Consensus theorem


In Boolean algebra, the consensus theorem or rule of consensus is the identity:
The consensus or resolvent of the terms and is. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If includes a term which is negated in , the consensus term is false; in other words, there is no consensus term.
The conjunctive dual of this equation is:

Proof

Consensus

The consensus or consensus term of two conjunctive terms of a disjunction is defined when one term contains the literal and the other the literal, an opposition. The consensus is the conjunction of the two terms, omitting both and, and repeated literals. For example, the consensus of and is. The consensus is undefined if there is more than one opposition.
For the conjunctive dual of the rule, the consensus can be derived from and through the resolution inference rule. This shows that the LHS is derivable from the RHS. The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS, then LHS = RHS.

Applications

In Boolean algebra, repeated consensus is the core of one algorithm for calculating the Blake canonical form of a formula.
In digital logic, including the consensus term in a circuit can eliminate race hazards.

History

The concept of consensus was introduced by Archie Blake in 1937, related to the Blake canonical form. It was rediscovered by Samson and Mills in 1954 and by Quine in 1955. Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "resolution principle".