# 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

- 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. - 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. - Fortnow, Lance (2009), "A simple proof of Toda's theorem",
*Theory of Computing*,**5**: 135–140, doi:10.4086/toc.2009.v005a007 - Köbler, Johannes; Schöning, Uwe; Torán, Jacobo (1992), "Graph isomorphism is low for PP",
*Computational Complexity*,**2**(4): 301–330, doi:10.1007/BF01200427.