# Is $L \subset 1NL$ when $L \neq NL$?

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 $$L=NL$$.

https://ecommons.cornell.edu/handle/1813/6253

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?

Define the language $$BACKPOINTER$$ to have words of length $$n+t\log n$$, divided to $$1+t$$ parts, one of length $$n$$ and the rest of length $$\log n$$, by commas such that $$BACKPOINTER=\{(x,p_1,\ldots,p_t)\mid x_{p_i}=1 \forall i\}$$. It should follow from some standard one-way communication complexity bound that $$BACKPOINTER$$ needs at least $$t$$ bits of memory stored after $$x$$ is read. So for any $$t\gg \log n$$, $$BACKPOINTER\in L\setminus 1NL$$.
ps. Note that $$BACKPOINTER\in 1co\text{-}NL$$.