A log-space Turing machine has a read-only input tape, a write-only output tape and uses at most $O(\log n)$ space in its read-write work tapes. The classes $L$ and $NL$ contain those languages which are decided by deterministic or nondeterministic log-space Turing machines, respectively. The two-way Turing machines may move their head on the input tape into two-way (left and right directions) while the one-way Turing machines are not allowed to move the input head on the input tape to the left.
Hartmanis and Mahaney have investigated the classes $1L$ and $1NL$ of languages recognizable by deterministic one-way log-space Turing machine and nondeterministic one-way log-space Turing machine, respectively. They have shown that $1NL \subseteq L$ if and only if $L=NL$.
See the paper here(it is free to download):
I wonder this question:
Is $L \subset 1NL$ when $L \neq NL$?
Moreover, I wonder this another question:
Is there any reference that shows whether at least one of the options $L \subset 1NL$ or $1NL \subset L$ or $L = 1NL$ should be true?
Thanks in advance!!!