Chapter 7: Time Complexity

For Your Enjoyment

Definition. A function f: Σ^* → Σ^* is a polynomial time computable function if some polynomial time Turing Machine M exists that halts with just f(w) on its tape, when started on input w.
Definition. Language A is polynomial time mapping reducible or simply polynomial time reducible to language B, written A ≤_P B, if a polynomial time computable function f: Σ^* → Σ^* exists where for every w, w ∈ A ⇔ f(w) ∈ B. The function f is called the polynomial time reduction of A to B.
Theorem. If A ≤_P B and B ∈ P, then A ∈ P.

Definition. A language B is NP-Complete if (1) B is in NP and (2) every A in NP is polynomial time reducible to B.
Theorem. If B is in NPC and B ∈ P, then P = NP.
Theorem. If B is in NPC and B ≤_P C for C in NP, then C is in NPC.

SAT = {<Φ> | Φ is a satisfiable Boolean formula.}
SAT was the first problem proved to be in NPC by Stephen Cook and Leonid Levin in 1971.
Theorem. SAT ∈ P iff P = NP.

3SAT = {<Φ> | Φ is a satisfiable 3cnf formula}. Every clause is a 3cnf (conjunctive normal form) has exactly 3 literals.
3SAT is an NPC problem.
How do you show that 3SAT is in NP?
How do you show that there is a polynomial time reduction from SAT to 3SAT?