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VLSI Design
Volume 14 (2002), Issue 1, Pages 13-21

Generalized Inclusive Forms—New Canonical Reed-Muller Forms Including Minimum ESOPs

Department of Electrical and Computer Engineering, Portland State University, 1800, 6th Avenue, Portland, OR 97207-0751, USA

Received 20 January 2000; Revised 4 October 2000

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper describes two families of canonical Reed-Muller forms, called inclusive forms (IFs) and their generalization, the generalized inclusive forms (GIFs), which include minimum ESOPs for any Boolean function. We outline the hierarchy of known canonical forms, in particular, pseudo-generalized Kronecker forms (PGKs), which led us to the discovery of the new families. Next, we introduce special binary trees, called the S/D trees, which underlie IFs and permit their enumeration. We show how to generate IFs and GIFs and prove that GIFs include minimum ESOPs. Finally, we present the results of computer experiments, which show that GIFs reduce the search space for minimum ESOP by several orders of magnitude, and this reduction grows exponentially with the number of variables.