Formula Synthesis (Examples)

Assume D = [●,○,●,○] is the set of desired truth values for an unknown formula D. Since, D has four values, D must have two atomic components. Allow A and B to be atomic components of D. If A = [○,○,●,●] and B = [○,●,○,●], then the minterms of A and B are (A ∧ B) = [○,●,●,●] , (A ∧ ¬B) = [●,○,●,●] , (¬A ∧ B) = [●,●,○,●] , (¬A ∧ ¬B) = [●,●,●,○]. Consequently, (A ∧ ¬B) and (¬A ∧ ¬B) are the proper minterms to choose. When combined into a disjunction [(A ∧ ¬B) ∨ (¬A ∧ ¬B)] = [●,○,●,○] = D.

Assume C is an unknown formula and C = [○,●,●,●,●,●,●,●]. C must have three atomic components because there are eight truth values in C. If E, F, and G are the atomic components of formula C, then [(E ∧ F) ∧ G] = [○,●,●,●,●,●,●,●] is the only minterm needed because C has only one true truth value. C is what is called a "consensus" formula because E, F, and G must all be true in order for C to be true.