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 a non-deterministic Turing machine with at most one accepting path for each input. UP contains P and is contained in NP. It is considered likely that either P ≠ UP or UP ≠ NP (or both), since otherwise P = NP, which is widely believed to be false. Most believe that both inequalities hold.

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

L = {x in {0,1}* | ∃! certificate, y with |y| = O(|x|^c) such that A(x,y) = 1}

Algorithm A verifies L in polynomial time.

Important complexity classes (more)
P • NP • co-NP • NP-C • co-NP-C • NP-hard • UP • #P • #P-C • L • NL • NC • P-C • PSPACE • PSPACE-C • EXPTIME • EXPSPACE • PR • RE • Co-RE • RE-C • Co-RE-C • R • BQP • BPP • RP • ZPP • PCP • IP • PH