P-complete
class in computational complexity theory
In computational complexity theory, a decision problem is P-complete (complete for the complexity class P) if it is in P and every problem in P can be reduced to it by an appropriate reduction.
The notion of P-complete decision problems is useful in the analysis of which problems are difficult to parallelize effectively and which problems are difficult to solve in limited space, specifically when stronger notions of reducibility than polytime-reducibility are considered.
The specific type of reduction used varies and may affect the exact set of problems. Generically, reductions stricter than polynomial-time reductions are used, since all languages in P (except the empty language and the language of all strings) are P-complete under polynomial-time reductions. If we use NC reductions, that is, reductions that can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effectively parallelized, under the unproven assumption that NC ≠ P. If we use the stronger log-space reduction, this remains true, but additionally we learn that all P-complete problems lie outside L under the weaker unproven assumption that L ≠ P. In this latter case the set P-complete may be smaller.
Motivation
The class P, typically taken to consist of all the "tractable" problems for a sequential computer, contains the class NC, which consists of those problems that can be efficiently solved on a parallel computer. This is because parallel computers can be simulated on a sequential machine.
It is not known whether NC = P. In other words, it is not known whether there are any tractable problems that are inherently sequential. Just as it is widely suspected that P does not equal NP, so it is widely suspected that NC does not equal P.
Similarly, the class L contains all problems that can be solved by a sequential computer in logarithmic space. Such machines run in polynomial time because they can only have a polynomial number of different configurations.
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