This paper considers the Reed-Muller transform for incompletely specified multiple-valued logic functions, which is obtained as the finite field polynomial representation. A new algorithm for dealing with single-variable functions is presented. It is applicable to finite fields of small sizes (two to four), which is of interest because only these fields are readily implementable with today's MVL technology. It is shown how such or any similar single-variable algorithm can be used to obtain a fast n-variable Reed-Muller transform based on this transform, a heuristic scheme is derived for dealing with incompletely specified functions. It has better computational properties than other methods and achieves the best results when applied to functions with a large number of unspecified points. >

A multiple-valued Reed-Muller transform for incompletely specified functions

Two algorithms are presented, the first is a technique to determine good, though not necessarily optimum, fixed polarity Reed-Muller expansions of completely specified boolean functions. The second algorithm determines the allocations of the ‘don't care’ terms of incompletely specified boolean functions resulting in optimum positive polarity Reed-Muller expansions. Additionally, investigations are made into combining these two techniques to determine fixed polarity Reed-Muller expansions of incompletely specified boolean functions. Results are presented which show the effectiveness of the techniques and comparisons are made with existing methods.

Optimization of Reed-Muller logic functions

Tabular techniques and algorithms for generating Kronecker expansions are presented. The Kronecker expansions, a subclass of mixed-polarity Reed-Muller expansions, can be generated from Boolean functions Kronecker expansions of different polarity. The techniques can be used for completely and incompletely specified Boolean functions. The algorithms were implemented in Pascal and fully tested. Examples are given to illustrate the procedures.

http://ieeexplore.ieee.org/iel1/2192/11318/00532853.pdf

Tabular techniques for generating Kronecker expansions

This paper presents a tabular technique for the calculation of fixed-polarity polynomial expressions for multiple-valued (MV) functions. The technique is derived from a generalization of the corresponding methods for fixed-polarity Reed-Muller (FPRM) expressions for switching functions. All useful features of tabular techniques for FPRMs (e.g. simplicity of the involved operations and high possibilities for parallelization of the calculation procedure) are preserved. The method can be extended to the calculation of coefficients in Kronecker expressions for MV functions.

/pdf/efficient-calculation-of-fixed-polarity-polynomial-3cmmspyr8f.pdf

Efficient calculation of fixed-polarity polynomial expressions for multiple-valued logic functions

Decomposition techniques are utilised for mixed polarity Reed-Muller minimisation, which lead to Reed-Muller programmable logic array implementations for Boolean functions. The proposed algorithm produces a simplified mixed polarity Reed-Muller format from the conventional sum-of-products input based on a top-down strategy. The output format belongs to the most general class of AND/XOR forms, namely exclusive-OR sum-of-products. This method is further generalised to very large multiple output Boolean functions. The developed decomposition method is implemented in the C language and tested with MCNC and IWLS'93 benchmarks. Experimental results show that the decomposition method can produce much better results than Espresso for many test cases. This efficient method offers compact Reed-Muller programmable logic array implementations with the added advantage of easy testability, in contrast to the conventional programmable logic array realisations.

Optimisation of Reed-Muller PLA implementations

Tabular techniques are described for the conversion between boolean expressions and Reed-Muller polynomials, and for the derivation of fixed polarities. The techniques are simple, systematic, and can be used manually or programmed on a computer. Further, they can be used for any number of variables and hence overcome map limitations. Computer programs have been developed to implement the algorithms.

Tabular techniques for Reed—Muller logic

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Tabular techniques for Reed—Muller logic