Non-trivial unary languages recognized by two-way one-counter machines.
Marzio De Biasi, Abuzer Yakaryılmaz · arXiv (Cornell University) · 2013
Finite automaton with one counter (CA) is a fundamental model in automata theory. It has been widely examined from different point of views since sixties. One recent significant result, for example, is that the equivalence problem of deterministic one-way CAs is NLcomplete [Stanislav Bohm, Stefan Goller, Petr Jancar. STOC 2013: 131140]. In the case of unary languages, on the other hand, we know little about the computational power of CAs. Since one-way nondeterministic pushdown automata, a generalization of one-way nondeterministic CAs, cannot recognize any nonregular unary language, it is interesting to focus on one-way alternating CAs (1ACAs) and two-way CAs (2CAs). Up to our knowledge, the only known unary non-regular languages recognized by 1ACAs and 2CAs are formed by the strings with exponential lengths. In this paper, we present a new programming technique for 2CAs on unary languages that allows to simulate multi-counter automata and space bounded Turing machines operating on unary or general alphabets. The idea is that a 2CA can take the input and the working memory of the simulated machine as the exponent of some integers encoded on unary inputs. Thus, once the 2CA becomes sure about the correctness of the encoding, it can start a two-counter simulation of the given machine. Here the second counter is simulated by the input head of the 2CA on the unary input. Based on this idea, we will present several new non-trivial unary languages recognized by deterministic, nondeterministic, alternating, and probabilistic 2CAs. In some cases, we use encodings on binary alphabet as well, in which we show that using a constant-size quantum memory can help to replace the encoding on binary alphabets with unary alphabets. keywords: automata theory, counter machines, unary languages, nondeterminism, alternation, randomization, quantum automata