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Negation Trees: A Unified Approach to Boolean Function Complementation
May 1996 (vol. 45 no. 5)
pp. 626-630

Abstract—Boolean function complementation is a basic operation of Boolean algebra. For functions given in SOP form, the complementation methods include the DeMorgan's Law, sharp [3], disjoint sharp[2], unate complementation [4], recursive [4], and Sasao's method [5].

This paper accomplishes three purposes: 1) It exposes an underlying unification of the existing complementation algorithms. It is proven that a) unate-complementation and sharp are the same as the DeMorgan Law algorithm; b) Sasao's algorithm is the same as disjoint sharp; c) disjoint sharp is a special case of the recursive method. 2) It proposes negation trees for the representation of functions and their complements, and 3) it gives faster algorithms for finding complements of functions in SOP form based on negation trees.

[1] R.E. Bryant, "Graph-Based Algorithms for Boolean Function Manipulation," IEEE Trans. Computers, Vol. C-35, No. 8, Aug. 1986, pp. 667-690.
[2] S.J. Hong,R.G. Cain, and D.L. Ostapko,"MINI: A Heuristic Approach for Logic Minimization," IBM J. Research and Development, pp. 443-458, Sept. 1974.
[3] J.P. Roth,Computer Logic, Testing and Verification. Computer Science Press, 1980.
[4] R.K. Brayton, G.D. Hachtel, C.T. McMullen, and A.L. Sangiovanni-Vincintelli, Logic Minimization Algorithms for VLSI Synthesis.Boston: Kluwer Academic, 1984.
[5] T. Sasao,"An Algorithm to Derive the Complement of a Binary Function with Multiple-Valued Inputs," IEEE Trans. Computers, vol. 34, no. 2, pp. 131-140, Feb. 1985.
[6] Y. Wang and C. McCrosky,"Negation Trees—A Unified Approach to Boolean Function Complementation," Dept. of Computational Science, Univ. of Saskatchewan, July, 1995.
[7] Y. Wang,"New Data Structures for Boolean Functions," Research Report 94-3, Dept. of Computational Science, Univ. of Saskatchewan, May, 1994.

Index Terms:
Boolean functions, unate functions, complementation algorithms, negation trees.
Yuke Wang, Carl McCrosky, "Negation Trees: A Unified Approach to Boolean Function Complementation," IEEE Transactions on Computers, vol. 45, no. 5, pp. 626-630, May 1996, doi:10.1109/12.509916
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