lec15

# lec15 - CS151 Complexity Theory Lecture 15 Arthur-Merlin...

This preview shows pages 1–12. Sign up to view the full content.

CS151 Complexity Theory Lecture 15 May 17, 2011

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
May 17, 2011 2 Arthur-Merlin Games IP permits verifier to keep coin-flips private necessary feature? GNI protocol breaks without it Arthur-Merlin game : interactive protocol in which coin-flips are public Arthur (verifier) may as well just send results of coin-flips and ask Merlin (prover) to perform any computation Arthur would have done
May 17, 2011 3 Arthur-Merlin Games Clearly Arthur-Merlin IP “private coins are at least as powerful as public coins” Proof that IP = PSPACE actually shows PSPACE Arthur-Merlin IP PSPACE “public coins are at least as powerful as private coins” !

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
May 17, 2011 4 Arthur-Merlin Games Delimiting # of rounds: AM[k] = Arthur-Merlin game with k rounds, Arthur (verifier) goes first MA[k] = Arthur-Merlin game with k rounds, Merlin (prover) goes first Theorem : AM[k] ( MA[k] ) equals AM[k] ( MA[k] ) with perfect completeness. Theorem : for all constant k 2 AM[k] = AM[2] .
May 17, 2011 5 MA and AM Two important classes: MA = MA[2] AM = AM[2] definitions without reference to interaction: L MA iff 9 poly-time language R x L 9 m Pr r [(x, m, r) R] = 1 x L 8 m Pr r [(x, m, r) R] ½ L AM iff 9 poly-time language R x L Pr r [ 9 m (x, m, r) R] = 1 x L Pr r [ 9 m (x, m, r) R] ½

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
May 17, 2011 6 MA and AM P NP coNP Σ 2 Π 2 AM coAM MA coMA
May 17, 2011 7 MA and AM Theorem : coNP AM PH = AM . Proof: – suffices to show Σ 2 AM (and use AM Π 2 ) – L Σ 2 iff 9 poly-time language R x L 9 y 8 z (x, y, z) R x L 8 y 9 z (x, y, z) R Merlin sends y 1 AM exchange decides coNP query: 8 z (x, y, z) R ? 3 rounds; in AM

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
May 17, 2011 8 MA and AM We know Arthur-Merlin = IP . “public coins = private coins” Theorem (GS): IP[k] AM[O(k)] stronger result implies for all constant k 2, IP[k] = AM[O(k)] = AM[2] So, GNI IP[2] = AM
May 17, 2011 9 Back to Graph Isomorphism The payoff: not known if GI is NP -complete. previous Theorems: if GI is NP -complete then PH = AM unlikely! Proof: GI NP -complete GNI coNP -complete coNP AM PH = AM

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Derandomization revisited L MA iff 9 poly-time language R x L 9 m Pr r [(x, m, r) R] = 1 x L 8 m Pr r [(x, m, r) R] ½ Recall PRGs: for all circuits C of size at most s : |Pr y [C(y) = 1] – Pr z [C(G(z)) = 1]| ≤ ε May 17, 2011 10 NP AM MA s e e d o utput s tring G t b its u  b its
Using PRGs for MA L MA iff 9 poly-time language R x L 9 m Pr r [(x, m, r) R] = 1 x L 8 m Pr r [(x, m, r) R] ½ produce poly-size circuit C such that C(x, m, r) = 1 , (x,m,r) 2 R • for each x, m can hardwire to get C x,m 9 m Pr y [C x,m (y) = 1] = 1 (“yes”) 8 m Pr y [C x,m (y) = 1] ≤ 1/2 (“no”) May 17, 2011 11

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 01/05/2012.

### Page1 / 46

lec15 - CS151 Complexity Theory Lecture 15 Arthur-Merlin...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online