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Parity-L, also known as $\oplus$L, 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. A recent related question was asked by Niel de Beaudrap.

My question is the following:

Do we know if NL $\subseteq$ $\oplus$L? Or are these two classes believed to be incomparable?

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Non-uniformly, NL is contained in parity-L: see http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/W94/proc.pdf

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