On the Power of Unambiguity in Logspace

We report progress on the \NL vs \UL problem. [-] We show unconditionally that the complexity class $\ReachFewL\subseteq\UL$. This improves on the earlier known upper bound $\ReachFewL \subseteq \FewL$. [-] We investigate the complexity of min-uniqueness - a central notion in studying the \NL vs \UL problem. We show that min-uniqueness is necessary and sufficient for showing $\NL =\UL$. We revisit the class $\OptL[\log n]$ and show that {\sc ShortestPathLength} - computing the length of the shortest path in a DAG, is complete for $\OptL[\log n]$. We introduce $\UOptL[\log n]$, an unambiguous version of $\OptL[\log n]$, and show that (a) $\NL =\UL$ if and only if $\OptL[\log n] = \UOptL[\log n]$, (b) $\LogFew \leq \UOptL[\log n] \leq \SPL$. [-] We show that the reachability problem over graphs embedded on 3 pages is complete for \NL. This contrasts with the reachability problem over graphs embedded on 2 pages which is logspace equivalent to the reachability problem in planar graphs and hence is in \UL.

💡 Research Summary

The paper tackles the long‑standing NL versus UL problem by exploring several interrelated concepts in logspace computation. First, it strengthens the known relationship between ReachFewL and FewL by proving unconditionally that ReachFewL ⊆ UL. ReachFewL consists of languages accepted by nondeterministic logspace machines that have only a few accepting paths on each input. The authors show that any ReachFewL computation can be simulated by an unambiguous logspace machine. The key technical tool is a logspace reduction that transforms an arbitrary directed graph into a “min‑unique” graph, i.e., a graph in which for every pair of vertices the shortest path (if it exists) is unique. Because this transformation can be carried out in logspace, the resulting graph admits a single accepting computation path, establishing the inclusion.

Next, the paper establishes that min‑uniqueness is both necessary and sufficient for proving NL = UL. In one direction, if every NL language can be reduced to a min‑unique instance, then an unambiguous machine can decide it, yielding UL = NL. Conversely, if UL = NL, then a standard construction can be used to enforce min‑uniqueness for any NL problem. Thus min‑uniqueness emerges as the precise structural property that captures the power of unambiguity in logspace.

The authors then revisit the optimisation class OptL