CMPT 407 - Complexity Theory Lecture 5: SAT, coNP, Search-to-Decision, NTime Hierarchy
نویسنده
چکیده
Proof. SAT is in NP (easy). To prove NP-hardness, we will show that Circuit-SAT is reducible to SAT. Let C be an arbitrary Boolean circuit with gates g1, . . . , gm, where g1, . . . , gn are input gates and gm is the output gate. For each gj, introduce a Boolean variable yj. For every i > n, define the Boolean formula gatei expressing that the value of yi is equal to the value of the gate gi. That is, if gate gi is an AND gate with inputs gi1 and gi2 , then gatei is True iff yi ≡ yi1 ∧ yi2 ; similarly, for OR, and NOT gates. Our final formula φC is defined as
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