Polynomial-Time Hierarchy 1 Stockmeyer 2 Wrathall Definitions

Polynomial-Time Hierarchy 1. Stockmeyer 2. Wrathall

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 є Θ+

The Hierarchy o The polynomial time hierarchy is where: and for k≥ 0 Also define

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

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

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

Proof continues o o Let L in Σk. P we will show that a proper R exists. L is decided by NDTM M with oracle for KєΣi-1 P. By induction Э relation S s. t. zєK iff Эw : (z, w)єS, SєΠi-2 P. R poly-balanced and poly decidable for L. n xєL iff Э acc. comput. of MK on x. n y records computation of MK. o o n o 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-1 P

Main Theorem Let L S+ be a language. For any k≥ 1, Lє if and only if there exist polynomials p 1, …, 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 p 1, …, pk

2 propositions 1. 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’} 2. For each k ≥ 1, is closed under poly-bounded existential quantification and is closed under polytime bounded universal quantification.

If for some k≥ 1 then for all j≥K o o o o Assume for some k≥ 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

1. 2. If If then for some k ≥ 1, then P ≠ NP contains infinitely main distinct classes, for all k ≥ 0. o Baker points out that NPPSPACE=PSPACE is an immediate consequence from Savitch’s theorem NSPACE(S(n)) DSPACE(S 2(N)). By induction on k we have for all k.

If for all k, o then Let k≥ 1 Bk={F(X 1, …, Xk)|F(X 1, …, Xk) is a boolean formula, and } o 1. 2. Bω is log-complete in PSPACE. Suppose A B and B є NPC. Then also A є NPC. PH PSPACE. If PSPACE PH then for some j, Since is closed under logspace reductions, implies that and then.