Sharp-P

#P , marked " sharp P" , is a class of complexity in the Théorie of complexity. It is the whole of the problems associated with the problems with decision with the class NP . Contrary to the majority of the most known classes of complexity, it is not a class of problems of decision but a class of problems of function.

A problem of the class NP is often of the form, " Are there solutions which satisfy certain constraints? " For example:

Y does it have a subset of a list of entireties whose sum is equal to zero? (Problem of the sum of a subset)
Y have does, in a graph given, a cycle Hamiltonien with a cost lower than 100? (Problem of the sales representative)
Y does it have assignement variables which checks a given formula (expressed in general under conjunctive normal Forme)? (Problem SAT)

The corresponding problems of the class #P raise the " question; How much y-a it " rather than " There is ". For example:

Combien there are subsets of a list of entireties whose sum is égaleà zero?
How much Hamiltoniens cycles of a given graph have a cost lower than 100?
How much assignements of variables satisfies a given formula?

More formally, a problem is in #P if there exists a Machine of Turing not-determinist in polynomial time which, for each authority I of the problem, has a number of acceptable final states exactly equal to the number of distinct solutions to the authority I .

Clearly, a problem #P must be at least as hard as the problem which corresponds to him in NP . If it is easy to count the answers, then it must be facil to determine if there are answers. It is enough for that to hope them to see sir the account is higher than zero. Consequently, the problem of the class #P correspondent with a Np-Complete problem must be Np-difficult .

Curiously, certain problems of #P which are regarded as difficult correspond to easy problems, class P . For more information on the subject, see Sharp-P-complete .

The class of problems of decision nearest to #P is PP , which is the soluble class of the problems by a machine of Turing not-determinist in polynomial time, whose majority (more half) of the final states are acceptable. This answers the most significant part of the problem of #P corresponding. The class of problems of decision ⊕P puts on the contrary the question concerning the least significant part of the problem #P corresponding.

A consequence of the Théorème of Toda is that a machine in polynomial time having an oracle of the class #P , ( P ^#P), can solve any problem of pH , i.e. of the polynomial Hiérarchie very whole. In fact, the machine at polynomial time needs only one request to oracle #P to solve any problem of pH . It is an indication of the extreme difficulty which there is to solve exactly of the problems #P - complete.

References

Complexity Zoo: #P

Random links: