Size of Unary One-Way Multi-head Finite Automata

One of the main topics of descriptional complexity is the question of how the size of the description of a formal language varies when being described by different formalisms. A fundamental result is the exponential trade-off between the number of states of nondeterministic (NFA) and deterministic finite automata (DFA) (see, for example, [12]). Additional exponential and double-exponential trade-offs are known, for example, between unambiguous and deterministic finite automata, between alternating and deterministic finite automata, between deterministic pushdown automata and DFA, and between the complement of a regular expression and conventional regular expressions. Beside these recursive tradeoffs, bounded by recursive functions, it is known that there also non-recursive trade-offs, which are not bounded by any recursive function. Such trade-offs have at first been shown to exists between context-free grammars generating regular languages and finite automata [12]. For a survey on recursive and non-recursive trade-offs we refer to [3, 5]. Unary languages, that is, languages defined over a singleton alphabet, are of particular interest, since in this case often better or more precise results than in the case of arbitrary alphabets can be obtained. For example, the tradeoff of 2 between an n-state NFA and DFA, is reduced to e √ n·ln(n)) in the unary case [1]. The descriptional complexity of unary regular languages has been studied in many ways. On the one hand, many automata models such as one-way finite automata, two-way finite automata, pushdown automata, or context-free grammars for unary languages are investigated and compared to each other with respect to simulation results and the size of the simulation (see, for example, [2, 11, 13, 15]). On the other hand, many results concerning the state complexity of operations on unary languages have been obtained (see, for example, [4, 7, 10, 14]). Here, we consider deterministic one-way multi-head finite automata accepting unary languages. Since it is known that every unary language accepted by a one-way multi-head finite automaton is semilinear and thus regular [6, 16], it is of interest to investigate the descriptional complexity of such devices in comparison with the models mentioned above. In detail, we establish upper and lower bounds for the conversion of k-head DFA to one-head DFA and one-head