Minimisation of fixed-polarity AND/XOR canonical networks

A method for extracting the cubes of the generalised Reed-Muller (GRM) form of a Boolean function with a given polarity is presented. The method does not require exponential space and time complexity and it achieves the lower-bound time complexity. The proof of the method's correctness constitutes the first half of the paper. Also, a separate heuristic algorithm to find the optimal polarity that requires the least number of cubes in the GRM representation is proposed. The algorithm is fast and derives the polarity for variables and extracts all cubes simultaneously. It is based on the concept of a Boolean centre for vertices, which emulates the centre of gravity concept in geometry. The experimental results for the heuristic algorithm agree strongly with the authors observations and analysis.

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