Contents

Consensus theorem

Theorem in Boolean algebra

Variable inputs			Function values
x	y	z	${\displaystyle xy\vee {\bar {x}}z\vee yz}$	${\displaystyle xy\vee {\bar {x}}z}$
0	0	0	0	0
0	0	1	1	1
0	1	0	0	0
0	1	1	1	1
1	0	0	0	0
1	0	1	0	0
1	1	0	1	1
1	1	1	1	1

InBoolean algebra, the consensus theoremorrule of consensus^[1] is the identity:

${\displaystyle xy\vee {\bar {x}}z\vee yz=xy\vee {\bar {x}}z}$

The consensusorresolvent of the terms ${\displaystyle xy}$ and ${\displaystyle {\bar {x}}z}$is${\displaystyle yz}$. 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 ${\displaystyle y}$ includes a term that is negated in ${\displaystyle z}$ (or vice versa), the consensus term ${\displaystyle yz}$ is false; in other words, there is no consensus term.

The conjunctive dual of this equation is:

${\displaystyle (x\vee y)({\bar {x}}\vee z)(y\vee z)=(x\vee y)({\bar {x}}\vee z)}$

Proof[edit]

${\displaystyle {\begin{aligned}xy\vee {\bar {x}}z\vee yz&=xy\vee {\bar {x}}z\vee (x\vee {\bar {x}})yz\\&=xy\vee {\bar {x}}z\vee xyz\vee {\bar {x}}yz\\&=(xy\vee xyz)\vee ({\bar {x}}z\vee {\bar {x}}yz)\\&=xy(1\vee z)\vee {\bar {x}}z(1\vee y)\\&=xy\vee {\bar {x}}z\end{aligned}}}$

Consensus[edit]

The consensusorconsensus term of two conjunctive terms of a disjunction is defined when one term contains the literal ${\displaystyle a}$ and the other the literal ${\displaystyle {\bar {a}}}$, an opposition. The consensus is the conjunction of the two terms, omitting both ${\displaystyle a}$ and ${\displaystyle {\bar {a}}}$, and repeated literals. For example, the consensus of ${\displaystyle {\bar {x}}yz}$ and ${\displaystyle w{\bar {y}}z}$is${\displaystyle w{\bar {x}}z}$.^[2] The consensus is undefined if there is more than one opposition.

For the conjunctive dual of the rule, the consensus ${\displaystyle y\vee z}$ can be derived from ${\displaystyle (x\vee y)}$ and ${\displaystyle ({\bar {x}}\vee z)}$ through the resolution inference rule. This shows that the LHS is derivable from the RHS (ifA → B then A → AB; replacing A with RHS and B with (y ∨ z) ). The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS (inpropositional calculus), then LHS = RHS (in Boolean algebra).

Applications[edit]

In Boolean algebra, repeated consensus is the core of one algorithm for calculating the Blake canonical form of a formula.^[2]

Indigital logic, including the consensus term in a circuit can eliminate race hazards.^[3]

History[edit]

The concept of consensus was introduced by Archie Blake in 1937, related to the Blake canonical form.^[4] It was rediscovered by Samson and Mills in 1954^[5] and by Quine in 1955.^[6] Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "resolution principle".^[7][8]

References[edit]

^ "Canonical expressions in Boolean algebra", Dissertation, Department of Mathematics, University of Chicago, 1937, ProQuest 301838818, reviewed in J. C. C. McKinsey, The Journal of Symbolic Logic 3:2:93 (June 1938) doi:10.2307/2267634 JSTOR 2267634. The consensus function is denoted ${\displaystyle \sigma }$ and defined on pp. 29–31.

Further reading[edit]

• Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.