Forgetting 1-Limited Automata

We introduce and investigate forgetting 1-limited automata, which are single-tape Turing machines that, when visiting a cell for the first time, replace the input symbol in it by a fixed symbol, so forgetting the original contents. These devices have the same computational power as finite automata, namely they characterize the class of regular languages. We study the cost in size of the conversions of forgetting 1-limited automata, in both nondeterministic and deterministic cases, into equivalent one-way nondeterministic and deterministic automata, providing optimal bounds in terms of exponential or superpolynomial functions. We also discuss the size relationships with two-way finite automata. In this respect, we prove the existence of a language for which forgetting 1-limited automata are exponentially larger than equivalent minimal deterministic two-way automata.