Homework III
CS6520
Computational Complexity
March 8, 2007
Due on Thursday, March 15th, at 6 pm. Collaboration is al-
lowed; please mention your collaborators.
1. The class MA is analogous to NP where the verifier can be a ran-
domized algorithm. There is an all powerful prover (called Merlin)
who gives a proof to the probabilistic polynomial time verifier (called
Arthur). Arthur uses a (private) random string r.
Define the class of languages MA2/3,1/3with two sided error as follows.
x∈L⇒ ∃y, Prr[V(x, y, r ) = 1] ≥2
3
x6∈ L⇒ ∀y, Prr[V(x, y, r ) = 1] ≤1
3
Here V(,) is a deterministic polynomial time verification procedure
and lengths of yare rare polynomially bounded in the length of x.
Similarly we define the class MA1,1/3with one-sided error as follows
x∈L⇒ ∃y, Prr[V(x, y, r ) = 1] = 1
x6∈ L⇒ ∀y, Prr[V(x, y, r ) = 1] ≤1
3
Show that MA2/3,1/3= MA1,1/3. That is, if a language has a MA-
protocol with two-sided error, then it also has a MA-protocol with
one-sided error. Hint: Use ideas from the proof of BPP ⊆Σ2.
2. Show that
PSPACE ⊆P/poly ⇒PSPACE = Σ2
1
