3.1.2 Determining Prime Implicants by Iterated Consensus

The tabular method discussed above is based on the application of the Quine's theorem. The tabular method is simple, however, it requires the minterm canonical form to start with, which is exponential in number of input variables (alphabets). To cater to this issue, iterated consensus method does not expand the product terms (of SOP) to minterms. Instead, consensus is made between all pairs of product terms. The consensus terms of all pairs of terms that are not contained in some other term are added to the equation. The new terms are compared to the existing terms and among themselves to see if more new consensus terms can be generated. All terms that are contained in some other term are removed. Once no more new terms can be generated, the SOP comprises of only prime implicants. Generation of prime implicants by iterated consensus follows from the following theorem (Theorem 2).

Definition 9: An SOP formula is complete sum if it comprises all the prime implicants of the function it represents.

Theorem 2 : A SOP formula is a complete sum if and only if:

•  No term includes any other term.

•  The consensus of any two terms of the formula either does not exist or is contained in some term of the formula.

We will not prove the theorem formally; for a formal proof, the reader is referred to [6]. We only highlight the basic philosophy behind the proof.

Suppose a SOP representing a function is not a complete sum, because there is one prime implicant of the function that does not appear in the SOP (and all other terms are prime implicants). Therefore, the remaining prime implicant must be covered by two or more of the prime implicants in the SOP. Suppose for simplicity let us assume that there are two such prime implicants . If we add the consensus term of , we get another term that covers the missing prime implicant.

The theorem suggests a simple procedure (steps given below) to generate all the primes of a function, called iterated consensus.

1.  Start from an arbitrary SOP formula and add the consensus terms of all pairs of terms that are not contained in some other term.

2.  The new terms are compared to the existing terms and among themselves to see if new consensus terms can be generated (and added). All terms that are contained in some other term are removed.

3.  Repeat step 2 until no more new terms are created.

The steps generate a complete sum (i.e., all prime implicants)

As an example, consider the SOP for the function (it is the same function used in the last section, however, now the don't care terms are made minterms).

Steps of iterative consensus for the function are discussed below.

1.  Consensus of (1 st two terms) generates , which contains . By eliminating and adding , we have

2.  Consensus of will generate , which contains . By eliminating and adding , we have .

3.  do not have consensus

4.  do not have consensus; do not have consensus

5.  generates . and are in By eliminating and and adding we have .

6.  Consensus between generates , which contains . Eliminating and adding we get .

Now is the complete sum as no more new terms can be created. It may be noted that are the prime implicants; same result was determined using tabular method, as discussed in the last sub-section (however, in that case comprised only don't cares).