Optimal state reductions of automata with partially specified behaviors

Nondeterministic finite automata with don't care states, namely states which neither accept nor reject, are considered. A characterization of deterministic automata compatible with such a device is obtained. Furthermore, an optimal state bound for the smallest compatible deterministic automata is provided. It is proved that the problem of minimizing deterministic don't care automata is NP-complete and PSPACE-hard in the nondeterministic case. The restriction to the unary case is also considered

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.Comment: In Proceedings NCMA 2023, arXiv:2309.0733

Pairs of Complementary Unary Languages with "Balanced" Nondeterministic Automata

For each sufficiently large N, there exists a unary regular language L such that both L and its complement L^c are accepted by unambiguous nondeterministic automata with at most N states while the smallest deterministic automata for these two languages require a superpolynomial number of states, at least $e^{\Omega(\sqrt[3]{N\cdot\ln^{2}\!N})}$. Actually, L and L^c are accepted by nondeterministic machines sharing the same transition graph, differing only in the distribution of their final states. As a consequence, the gap between the sizes of unary unambiguous self-verifying automata and deterministic automata is also superpolynomial