Lecture 18

Cook-Levin Theorem

Theorem $$SAT$$ is $$NP-complete$$

Proof First, $$SAT\in NP$$ because we can verify in polynomial time that a given assignment cause a formula to evaluate to True.

Next we want to show that $$SAT$$ is $$NP-hard$$. Let $$L\subseteq \{0,1\}^*$$ be any language in $$NP$$, we want to show that $$L\leq_p SAT$$. This means that we want to construct a function $$f$$ that on input $$x\in\{0,1\}^*$$, outputs $$f(x)=\varphi x$$ such that $$\varphi x$$ is satistiable iff $$x\in L$$ and $$f$$ can be computed in polynomial-time.

Let $$M$$ be a NTM with polynomial running time that decides $$L$$. We can assume:

1. $$M$$ never goes to the left of the fiist symbol of $$x$$ in original input
2. We know that the running time of $M4 is bounded by $$t(n)=an^b$$ for some constant $$a,b\in\mathbb{N}$$. Tableau q0 0111101 q1 1111101 q2 1111101 ... qacc xxxxxxx To build $$\varphi x$$, we start by defining variables $$z_{i,j,s}=1$$ if cell $$(i,j)$$ of Tableau is $$s$$, and $$0$$ otherwise. There are $$3(q+1)$$ possible symbols in any cell whem $$M$$ has $$q$$ states and $$t(n)xf(n)$$ cell in the Tableau. So the total number of variables in polynomial in $$n$$. I. Create formula that is satisfied only when variable encode a Tableau.$^{(enc)}{i,j}(z)={V_s z{i,j,s})({s,t}(\hat{z{i,j,t}))

Define $$\varphi^{enc}-\land_{i,j\in t(n)}\varphi^{enc}_{i,j}$$

1. Check that each row is a valid configuration. (exercise: find formula $$\varphi^{conf}$$ that does this)

2. Check initial configuration

1. $$\varphi^{init}_{j}=Z_{1,1,\binom{x_1}{q_0}}$$ if $$j=1$$
2. $$\varphi^{init}_{j}=Z_{1,j,x_j}$$ if $$j<j\leq n$$
3. $$\varphi^{init}_{j}=Z_{i,j,-}$$ if $$j>n$$

$$\varphi^{init}=\land_j\varphi^{init}_j$$

1. Check that $$M$$ reaches the accept state $$\varphi^{accept}=(Exercise)$$

2. Check that the transitions in the Tableau are valid.

Key idea: enforce valid transitions on $$2\times 3$$ windows.

$$\varphi^{trans}=(Exercise)$$

Define $$\varphi x=\varphi^{enc}\land\varphi^{init}\land\varphi^{acc}\land\varphi^{trans}$$

Theorem There is a language $$L\in NP$$ for which any algorithm runs in time $$c\cdot n$$.