If $mathsf{L = NL}$, then there is a logspace algorithm that solves the decision version of 2-SAT.
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Is $mathsf{L = NL}$ known to imply that there is a logspace algorithm to obtain a satisfying assignment, when given a satisfiable 2-SAT instance as input?
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If not, what about algorithms which use sub-linear space (in the number of clauses)?
