  Single-tape nondeterministic Turing machines that are allowed to replace the
symbol in each tape cell only when it is scanned for the first time are also
known as 1-limited automata. These devices characterize, exactly as finite
automata, the class of regular languages. However, they can be extremely more
succinct. Indeed, in the worst case the size gap from 1-limited automata to
one-way deterministic finite automata is double exponential.
  Here we introduce two restricted versions of 1-limited automata, once-marking
1-limited automata and always-marking 1-limited automata, and study their
descriptional complexity. We prove that once-marking 1-limited automata still
exhibit a double exponential size gap to one-way deterministic finite automata.
However, their deterministic restriction is polynomially related in size to
two-way deterministic finite automata, in contrast to deterministic 1-limited
automata, whose equivalent two-way deterministic finite automata in the worst
case are exponentially larger. For always-marking 1-limited automata, we prove
that the size gap to one-way deterministic finite automata is only a single
exponential. The gap remains exponential even in the case the given machine is
deterministic.
  We obtain other size relationships between different variants of these
machines and finite automata and we present some problems that deserve
investigation.
