Definition
A probabilistically checkable proof system with completeness c(n) and soundness s(n) over alphabet Σ for a decision problem L, where 0 ≤ s(n) ≤ c(n) ≤ 1, is a randomized oracle Turing Machine V (the verifier) that, on input x and oracle access to a string π ∈ Σ* (the proof), satisfies the following properties:
- Completeness: If x ∈ L then for some π, Vπ(x) accepts with probability at least c(n),
- Soundness: If x ∉ L then for every π, Vπ(x) accepts with probability at most s(n).
The randomness complexity r(n) of the verifier is the maximum number of random bits that V uses over all x of length n.
The query complexity q(n) of the verifier is the maximum number of queries that V makes to π over all x of length n.
The verifier is said to be non-adaptive if it makes all its queries before it receives any of the answers to previous queries.
The complexity class PCPc(n), s(n) is the class of all decision problems having probabilistically checkable proof systems over binary alphabet of completeness c(n) and soundness s(n), where the verifier is nonadaptive, runs in polynomial time, and it has randomness complexity r(n) and query complexity q(n).
The shorthand notation PCP is sometimes used for PCP1, ½. The complexity class PCP is defined as PCP1, ½.
Read more about this topic: Probabilistically Checkable Proof
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