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The complexity class PCP c(n), s(n) [r(n), q(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 PCP theorem states that NP = PCP[O(log n), O(1)],. where PCP[r(n), q(n)] is the class of problems for which a probabilistically checkable proof of a solution can be given, such that the proof can be checked in polynomial time using r(n) bits of randomness and by reading q(n) bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1/2.
If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP-complete problem, once a problem has been proven to be NP-complete, this is widely regarded as a sign that a ...
A strongly-polynomial time algorithm is polynomial in both models, whereas a weakly-polynomial time algorithm is polynomial only in the Turing machine model. The difference between strongly- and weakly-polynomial time is when the inputs to the algorithms consist of integer or rational numbers. It is particularly common in optimization.
The complexity class NP may be viewed as a very simple proof system. In this system, the verifier is a deterministic, polynomial-time machine (a P machine). The protocol is: The prover looks at the input and computes the solution using its unlimited power and returns a polynomial-size proof certificate.
Propositional proof system can be compared using the notion of p-simulation. A propositional proof system P p-simulates Q (written as P ≤ p Q) when there is a polynomial-time function F such that P(F(x)) = Q(x) for every x. [1] That is, given a Q-proof x, we can find in polynomial time a P-proof of the same tautology.
A propositional proof system is given as a proof-verification algorithm P(A,x) with two inputs.If P accepts the pair (A,x) we say that x is a P-proof of A.P is required to run in polynomial time, and moreover, it must hold that A has a P-proof if and only if A is a tautology.
A problem is polynomial-time Turing-reducible to a problem if, given a subroutine that solves in polynomial time, one could write a program that calls this subroutine and solves in polynomial time. This contrasts with many-one reducibility, which has the restriction that the program can only call the subroutine once, and the return value of the ...