COMS W3261
Computer Science Theory
Lecture 22: December 5, 2011
SAT is NP-Complete

Outline

A truth assignment for a boolean expression assigns either the value true (1) or the value false (0) to each of the variables in the expression.
The value E(T) of an expression E given a truth assignment T is the result of evaluating E with each variable x in E replaced by T(x).
A truth assignment T satisfies E if E(T) = 1.
An expression E is satisfiable if there exists a truth assignment T that satisfies E.
The satisfiability problem (SAT) is to determine whether a given boolean expression is satisfiable.
SAT is the set of (encoded) boolean expressions that are satisfiable.

Given a boolean expression E of length n, a multitape nondeterministic Turing machine can guess a truth assignment T for E in O(n) time.
The NTM can then evaluate E using the truth assignment T in O(n²) time.
If E(T) = 1, then the NTM accepts E.

We must also show that if L is any language in NP, then there is a polynomial-time reduction of L to SAT:

If a NTM M accepts an input w of length n in p(n) time, then M has a sequence of moves such that

α₀ is the initial ID of M with input w.
α₀ |– α₁ |– … |– α_k where k ≤ p(n).
α_k is an ID with an accepting state.
Each α_i consists only of nonblanks unless α_i ends in a state and a blank, and extends from the initial head position to the right.

From M and w we can construct in polynomial time a boolean expression E_M,w that is satisfiable iff M accepts w within p(n) moves. See HMU, Sect. 10.2.3, pp. 440-446 for details.