Bogdan J. Falkowski

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

A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC-cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from nondisjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. The programs for both algorithms use advantages of C language to speed up the execution. The comparison of different versions of the algorithm has been carried out. The algorithm and its implementation provide the fastest and most comprehensive program (having many options) known to the authors for the calculation of the Rademacher-Walsh transform. It successfully overcomes all drawbacks in the calculation of the transform from the design automation system based on spectral method-the SPECSYS system from Drexel University, which uses fast Walsh transform. >

A family of all essential Radix-2 addition/subtraction multi-polarity transforms: algorithms and interpretations in Boolean domain

By investigating some families of elementary order-2 matrices, new transforms of real vectors are introduced. When used for Boolean function transformations these transforms are one-to-one mappings in a binary/ternary vector space. The concept of different polarities of considered arithmetic and adding transforms is introduced. The links of arithmetic and adding transforms with classical logic design are discussed.<<ETX>>

Algorithm for the generation of disjoint cubes for completely and incompletely specified boolean functions

ABSTRACT A new algorithm is shown that generates the representation of completely or incompletely specified boolean functions in the form of arrays of disjoint ON- and DC- ( if any) cubes. A peculiar feature of the algorithm, which speeds up its execution, is the fact that in comparison to known algorithms it minimizes the number of usages of cube calculus operations.

One more way to calculate the Hadamard-Walsh spectrum for completely and incompletely specified boolean functions

A new algorithm is described for calculation of the forward Hadamard-Walsh transform of completely and incompletely specified boolean functions. It makes use of the properties of the disjoint cubes array representation of boolean functions.

Algorithms for the calculation of Hadamard-Walsh spectrum for completely and incompletely specified Boolean functions

The authors describe two algorithms for the calculation of the forward Hadamard-Walsh transform for completely and incompletely specified Boolean functions. The first method is based on direct manipulation from Karnaugh maps. The conversion starts from Karnaugh maps and results in Hadamard-Walsh spectral coefficients. The second algorithm makes use of the properties of a disjoint-cube-array representation of Boolean functions.<<ETX>>

Integration of logic synthesis and high-level synthesis into the DIADES design automation system

A description is presented of the high-level and logic synthesis stages in the digital design automation system DIADES. High level design, namely, data path synthesis, and control unit synthesis start from a parallel program graph, the form of description that includes both the control-flow and the data-flow graph. While the data path is allocated and scheduled, the control unit is designed to be composed of either microprogrammed units or finite-state machines. The latter are minimized in two dimensions (states and inputs), assigned and realized in logic. Several logic synthesis procedures, respective to various design styles and methodologies, can be used to design combinational parts of state machines, microprogrammed units, and data path logic.<<ETX>>

Two learning methods for a tree-search combinatorial optimizer

Several combinatorial problems of logic synthesis and other CAD problems have been solved in a uniform way using a general-purpose tree-searching program MULT-II. Two learning methods that have been implemented to improve the programs efficiency are presented. A weighted heuristic function, used to evaluate operators, is applied during a solution tree search. The optimal vector of coefficients for this function is learned in a simplified perceptron scheme. By using the second learning method, the similarity of shapes among the solution cost improvement curves is used to define the termination moment of the search process. The amplification effect of the concurrent action of both these methods has been observed.<<ETX>>

A fast computer algorithm for the generation of disjoint cubes for completely and incompletely specified Boolean functions

An algorithm is presented that generates an array of disjoint cubes describing completely and incompletely specified Boolean functions. A peculiar feature of the algorithm, which speeds up its execution, is that in comparison to known algorithms it minimizes the number of usages of cube calculus operations. Pointer structures and array allocations of the C language are effectively used in its implementation.<<ETX>>

One more way to calculate generalized Reed-Muller expansions of boolean functions

A new algorithm is given that converts disjoint cube representation of boolean functions into fixed-polarity generalized Reed-Muller expansions (GRME). Since the known fast algorithm that generates the GRME based on the factorization of the Reed-Muller transform matrix always starts from the truth table (minterms) of a boolean function, the method described has the advantages that come from requiring a smaller computer memory. Moreover, for the boolean functions described by only a few disjoint cubes, the method is much more efficient than the fast algorithm. The algorithm allows either the calculation of only selected Reed-Muller coefficients, or all the coefficients can be calculated in parallel.

Essential relations between classical and spectral approaches to analysis, synthesis, and testing of completely and incompletely specified Boolean functions

The links between spectral and classical methods used for the design of digital circuits are described. The real meaning of spectral coefficients from R and S Walsh spectra in classical logic terms is shown. The presented relations can be used for the calculation of Walsh-type transforms. The methods are based on direct manipulations on Karnaugh maps. The conversion starts from Karnaugh maps and results in the spectral coefficients.<<ETX>>