## W-transformations -- definitions

We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of integers, and $\ C\ $ stands for an arbitrary finite set of cardinality $\ |C|>1$, where $\ C\ $ is called a set of colors.

Let any non-empty finite set $\ W\subseteq \mathbb Z\ $ be called a window; set its diameter as

$$\ \delta(W):=1+\max W-\min W$$

Mappings $\ F: C^W\rightarrow C\ $ are called $W$-functionals or simply functionals. Define the induced transformation $\ T_F:C^\mathbb Z\rightarrow C^\mathbb Z\ $ by formula:

$$ (T_F(f))(p)\ :=\ F((f\circ S_p)|W)$$

for every $\ f\in C^\mathbb Z\ $ (i.e. $\ f:\mathbb Z\rightarrow C$) and $\ p\in \mathbb Z$, where the shift $\ S_p :\mathbb Z\rightarrow \mathbb Z\ $ is given by:

$$ \forall_{z\in\mathbb Z}\quad S_p(z) := p+z $$

That's what finite window transformations are, or $W$-transformations for any particular window $\ W$.

## M.Nasu's Theorem

The following indecomposability theorem was proved by **M.Nasu**:

**THEOREM** (M.Nasu) Let $\ C\ $ be an arbitrary finite set such that $\ |C|>1$. Let $\ d\ $ be an arbitrary natural number ($\ n=1\ 2\ \ldots$). Then there exists a $\ W$-transformation $\ T\ $ of diameter $\ \delta(W)=d\ $ such that there does not exist any finite sequence of window transformations $\ T_1\ldots T_n\ $ such that

$$ T\ =\ T_n\circ\ldots\circ T_1$$

and the diameters of transformations $\ T_k\ $ are smaller than $\ d\ $ for every $\ k=1\ldots n\ $ and $\ n=1\ 2\ \ldots$.

## Questions--concept: allow for memory.

Let's continue to assume that $\ C\ $ is an arbitrary finite set (of colors) such that $\ |C|>1$. Let $\ D:=A\times C\ $ where $\ A\ $ is another finite set. We obtain an open problem below for each $\ A\ $ such that

$$ 2\ \le\ |A|\ <\ 2\times |C|$$

Consider the cartesian projection $\ \pi_C: D\rightarrow C$. First let's define processing of an image transformation $\ T:C^{\mathbb Z}\rightarrow C^{\mathbb Z}$. It is a sequence $\ T_k:D^{\mathbb Z}\rightarrow D^{\mathbb Z}\ $ for a natural $\ n$, and $\ k=1\ldots n$, such that for every $\ f:\mathbb Z\rightarrow C\ $ and $\ g:\mathbb Z\rightarrow D\ $ subjected to $\ \pi_C\circ g = f\ $ the following holds:

$$ \pi_C\circ ((T_n\circ\ldots\circ T_1)(g))\ =\ T(f)$$

**PROBLEM** Let $\ d\ $ be an arbitrary natural number $\ 1\ 2\ \ldots$. Show that there exists a $\ W$-transformation $ T:C^{\mathbb Z}\rightarrow C^{\mathbb Z}\ $ of diameter $\ \delta(W)=d\ $ such that there does not exist any sequence of window transformations $\ T_k:D^{\mathbb Z}\rightarrow D^{\mathbb Z}\ $ which processes transformation $\ T$, with all diameters of the mentioned sequence transformations smaller than $\ d$.