Polynomial-Time Hierarchy

1. Polynomial-Time Hierarchy 1. Stockmeyer 2. Wrathall

2. Definitions Let A Θ+and B Δ+for finite alphabets Θ and Δ. A transforms to B within logspace via f (A B via f) iff f is a transformation, f:Θ+→Δ+, such that f є logspace and xєA↔f(x)єB for all x єΘ+

3. The Hierarchy • The polynomial time hierarchy is where: and for k≥0 Also define

4. 2 notes • Note that and . Since obviously BєPB andfor any set B, the P-hierarchy has the following structure: • Also

5. Lemmas • Let L a language and i≥1. L in ΣkP iff there is a poly-balanced relation R s.t. the language{x,y: (x,y)єR} is in Πk-1P and L={x: Эy s.t. (x,y) єR} • Let L a language and i≥1. L in ΠkP iff there is a poly-balanced relation R s.t. the language{x,y: (x,y)єR} is in Σk-1P and L={x: for all y with |y|≤|x|k, (x,y) єR}

6. Proof • ΠkP=co ΣkPso it suffices to prove it for ΣkP . • For i=1 it holds. • Let i>1 and R exists. • NDTM M choses a y nondet. And with a Σi-1P oracle decides if (x,y) not in R (since R in Πi-1P)

7. Proof continues • Let L in ΣkP we will show that a proper R exists. • L is decided by NDTM M with oracle for KєΣi-1P. • By induction Э relation S s.t. zєK iff Эw : (z,w)єS, SєΠi-2P. • R poly-balanced and poly decidable for L. • xєL iff Э acc. comput. of MK on x. • y records computation of MK. • Some steps are queries to K. • For each yes query (zi) y will contain the certificate wi s.t. (zi,wi)єS. • (x,y)єR iff y records an acc computation of M with a certificate wi for each yes querry zi in computation.\ • (x,y)єR can be checked in Πi-1P

8. Main Theorem Let L S+ be a language. For any k≥1, Lє if and only if there exist polynomials p1,…,pk and a language L’ є P such that for all x є S+, x є L iff Dually, L є if and only if x є L iff for some L’ є P and polynomials p1,…,pk

9. 2 propositions • For any k ≥ 1, a language L S+ is in iff there exist a homomorphism h:S*→T*, a language L’ T+ in and a polynomial p(n) such that L=h(L’) and for any x є L’, |x|≤p(|h(x)|), That is ={h(L’): L’ є , h a homomorphism that performs poly-bounded erasing on L’} • For each k ≥ 1, is closed under poly-bounded existential quantification and is closed under poly-time bounded universal quantification.

10. If for some k≥1 then for all j≥K • Assume for somek≥1 • By induction on j we will prove it • For j=k it stands • Assume that for some j>k we will show that • From previous theorem: There is a 2-ary relation R and a polynomial p such that for all x, xєA iff • By induction we have R . A because for k≥1, is closed under the operation of poly-bounded existential quantification over variables of relations (prop 2). • Thus and by definition

11. If for some k ≥ 1, then P ≠ NP • If contains infinitely main distinct classes, then for all k ≥ 0. Baker points out that NPPSPACE=PSPACE is an immediate consequence from Savitch’s theorem NSPACE(S(n)) DSPACE(S2(N)). By induction on k we have for all k.

12. If for all k, then • Let k≥1 Bk={F(X1,…,Xk)|F(X1,…,Xk) is a boolean formula, and } • Bω is log-complete in PSPACE. • Suppose A B and B єNPC. Then also A єNPC. PH PSPACE. If PSPACE PH then for some j, Since is closed under logspace reductions, implies that and then .