CONSENSUS THEOREM Choopan Rattanapoka

Introduction to The Consensus Theorem The consensus theorem is very useful in simplifying Boolean expressions. Given an expression of the form XY + X’Z + YZ then term YZ is redundant and can be eliminated to form the equivalent expression XY + X’Z The eliminated term is referred to as the consensus term.

Consensus Term Given a pair of terms for which a variable appears in one term and the complement of that variable in another . The consensus term is formed by multiplying the two original terms together, leaving out the selected variable and its complement. Example : AB and A’C , consensus is BC ABD and B’DE’, consensus is (AD)(DE’)  ADE’

The consensus theorem (1) The consensus theorem can be stated as follows: XY + X’Z + YZ = XY + X’Z Proof XY + X’Z + YZ  XY + X’Z + (X + X’)YZ  XY + X’Z + XYZ + X’YZ  (XY + XYZ) + (X’Z + X’YZ)  XY(1 + Z) + X’Z(1 + Y)  XY + X’Z

The consensus theorem (2) Example : Simplify this expression A’B’ + AC + BC’ + B’C + AB Ans : A’B’ + AC + BC

The consensus theorem (3) The dual form of the consensus theorem is (X + Y)(X’ + Z)(Y + Z) = (X + Y)(X’ + Z) Example : (A + B + C’)(A + B + D’)(B + C + D’) The Consensus of (A + B + C’) and (B + C + D’) is (A + B + D’) Thus, we can eliminate the consensus term Answer : (A + B + C’)(A + B + D’)

Consensus Term Eliminating Order (1) Attention The final result obtained by application of the consensus theorem may depend on the order in which terms are eliminated. Example : A’C’D + A’BD + BCD + ABC + ACD’  Eliminate BCD terms (consensus of A’BD , ABC)  A’C’D + A’BD + ABC + ACD’ (No more eliminated term.)

Consensus Term Eliminating Order (2) Same Example : A’C’D + A’BD + BCD + ABC + ACD’  Eliminate A’BD terms (consensus of A’C’D , BCD)  A’C’D + BCD + ABC + ACD’  Eliminate ABC terms (consensus of BCD, ACD’)  A’C’D + BCD + ACD’ (no more eliminated term)

Trick to use consensus theorem Sometimes it is impossible to directly reduce an expression to a minimum number of terms by simply eliminating terms. It may be necessary to first add a term using the consensus theorem and then use the added term to eliminate other terms.

Example F = ABCD + B’CDE + A’B’ + BCE’ Consensus of ABCD and B’CDE  ACDE Consensus of A’B’ and BCE’  ACE’ But none of them appear in the original expression. However, if we first add the consensus ACDE to F F = ABCD + B’CDE + A’B’ + BCE’ + ACDE Consensus of ACDE and A’B’  B’CDE Consensus of ACDE and BCE’  ABCD Thus, F = A’B’ + BCE’ + ACDE

Exercise 1 Simplify each of the following expressions using only the consensus theorem BC’D’ + ABC’ + AC’D + AB’D + A’BD’ (reduce to 3 terms) W’Y’ + WYZ + XY’Z + WX’Y (reduce to 3 terms)

Algebraic Simplification (1) Combining terms XY + XY’ = X Example : abc’d’ + abcd’ = abd’ (X = abd’, Y = d) Complex example : ab’c + abc + a’bc  (X + X = X) ab’c + abc + abc + a’bc ac + bc

Algebraic Simplification (2) Eliminating terms X + XY = X Example : a’b + a’bc = a’b (X = a’b, Y = c) XY + X’Z + YZ = XY + X’Z (consensus theorem) a’bc’ + bcd + a’bd = a’bc’ + bcd (X = c, Y = bd, Z = a’b)

Algebraic Simplification (3) Eliminating literals X + X’Y = X + Y Simply factoring may be necessary before the theorem is applied Example : A’B + A’B’C’D’ + ABCD’ = A’(B + B’C’D’) + ABCD’ = A’(B + C’D’) + ABCD’ = A’B + AC’D’ + ABCD’ = B(A’ + ACD’) + AC’D’ = B(A’ + CD’) + AC’D’ = A’B + BCD’ + AC’D’

Algebraic Simplification (4) Adding redundant terms. Redundant terms can be introduced in several ways such as adding xx’ multiplying by (x + x’) Adding yz to xy+x’z Adding xy to x Example : WX + XY + X’Z’ + WY’Z’ = A’(B + B’C’D’) + ABCD’ = WX + XY + X’Z’ + WY’Z’ + WZ’ (add WZ’ by consensus term) = WX + XY + X’Z’ + WZ’ (WZ’ + WY’Z’  WZ’) = WX + XY + X’Z’ (eliminate WZ’ [consensus of WX and X’Z’])

TODO Simplify to a sum of three terms: A’C’D’ + AC’ + BCD + A’CD’ + A’BC + AB’C’ A’B’C’ + ABD + A’C + A’CD’ + AC’D + AB’C’