Consider the class of problems $\mathsf{StreamL}$ which can be solved in logarithmic space reading the input in a single pass from left to right. In other words:
$L \in \mathsf{StreamL}$ if there exists a Turing machine $M$ which decides $L$, where:
There are two tapes, the read-only input tape and the working tape
$M$ moves only to the left on the input tape, and uses at most $O(\log n)$ space on the working tape.
Has this class been studied?
My assumption is that the answer is yes, but I'm not yet aware of a definition of the class in the literature.
Most literature on streaming algorithms that I am aware of considers the complexity of solving specific algorithmic problems, and does not tackle structural complexity i.e. defining classes such as the above and determining their relationships.
There is also a large body of work on communication complexity classes. In this domain there is a relevant class called $\mathsf{P}^{cc}$ (see Babai, Frankl, and Simon 1986: Complexity classes in communication complexity theory), which contains functions of two variables $f(x,y)$ where they can be solved using a small amount of communication between $x$ and $y$. This is related to $\mathsf{StreamL}$ above (for functions of two variables, $\mathsf{StreamL}$ is contained in $\mathsf{P}^{cc}$), but the class above is not limited to functions of two variables and enforces a stricter computational requirement.
The obvious inclusions are $\mathsf{REG} \subseteq \mathsf{StreamL} \subseteq \mathsf{L}$, and no apparent inclusion either way between $\mathsf{StreamL}$ and $\mathsf{NC}^1$.
