On NL vs UL

Just for recap: a (decision) problem is in NL if some logspace-bounded nondeterministic machine can decide it. It's in UL if in addition the machine has at most one accepting path for each input. (U is for unambigous here.)

I have not that deep insight in the topic, but it seems to me that NL = UL is highly possible. However, to understand the problem statement of this post, you don't have to know anything about nondeterminism or unambiguity; it suffices to know what logspace computability means.
In order to show this it suffices to define an algorithm f that

• takes a directed graph G=(V,E) as input;
• assigns a positive integer weight w(e) to each edge e of G;
• such that w(e) is bounded by some polynomial of |V|;
• such that for each pair (s,t) of nodes with t being reachable from s there exists a unique path from s to t having minimum weight;
• finally, f has to be computable in logspace.

E.g, without the polynomially-boundedness condition a good f could be one which assigns distinct powers of two to the edges, i.e. edge i has weight 2ⁱ.

The thing would work since the "min-unique" problem (given (G,s,t), say "yes" if t is reachable from s via a unique path, say "no" if t is not rechable from s at all, say anything if there are more s-t paths) is in UL and UL is closed under logspace reductions, thus one has "only" to reduce the NL-complete reachability problem (in dags, or in strictly layered graphs, whatever) to min-uniqueness. Having these weights in logspace, each weighted edge e can be replaced by a path of length w(e), concluding the reduction.

For more details see Raghunath Tewari's PhD dissertation from 2011.