Editing P-complete (section)

== 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 which 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'''&nbsp;=&nbsp;'''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 (complexity)|L]]''' contains all problems that can be solved by a sequential computer in logarithmic space. Such machines run in polynomial time because they can have a polynomial number of configurations. It is suspected that '''L'''&nbsp;≠&nbsp;'''P'''; that is, that some problems that can be solved in polynomial time also require more than logarithmic space.

Similarly to the use of [[NP-complete]] problems to analyze the '''P'''&nbsp;=&nbsp;'''NP''' question, the '''P'''-complete problems, viewed as the "probably not parallelizable" or "probably inherently sequential" problems, serves in a similar manner to study the '''NC'''&nbsp;=&nbsp;'''P''' question. Finding an efficient way to parallelize the solution to some '''P'''-complete problem would show that '''NC'''&nbsp;=&nbsp;'''P'''. It can also be thought of as the "problems requiring superlogarithmic space"; a log-space solution to a '''P'''-complete problem (using the definition based on log-space reductions) would imply '''L'''&nbsp;=&nbsp;'''P'''.

The logic behind this is analogous to the logic that a polynomial-time solution to an '''NP'''-complete problem would prove '''P'''&nbsp;=&nbsp;'''NP''': if we have a '''NC''' reduction from any problem in '''P''' to a problem A, and an '''NC''' solution for A, then '''NC'''&nbsp;=&nbsp;'''P'''. Similarly, if we have a log-space reduction from any problem in '''P''' to a problem A, and a log-space solution for A, then '''L'''&nbsp;=&nbsp;'''P'''.