Polynomial-Time Hierarchy

1. Stockmeyer2. 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+

1

( )k

k

P DTIME n

1

( )k

k

NP NTIME n

1

( )k

k

PSPACE DSPACE n

log

The Hierarchy The polynomial time hierarchy is

where:

and for k≥0

Also define

0 0 0p p p P

1

1

1

pk

pk

pk

pk

pk

pk

NP

coNP

P

{ , , , 0}p p pk k k k

0

pkk

PH

2 notes Note that and . Since obviously BєPB

and for any set B, the P-hierarchy has the following structure:

Also

1p NP 1

p coNP B B BP NP coNP

1 1 1p p p p pk k k k k

1pkp

k NP

Lemmas 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 ΠkP 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

ΠkP =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-1P oracle decides if (x,y) not in R

(since R in Πi-1P)

Proof continues 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-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-1

P

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 iffDually, L є if and only if

x є L ifffor some L’ є P and polynomials p1,…,pk

pk

pk

1 21 2 1( ) ( ) ...( ) [ , ,..., ']kp p k p ky y Qy x y y L

1 21 2 1( ) ( ) ...( ) [ , ,..., ']kp p k p ky y Qy x y y L

2 propositions1. 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.

pk

pk

1pk

1pk

pk

pk

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

p pk k

p pk k

p p pj j k

1 1p p pj j k

p pj k

1pj

( )[ (| |) ( , )]y y p x R x y pk p

k pk

p pj k p p

j k

1. If for some k ≥ 1, then P ≠ NP2. 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.

0p p

k { : 0}p

k k

pk PSPACE

PH PSPACE

1p pk k

If for all k, then Let k≥1

Bk={F(X1,…,Xk)|F(X1,…,Xk) is a boolean formula, and

} 1. Bω is log-complete in PSPACE.

2. 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 .

pm

1p pk k PH PSPACE

1k

k

B B

1 2 1( )( )...( )[ ( ,..., ) 1]k kX X QX F X X

pjB

pjB

pj

pjPSPACE 1

p pj j