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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).
Efficiency: P runs in polynomial time. In general, a proof system for a language L is a polynomial-time function whose range is L. Thus, a propositional proof system is a proof system for TAUT. Sometimes the following alternative definition is considered: a pps is given as a proof-verification algorithm P(A,x) with two inputs.
It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".
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.
A generalization of P is NP, which is the class of decision problems decidable by a non-deterministic Turing machine that runs in polynomial time. Equivalently, it is the class of decision problems where each "yes" instance has a polynomial size certificate, and certificates can be checked by a polynomial time deterministic Turing machine.
A natural example of a problem in co-RP currently not known to be in P is Polynomial Identity Testing, the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, x·x − y·y − (x + y)·(x − y) is the zero-polynomial while x·x + y·y is not.