Abstract
A new algorithm is given that converts a reduced representation of Boolean functions in the form of disjoint cubes to Generalized Adding and Arithmetic spectra. Since the known algorithms that generate Adding and Arithmetic spectra always start from the truth table of Boolean functions the method presented computes faster with a smaller computer memory. The method is extremely efficient for such Boolean functions that are described by only few disjoint cubes and it allows the calculation of only selected spectral coefficients, or all the coefficients can be calculated in parallel.
Highlights
Manipulations and calculations of discrete functions is a fundamental task in many areas of Computer Science and Engineering ranging from applied mathematics, linear integer programming to artificial intelligence and in Computer-Aided Design of digital circuits [1,2,3,4,5,6,7]
The new algorithm was implemented in C ++ and run on a Sun SPARC IPC 2/475 station
The number of disjoint cubes obtained and the time required for their calculation are given in the fourth and fifth columns, respectively
Summary
Manipulations and calculations of discrete functions is a fundamental task in many areas of Computer Science and Engineering ranging from applied mathematics, linear integer programming to artificial intelligence and in Computer-Aided Design of digital circuits [1,2,3,4,5,6,7]. With detailed investigations of operations on Boolean functions and variables in spectral domain of Arithmetic transform [22], such operations may be performed directly on decision diagrams, Different decision diagrams [6, 20, 21, 23] have proved to be very convenient data structures for majority of discrete functions representations permitting manipulations and calculation with large discrete functions efficiently in terms of space and time They are frequently used to represent data structures in modern CAD VLSI systems. The new algorithm has allowed practical applications of both Adding and Arithmetic transforms in real life problems’ sizes of analysis, synthesis, testing and probabilistics of Boolean functions for CAD systems using cubical ather than graph based representations of discrete functions, The algorithm introduced is valid for completely specified Boolean functions and for incompletely specified functions, since don’t care minterms can be represented in the form of disjoint cubes as input data to the algorithm. For an n variable Boolean function, there are 2 different polarities and there are 2 different canonical Arithmetic and Adding expansions of a Boolean functions
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