a counter helps for AfAs by showing that $ {\mathtt{MANYTWINS}}$ can be recognized by AfCAs with one-sided bounded-error in realtime read mode.

In the next section, we provide the necessary background. Our main results are given in Sections 3, 4 and 5, respectively. Section 6 concludes the paper.

Background
==========

We assume the reader to have the knowledge of automata theory, and familiarity with the basics of probabilistic and quantum automata. We refer [@SayY14A] and [@AY15A] for the quantum models.

Throughout the paper, the input alphabet is denoted as $ \Sigma $ not including the left end-marker (${\mbox{\textcent}}$) and the right end-marker (${\$}$). The set $ {\widetilde{\Sigma}}$ denotes $ \Sigma \cup \{{\mbox{\textcent}},{\$}\} $. For a given input $ w \in \Sigma^* $, $ |w| $ is the length of $w$, $w[i]$ is the $i$-th symbol of $w$, and $ {\tilde{w}}= {\mbox{\textcent}}w {\$}$. For any given string $ w \in \{1,2\}^* $, $ e(w) $ denotes the encoding of $ w $ in base-3. The value $ \overline{1} $ in a vector represents the value that makes the vector summation equal to 1, i.e., if the summation of all other entries is $ x $, then it represents the value $ 