Interactive Proofs. Slides by Dana Moshkovitz. Adapted from Oded Goldreich’s course lecture notes. Outline. Proof systems: NP revisited. Interactive proofs The complexity class IP Example: An interactive proof for Graph Non-Isomorphism IP=PSPACE Public coins.
Slides by Dana Moshkovitz.
Adapted from Oded Goldreich’s course lecture notes.
(z)=falseProof Systems Back to NP
We will demonstrate this process for 3SAT:
We would like to check the membership of a given formula:
The verifier simply needs to check the truth value of the formula under the assignment it received in order to find out whether the prover was right. This merely takes polynomial time.
polynomial in the number of variables
The prover must convince the verifier this formula is satisfiable, so it sends it an assignment, which supposedly satisfies the formula. It is not difficult for the mighty prover to find such, if such exists.
Make sure you understand why does the the proof system we presented for 3SAT satisfy these properties.
GNI or Randomness?Motivation
Make sure you understand why could we assume,
without loss of generality, that V1=V2.
The common input or Randomness?An Interactive Proof for GNI Simulation
The 2nd Graph
We shall prove next a rather surprising result, stating
To do so, we will prove the following two claims:
1. IPPSPACE this will follow if we can simulate every interactive proof using polynomial space.
2. PSPACEIP this will follow if we can exhibit a PSPACE-complete language which is in IP.
Do you remember that verifier strategy? or Randomness?
½ or Randomness?
2An Optimal prover Example
Prob. For acceptance:
With each CNF formula , we associate a polynomial p in the following manner:
The bottom Line: (x1,...,xn) is false iff p(x1,...,xn)=0
Note that in the resulting polynomial the degree of each variable is at most n (the number of variables in the formula).
Why is this definition of f
the same as the previous one
for boolean input?
Note, that we reorder
the inputs to the
functions, so the
variable yi+1 is the last
Take this formula:
We build ’
Now we can use
computation in order
f or Randomness?0()
A setting of the
variables to the
rAn Interactive Proof for TQBF
. . .
. . .
A family of hash functions H is 2-universal,
if whenever h is chosen uniformly from H,
(h(x),h(y)) is also uniformly distributed.
Can you prove that the functions we defined are
Pr[at least one element in the size N set is mapped to ]
= Pr[E1... EN]
i Pr[Ei]- i<j Pr[Ei,Ej]
We used a 2-universal hash family