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 au- tomata, 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.