One-way multihead finite automata and 2-bounded languages

Abstract

LanguagesL _n ={1^x2^ix:i, x ∈ ℕ, 1≤i≤n} were used to show that, for eachk, one-way non-sensing deterministic finite automata (1-MFA) withk+1 heads are more powerful than such automata withk heads, even if we consider only 2-bounded languages (Chrobak). Fork ∈ ℕ letf(k) be the maximal numbern such that languageL _n can be recognized by a 1-MFA withk heads. We present a precise inductive formula forf(k). It may be shown that, fork≥3,

$$\frac{{(2k - 5)! \cdot (k - 2) \cdot (k - 1)}}{{2^{k - 3} }} \leqslant f(k) \leqslant \frac{{(2k - 5)! \cdot (k - 2) \cdot (k - 1) \cdot 3k^2 }}{{2^{k - 3} }}$$

that is,f(k)≈k ^2k. The proof is constructive in the sense that it shows how to construct ak-head automaton recognizingL _f(k) . This is a solution of the problem stated by Chrobak.