# UP (complexity)

In complexity theory, UP ("Unambiguous Non-deterministic Polynomial-time") is the complexity class of decision problems solvable in polynomial time on an unambiguous Turing machine with at most one accepting path for each input. UP contains P and is contained in NP.

A common reformulation of NP states that a language is in NP if and only if a given answer can be verified by a deterministic machine in polynomial time. Similarly, a language is in UP if a given answer can be verified in polynomial time, and the verifier machine only accepts at most one answer for each problem instance. More formally, a language L belongs to UP if there exists a two-input polynomial-time algorithm A and a constant c such that

if x in L , then there exists a unique certificate y with ${\displaystyle |y|=O(|x|^{c})}$ such that ${\displaystyle A(x,y)=1}$
if x is not in L, there is no certificate y with ${\displaystyle |y|=O(|x|^{c})}$ such that ${\displaystyle A(x,y)=1}$
algorithm A verifies L in polynomial time.

UP (and its complement co-UP) contain both the integer factorization problem and parity game problem; because determined effort has yet to find a polynomial-time solution to any of these problems, it is suspected to be difficult to show P=UP, or even P=(UPco-UP).

The Valiant-Vazirani theorem states that NP is contained in RPPromise-UP, which means that there is a randomized reduction from any problem in NP to a problem in Promise-UP.

UP is not known to have any complete problems.[1]

## References

Lane A. Hemaspaandra and Jorg Rothe, Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets, SIAM J. Comput., 26(3) (June 1997), 634–653