Consensus theorem

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

In Boolean algebra, the consensus theorem or rule of consensus^[1] is the identity:

Karnaugh map of AB ∨ AC ∨ BC. Omitting the red rectangle does not change the covered area.

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

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

The conjunctive dual of this equation is:

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

Proof

{\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

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

For the conjunctive dual of the rule, the consensus $y\vee z$ can be derived from $(x\vee y)$ and $({\bar {x}}\vee z)$ through the resolution inference rule. This shows that the LHS is derivable from the RHS (if A → 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 (in propositional calculus), then LHS = RHS (in Boolean algebra).

Applications

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

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

History

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

^ Frank Markham Brown [d], Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 44
^ ^a ^b Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 81
^ Rafiquzzaman, Mohamed (2014). Fundamentals of Digital Logic and Microcontrollers (6 ed.). John Wiley & Sons. p. 65. ISBN 978-1118855799.
^ "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 $\sigma$ and defined on pp. 29–31.
^ Edward W. Samson, Burton E. Mills, Air Force Cambridge Research Center, Technical Report 54-21, April 1954
^ Willard van Orman Quine, "The problem of simplifying truth functions", American Mathematical Monthly 59:521-531, 1952 JSTOR 2308219
^ John Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", Journal of the ACM 12:1: 23–41.
^ Donald Ervin Knuth, The Art of Computer Programming 4A: Combinatorial Algorithms, part 1, p. 539