Counting problem (complexity) - Alchetron, the free social encyclopedia

In computational complexity theory and computability theory, a counting problem is a type of computational problem. If R is a search problem then

c R ( x ) = | { y ∣ R ( x , y ) } |

is the corresponding counting function and

# R = { ( x , y ) ∣ y ≤ c R ( x ) }

denotes the corresponding counting problem.

Note that c_R is a search problem while #R is a decision problem, however c_R can be C Cook reduced to #R (for appropriate C) using a binary search (the reason #R is defined the way it is, rather than being the graph of c_R, is to make this binary search possible).

Counting complexity class

If NC is a complexity class associated with non-deterministic machines then #C = {#R | R ∈ NC} is the set of counting problems associated with each search problem in NC. In particular, #P is the class of counting problems associated with NP search problems.

References

Counting problem (complexity) Wikipedia

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