Logic Synthesis with AND and OR Gates in Multi-Levels:General Division.

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General Division

Rewriting of a logic expression using parentheses can be done by the following division. The division is based on the use of sub-expressions that can be found in the given logic expression. The given logic expression in a sum-of-products can be rewritten with parentheses if it has common sub-expressions. For example, the logic expression in Eq. 32.1 can be converted to the following expression, using a sub- expression (a ∨ b):

Logic Synthesis with AND and OR Gates in Multi-Levels-0398

This rewriting can be regarded symbolically as division. Rewriting of the expression in Eq. 32.1 into the one in Eq. 32.3 may be regarded as division with the divisor x = a ∨ b, the quotient q = cd ∨ ce and the remainder r = ab. Then, the expression f can be represented as follows:

Logic Synthesis with AND and OR Gates in Multi-Levels-0399

The division is symbolically denoted as f/x. Generally, the quotient should not be 0, but the remainder may be 0.

The division, however, may yield many different results because there are many possibilities in choosing a divisor and also the given logic function can be written in many different logic expressions, as explained in the following.

Division can be repeated on one given logic expression. Suppose f = ab ∨ ac ∨ ba ∨ bc ∨ ca ∨ c b is given. Repeating division three times, choosing successively b ∨ c, a ∨ c, and a ∨ b as divisors, the following result is derived:

f = x1a ∨ x2b ∨ x3c

with divisors x1 = b ∨ c , x2 = a ∨ c , and x3 = a ∨ b.

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