r r a y specification of incompletely specified Boolean functions. T h e importance of representing Boolean functions a s a r r a y s of disjoint ON-a n d Ix-cubes has been pointed out, a n d a n efficient new algorithm to generate disjoint cubes f r o m nondisjoint ones has been designed. The t r a n s f o r m algorithm makes use of the properties of a n a r r a y of disjoint cubes a n d allows the determination of the spectral coefficients in a n independent way. The programs for both algorithms use advantages of C language to speed u p the execution. The comparison of different versions of the algorithm has been carried out. T h e presented algorithm a n d its implementation is the fastest a n d most comprehensive program (having many options) known t o us for the calculation of R ademacher-Walsh transform. It successfully overcomes all drawbacks in the calculation of the t r a n s f o r m f r o m the design automation system based on spectral methods-the SPECSYS system from Drexel University that uses Fast Walsh T r a n sform.

Low Training Strength High Capacity Classifiers for Accurate Ensembles Using Walsh Coefficients

“…For a two-class supervised learning problem of  training patterns, the target label given to each pattern Realistic learning problems are ill-posed [8], and therefore  may be partially specified, noisy and possibly contradictory. Relationships for computing spectral coefficients for partially specified Boolean functions, are proved in [9], for which the context is logic circuit design. The relevant ideas are presented here using different terminology, specifically minterms interpreted as patterns.…”

“…The equations (3) and (4) n is the number of class 0 patterns (false minterms in [9]) for which both  and  have the logical value 0. Corresponding definitions follow for 01 n and 10 n .…”

“…In [9], the concept of a standard trivial function  is introduced. Each spectral coefficient gives a correlation value between the Boolean function  and  .…”

“…Corresponding definitions follow for 01 n and 10 n . According to [9], all spectral coefficients where l may be i or ij. Substitution of (5) into (6) gives various equivalent formulae, but the advantage of (6) is that it is not necessary to include unspecified patterns  darkly shaded region) according to [7], which for simplicity is restricted to one dimension (x).…”

Spectral Response of Ternary Logic Netlists

“…Methods that have been devised to represent switching functions in a more compact manner include decision diagrams [3] and cube lists [4]. Unfortunately, these structures continue to have a worst case size of O(r n ) where r is the number of logic values and n is the number of logic network inputs.…”