UP (complexity)

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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.^[1] 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=(UP ∩ co-UP).

The Valiant–Vazirani theorem states that NP is contained in RP^Promise-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.^[2]

• Hemaspaandra, Lane A.; Rothe, Jörg (June 1997). "Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets". SIAM Journal on Computing. 26 (3): 634–653. arXiv:cs/9907033. doi:10.1137/S0097539794261970. ISSN 0097-5397.