= Solution
\b[Exactly the <regular languages>.] A <read-only-from-left Turing acceptor> has only its current control state as usable memory: it never changes a tape symbol, and its head never returns to an earlier cell. This is the behaviour of a <deterministic finite automaton> while it reads the <input>.
To make the correspondence precise, retain the control states and the transitions on the <input> <alphabet>. Replace accepting and rejecting halts by absorbing states, so that halting before the end of a <word over an alphabet> has the same effect on every continuation. For each remaining state $q$, follow the transitions on blank symbols and mark $q$ accepting in the <deterministic finite automaton> exactly when this blank-only continuation eventually accepts. This marking is a finite calculation: either a halting state is reached or a state repeats, after which the blank-only run loops. The resulting <deterministic finite automaton> accepts exactly the same <formal language> as the <read-only-from-left Turing acceptor>. If the latter is a decider, the blank-only run from every reachable end-of-input state halts.
Conversely, simulate any <deterministic finite automaton> by a <read-only-from-left Turing acceptor>, using its state transitions while reading the <input>. At the first blank symbol, halt and accept or reject according to the current <deterministic finite automaton> state. This machine halts on every <input> and decides the required <regular language>.
$$
\boxed{\text{ROFL-decidable languages}=\text{regular languages}.}
$$
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