In computational complexity theory, the complexity class FP is the set of function problems which can be solved by a deterministic Turing machine in polynomial time; it is the function problem version of the decision problem class P. Roughly speaking, it is the class of functions that can be efficiently computed on classical computers without randomization. In computer science, computational complexity theory is the branch of the theory of computation that studies the resources required during computation to solve a given problem. ... In computational complexity theory, a complexity class is a set of problems of related complexity. ... In computational complexity theory, a function problem is a problem other than a decision problem, that is, a problem requiring a more complex answer than just YES or NO. Notable examples include the traveling salesman problem which asks for the route taken by the salesman, and the integer factorization problem... The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ... In computational complexity theory, Polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ...

The difference between FP and P is that problems in P have one-bit, yes/no answers, while problems in FP can have any output that can be computed in polynomial time. For example, adding two numbers is an FP problem, while determining if their sum is odd is in P. More complex is the relationship between FP and FNP. FNP is defined as follows: In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is a bit of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains: A binary relation P(x,y...

A binary relation P(x,y) is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y.

FP may similarly be defined as: In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...

A binary relation P(x,y) is in FP if and only if there is a deterministic polynomial time algorithm that, given x, can find some y such that P(x,y) holds.

That is, instead of merely verifying y, it must find its value. This is similar to the computation/verification relationship between P and NP; it also shows that FP is contained in FNP. In fact, FP = FNP if and only if P = NP. Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal. ... Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal. ...

Polynomial-time function problems are fundamental in defining polynomial-time reductions, which are used in turn to define the class of NP-complete problems. In computational complexity theory a polynomial-time reduction is a reduction which is computable by a deterministic Turing machine in polynomial time. ... In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use...

Because a machine that uses logarithmic space has at most polynomially many configurations, FL, the set of function problems which can be calculated in logspace, is contained in FP. It is not known whether FL = FP; this is analogous to the problem of determining whether the decision classes P (complexity) and L (complexity) are equal. In computational complexity theory, the complexity class FL is the set of function problems which can be solved by a deterministic Turing machine in a logarithmic amount of memory space. ... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... In computational complexity theory, L is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a logarithmic amount of memory space. ...

References

• Complexity Zoo: FP