with value $ \delta(s,\sigma,\theta,s',d) $: it switches to the state $ s' \in S $ and updates the value of the counter by $ d \in \{ -1,0,+1 \} $. To be a well-formed affine machine, the transition function must satisfy that, for each triple $ (s,\sigma,\theta) \in S \times {\widetilde{\Sigma}}\times \{Z, NZ\} $, $$\sum\limits_{s' \in S, d\in \{-1,0,+1\}} \delta(s,\sigma,\theta,s',d) = 1.$$ Remark that the value of the counter can be updated by a value in $ \{-t,\ldots,+t\} $ for some $ t>1 $ but this does not change the computational power of the model (see [@YFSA12A] for more details).

Any classical configuration of $ A $ is formed by a pair $ (s,c) \in S \times \mathbb{Z} $, where $ s $ is the deterministic state and $ c  $ is the value of the counter. Let $w \in \Sigma^*$ be the given input and $ m = | {\mbox{\textcent}}w {\$}| $. Since all possible values of the counter are in $ \{ -m,\ldots,m \} $, the total number of classical configurations is $ N = m |S| $. We denote the set $ \{ (s,c) \mid s \in S, c \in \{-m,\ldots,m\} \} $ for $ w $ 