Lower Bounds on Size of Periodic Functions in Exclusive-OR Sum-of-Products Expressions

Yasuaki NISHITANI  Kensuke SHIMIZU  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.3   pp.475-482
Publication Date: 1994/03/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on the 6th Karuizawa Workshop on Circuits and Systems)
Category: Computer Aided Design (CAD)
Keyword: 
exclusive-OR sum-of-products,  size of circuits,  lower bound,  logic minimization,  logic design,  

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Summary: 
This paper deals with the size of switching functions in Exclusive-OR sum-of-products expressions (ESOPs). The size is the number of products in ESOP. There are no good algorithms to find an exact minimum ESOP. Since the exact minimization algorithms take a time in double exponential order, it is almost impossible to minimize ESOPs for an arbitrary n-variable functions with n5. Then,it is necessary to study the size of some concrete functions. These concrete functions are useful for testing heuristic minimization algorithms. In this paper we present the lower bounds on size of periodic functions in ESOPs. A symmetric function is said to be periodic when the vector of weights of inputs X such that f(X)1 is periodic. We show that the size of a 2t+1-periodic function with rank r is proportional to n2t+r, where t0 and 0r2t, i.e., in polynomial order,and thet the size of a (2s+1)2t-periodic function with s0 and t0 is greater than or equal to (3/2)n-(2s+1)2t, i.e., in exponential order. The concrete function the size of which is greater than or equal to 32(3/2)n-8 is presented. This function requires the largest size among the concrete functions the sizes of which are known. Some results for non-periodic symmetric functions are also given.