Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows.
Is there a result comparable to the Karp-Lipton theorem starting from the assumption $Linmathsf{NC}^1/mathsf{poly}$ with $L$ an $mathsf L$-complete language (under, say, $mathsf{AC}^0$ reductions)? By “comparable” I mean that it derives non-trivial consequences (perhaps considered unlikely) from the assumption.
I am having trouble finding in the literature (or online) a discussion of the relationship between $mathsf{NC}^1$ and logarithmic space going beyond “the former is included in the latter and it is open whether the inclusion is strict”. More specifically, I could find only two three pieces of evidence against equality of the two classes:
- Barrington’s theorem and the characterization of non-uniform deterministic logspace in terms of branching programs gives us $mathsf{NC}^1/mathsf{poly}=mathsf{L}/mathsf{poly}$ iff bounded-width polysize branching programs are as expressive as arbitrary polysize branching programs, which I guess would be highly surprising (especially considering how surprising Barrington’s theorem itself is).
- Markus Holzer [Hol02] proved that $mathsf{NC}^1/mathsf{poly}=mathsf{L}/mathsf{poly}$ iff one-head two-way non-uniform deterministic finite automata have the same expressive power whether they are oblivious or not (“oblivious” means that the movement of the head during the computation depends only on the length of the input, not on the input itself). Oblivious polytime Turing machines do have the same power as non-oblivious ones, but I guess it is hard to see how that simulation may be done in the much more restricted framework of finite automata.
- Edit: there is a paper by Allender et al. [ABCDR09] in which a number of reachability problems for certain classes of graphs are shown to be hard for $mathsf{NC}^1$ under $mathsf{AC}^0$ reductions, whereas the same problems are not known to be hard for $mathsf{L}$. As stated by the authors, “this gives a cluster of natural problems that are candidates for having complexity intermediate between $mathsf{NC}^1$ and $mathsf{L}$”.
Besides the above points and the usual empirical evidence ($mathsf{L}$-complete problems do not seem to have logarithmic-depth bounded fan-in circuits), is there any other evidence against $mathsf{NC}^1=mathsf{L}$?
[Hol02] Markus Holzer. Multi-head finite automata: data-independent versus data-dependent computations. Theor. Comput. Sci. 286(1):97-116 (2002).
[ABCDR09] Eric Allender, David A. Mix Barrington, Tanmoy Chakraborty, Samir Datta and Sambuddha Roy. Planar and Grid Graph Reachability Problems. Theory Comput. Syst. 45(4):675–723 (2009).
