lecture8 - ORACLE TM CSci 5403 COMPLEXITY THEORY An oracle...

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1 COMPLEXITY THEORY CSci 5403 LECTURE VIII: ALTERNATING QUANTIFIERS ORACLE TM An oracle Turing Machine is a TM with a query Tape and three states q ? , q yes , q no . FINITE STATE CONTROL INPUT WORK QUERY S With an oracle for set S, it goes in one step from state q ? with x on the query tape to q x 2 S . ORACLE COMPLEXITY We denote M running with oracle S by M S . For fixed O, we can define complexity classes NP O , P O , PSPACE O , etc: S 2 P O iff 9 poly(n)-time M . L(M O ) = S S 2 NP O iff 9 poly(n)-time V, poly q. S = { x | 9 y 2 {0,1} q(|x|) . V O (x,y) accepts } We can also define complexity classes relative to other classes: S 2 P NP iff 9 A 2 NP, poly(n)-time M. L(M A ) = S. ORACLE ARITHMETIC P NP = L 9 M, 9 A 2 NP. M A decides L in polynomial time. P SAT = L 9 M. M SAT decides L in polynomial time. Claim. P NP = P SAT . (why?) Quiz. We know that P PSPACE = NP PSPACE . Does P NP = NP NP ?
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2 P NP vs NP NP Does P NP = NP? Notice that coNP µ P NP . Consider the following problems: C C has a smaller, equivalent circuit C’. 9 X 2 {0,1} n . φ (X,y 1 ,…,y n ) is a tautology Claim. NMIN-CIRCUIT, QSAT 2 2 NP NP . NMIN-CIRCUIT = QSAT 2 = φ (x 1 …x n ,y 1 …y n ) THE POLYNOMIAL HIERARCHY Definition. Let Δ 0 P = Σ 0 P = Π 0 P = P. Define the “hierarchy” of complexity classes: Δ 1 = P Σ 1 = NP Π 1 = coNP Δ 2 = P NP Σ 2 = NP NP Π 2 = coNP NP Δ i = P Σ i-1 Σ i = NP Σ i-1 Π i = coNP Σ i-1 PH = [ i Σ i THE POLYNOMIAL HIERARCHY P NP coNP Δ 2 Σ 2 Π 2 Δ 3 Σ 3 Π 3 Δ 4 Σ 4 Π 4 NMIN-CIRCUIT 2 Σ 2 : guess a circuit C’ accept if 8 x. C(x)=C’(x). MIN-CIRCUIT 2 Π 2 : for all |C’| < |C|: accept if 9 x .(C’(x) C(x)) Each path is a circuit Each node is an OR Each leaf is an oracle call to CIRCUIT-TAUT Each path is a circuit Each node is an AND
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