In complexity theory, UP ("Unambiguous Non-deterministic Polynomial-time") is the complexity class of decision problems solvable in polynomial time on a 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 |y| = O(|x|c) such that A(x,y) = 1
- if x is not in L, there is no certificate y with |y| = O(|x|c) such that 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).
Lane A. HEMASPAANDRA and Jorg Rothe, Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets, SIAM J. Comput., 26(3), 634–653