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]

An example of a ⊕P-complete problem (under many-one reductions) is ⊕SAT: given a Boolean formula, is the number of its satisfying assignments odd? This follows from the Cook–Levin theorem because the reduction is parsimonious.[2]

⊕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.[3]

While Toda's theorem shows that PPP 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.[4]

⊕P contains the graph automorphism problem, and in fact this problem is low for ⊕P.[5] 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. ^ Abhishek Shetty, Raghav Malhotra, and Chandan Saha. Lecture 26 scribe notes from Computational Complexity Theory class, 2015.
  3. ^ 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.
  4. ^ Fortnow, Lance (2009), "A simple proof of Toda's theorem", Theory of Computing, 5: 135–140, doi:10.4086/toc.2009.v005a007
  5. ^ Köbler, Johannes; Schöning, Uwe; Torán, Jacobo (1992), "Graph isomorphism is low for PP", Computational Complexity, 2 (4): 301–330, doi:10.1007/BF01200427.