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

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

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

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

Proof continues  Let L in Σ k P we will show that a proper R exists.  L is decided by NDTM M with oracle for Kє Σ i-1 P.  By induction Э relation S s.t. z єK iff Эw : (z,w)єS, Sє Π i-2 P.  R poly-balanced and poly decidable for L. x єL iff Э acc. comput. of M K on x. y records computation of M K.  Some steps are queries to K.  For each yes query (z i ) y will contain the certificate w i s.t. (z i,w i )єS. (x,y)єR iff y records an acc computation of M with a certificate w i for each yes querry z i in computation.\  (x,y)єR can be checked in Π i-1 P

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

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 poly- time bounded universal quantification.

If for some k≥1 then for all j≥K  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.If for some k ≥ 1, then P ≠ NP 2.If contains infinitely main distinct classes, then for all k ≥ 0.  Baker points out that NP PSPACE =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, then  Let k≥1 B k ={F(X 1,…,X k )|F(X 1,…,X k ) is a boolean formula, and }  1.B ω is log-complete in PSPACE. 2.Suppose A B and B є NP C. Then also A є NP C. PH PSPACE. If PSPACE PH then for some j, Since is closed under logspace reductions, implies that and then.