Parity P

In computational complexity theory, the complexity class ⊕P (pronounced "parity P") is the class of decision problems solvable by a nondeterministic Turing machine in polynomial time, where the acceptance condition is that the number of accepting computation paths is odd. An example of a ⊕P problem is "does a given graph have an odd number of perfect matchings?" The class was defined by Papadimitriou and Zachos in 1983.^[1]

⊕P is a counting class, and can be seen as finding the least significant bit of the answer to the corresponding #P problem. The problem of finding the most significant bit is in PP. PP is believed to be a considerably harder class than ⊕P; for example, there is a relativized universe (see oracle machine) where P = ⊕P ≠ NP = PP = EXPTIME, as shown by Beigel, Buhrman, and Fortnow in 1998.^[2]

While Toda's theorem shows that P^PP contains PH, P^⊕P is not known to even contain NP. However, the first part of the proof of Toda's theorem shows that BPP^⊕P contains PH. Lance Fortnow has written a concise proof of this theorem.^[3]

⊕P contains the graph isomorphism problem, and in fact this problem is low for ⊕P.^[4] It also trivially contains UP, since all problems in UP have either zero or one accepting paths. More generally, ⊕P is low for itself, meaning that such a machine gains no power from being able to solve any ⊕P problem instantly.

The ⊕ symbol in the name of the class may be a reference to use of the symbol ⊕ in Boolean algebra to refer the exclusive disjunction operator. This makes sense because if we consider "accepts" to be 1 and "not accepts" to be 0, the result of the machine is the exclusive disjunction of the results of each computation path.

References

1. C. H. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings of the 6th GI Conference in Theoretical Computer Science, Lecture Notes in Computer Science, volume 145, Springer-Verlag, pp. 269-276. 1983.
2. R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of ACM STOC'98, pp. 203-208. 1998.
3. Fortnow, Lance (2009), "A simple proof of Toda's theorem", Theory of Computing 5: 135–140, doi:10.4086/toc.2009.v005a007
4. Köbler, Johannes; Schöning, Uwe; Torán, Jacobo (1992), "Graph isomorphism is low for PP", Computational Complexity 2 (4): 301–330, doi:10.1007/BF01200427.