Pcp Verifiers

Definition 1 ((r, q)-verifier) An algorithm V is an (r, q)-verifier for a language L if on input strings x and y: step 1 V reads input x and draws r random bits ρ ∈ {0, 1}. step 2 V decides (after a polynomial-time computation in |x|) on q indexes: i1, . . . , iq and on a predicate φ : {0, 1} → {0, 1}. step 3 V outputs φ(y[i1], . . . , y[iq]). and the following holds, 1. (Completeness) If x ∈ L then ∃y s.t. Prρ[V er(x, ρ) = 1] = 1. 2. (Soundness) If x / ∈ L then ∀y Prρ[V er(x, ρ) = 1] ≤ 1/2. Notes: 1. V er is the notation for the fact that that the verifier can access any index in y in one step (see also previous lecture). 2. This is an equivalent definition to the one given in the previous lecture. 3. The above definition is called a non-adaptive verifier. An adaptive verifier accesses y one bit at a time and may adapt its computation according to the read bit. In our definition V has a fixed predicate after step 2, and simply outputs that predicate in step 3. 4. The length of the proof y is at most 2q since there are 2 possible random strings of length r and q possible accesses for each string. Therefore if r is c log n(n = |x|) for some constant c, the proof y is of length at most nq, i.e. polynomial in n.