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Parity-P is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero number of acceptance paths). Thus Parity-P is basically PP's stunted younger sibling: while PP counts whether or not the number of accepting paths of an NP-machine is a majority or not (i.e. the most-significant bit of that quantity), Parity-P indicates the least-significant bit of the number of accepting paths.

Like NP, Parity-P contains UP (which contains P, "probably" strictly so); and like NP, Parity-P is contained in PSPACE.

Question. What are the best known joint upper and lower bounds for NP and Parity-P?

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By Valiant-Vazirani, NP is contained in BP dot Parity-P (which obviously contains Parity-P). Moreover, Toda showed that PH is in BP dot Parity-P which is in P^(#P) (which is in PSPACE).

For lower bounds, I think both classes contain a class known as FewP which contains UP and is like NP but you ask that strings in the language have at most polynomially many accepting paths.

[Update: corrected typo BPP instead of BP]

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    $\begingroup$ A corollary of the containment PH in BPP dot Parity-P, is that Parity-P is not contained in the Poly Hierarchy unless that Hierarchy collapses. $\endgroup$ – Andy Drucker Aug 17 '10 at 22:21
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    $\begingroup$ This follows because, if Parity-P is in Sigma_k-P, then PH is in BPP dot Sigma_k-P, which is contained in Pi_(k+1)-P. (this last containment follows from a straightforward 'operator' generalization of the result that BPP is in Sigma_2 P intersect Pi_2 P.) $\endgroup$ – Andy Drucker Aug 17 '10 at 22:27
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    $\begingroup$ I think it's considered plausible that BPP dot Parity-P is contained in P^(Parity-P). If this is true, then PH is contained in P^(Parity), which is contained in (Parity-P)^(Parity-P), which actually equals Parity-P. What I'm not sure of is whether any papers on hardness vs. randomness give a hypothesis which implies BPP dot Parity-P contained in P^(Parity-P). $\endgroup$ – Andy Drucker Aug 17 '10 at 22:39
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    $\begingroup$ Finally, Parity-P is distinguished from NP and other PH classes in that it's known to have worst-case-to-average-case reductions. That is, if Parity-P is not in P, then it contains distributional problems that are average-case hard. See Feigenbaum-Fortnow, "Random-self-reducibility of complete sets". $\endgroup$ – Andy Drucker Aug 17 '10 at 22:42
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    $\begingroup$ Here's the general idea: let C be a complexity class. A language L is in (BPP dot C) if there exists a language S in C, consisting of encoded pairs (x, r), such that: -if x is in L, then for 2/3 of all r, the pair (x, r) is in S; -if x is not in L, then for 2/3 of all r, the pair (x, r) is not in S. (Technically, the length of r depends on x and is required to be at most some polynomial in |x|.) $\endgroup$ – Andy Drucker Aug 18 '10 at 17:04

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