\documentclass{amsart}
\usepackage{amssymb,amsfonts,latexsym,amsmath}
\begin{document}
\title{On a class of threshold transformations having
single-cycle attractors}
\author{Takao Ueda}
\begin{abstract} The first step is to construct a class of
skew-circular one-to-one threshold transformations of the Boolean
$n$-space having single cycles of length $2n$ by means of [\; ]-
representations. Then $2n$-dimensional threshold transformations
having single $2n$-cycle attractors are constructed through
expansion and neighborhood functions. The construction of $n$-
dimensional transformations having single $2n$-cycle attractors
is carried out through partial neighborhood functions. For the
proof of attractiveness, newly-introduced extended
representations and decomposition of transformations are
employed.
\end{abstract}
\maketitle
\section*{1. Introduction}
The mathematical nature of neural networks is fundamentally
characterized by threshold transformations. Yet we have few
results on the transformations, because of the difficulties of
non-linearity [1,3,4,5,6,7,8]. The author's combinatorial
approach is based on the [\, ]-representations of Boolean
transformations introduced in [4]. In particular, [4,5,6] were
concerned with construction and analysis of one-to-one threshold
transformations. Dynamical systems generated by threshold
transformations were the concern of [7], in which it was proved
that there existed a transformation having a single-cycle
attractor of any length and the basin of attraction was the whole
space. In this article we construct a class of $n$-dimensional
threshold transformations having single-cycle attractors of
length $2n$. In contrast to [7], the basin of attraction is
limited to a close neighborhood of the attractive cycles.
The background concepts of this preliminary section are
found in good introductory textbooks on discrete mathematics
such as [2,9]. Those directly relevant to this article are
described in [8]. Definitions on finite-space dynamical systems
are given in Appendix of this article.
The difference of sets is denoted by $\backslash$ , and
$A\backslash\{ q\}$ is denoted by $A\backslash q$ for a
one-element set. The cardinality of a set $A$ is denoted by
$|A|$. $\mathbf{N} =\{ 1, 2, . . . , n\}$ is the residue class
ring with $n$ as the zero element. The symmetric group on
$\mathbf{N}$ is denoted by $\mathrm{SYM}(\mathbf{N})$.
Let $\mathbf{Q}$ be the minimal Boolean algebra $\{ 0,
1\}$ with the operations, identity $I_{\mathbf{Q}}$,
complementation $\neg $, AND $\cdot $, and OR $\vee $.
$ \mathbf{Q}^{\mathbf{N}}$, the set of functions:
$\mathbf{N} \rightarrow \mathbf{Q}$, is simply denoted by
$\mathbf{Q}^{n}$ hereafter, and $\mathbf{Q}^{\{ i\} }$ is
identified with $\mathbf{Q}$. Let $L \subseteq M
\subseteq \mathbf{N}$. Then the \emph{projection}
$P_{L}: \mathbf{Q}^{M} \rightarrow \mathbf{Q}^{L}$ is
defined by
$$
(p_{L}x)_{j} = x_{j} \quad \mbox{for every $j \in
L$ for every $x \in \mathbf{Q}^{M}$.}
$$
If $i \in M$, then the \emph{projection} $p_{i}:
\mathbf{Q}^{M} \rightarrow \mathbf{Q}$ is defined by
$$
p_{i}x = x_{i } \quad \mbox{for every $x \in \mathbf{Q}^M$.}
$$
A function: $\mathbf{Q}^{M} \rightarrow \mathbf{Q}$ is called
a \emph{Boolean function}, and a function: $\mathbf{Q}^{M}
\rightarrow \mathbf{Q}^{M}$ is called a \emph{Boolean
transformation}. For a set of Boolean functions, let $S_m\{.\}$
denote the disjunction of all conjunctions of $m$ elements of
$\{.\}$. For example, $S_2\{p_1,p_2,p_3\} = p_1\cdot p_2\vee
p_1\cdot p_3\vee p_2\cdot p_3$. The Polya action $h$ of
$\mathrm{SYM}(\mathbf{N})$ on $\mathbf{Q}^{n}$
associates a permutation $\tau $ of $\mathbf{N}$ with a
permutation of coordinates of $\mathbf{Q}^{n}$ by
$$
(h\tau )(x_{1}, x_{2}, . . , x_{n})
= (x_{\tau ^{-1}1}, x_{\tau ^{-1}2},..,x_{\tau ^{-1}n}).
$$
We omit the Polya action $h$ hereafter and write $\tau x$ in
place of $(h\tau )x$ for an element $x$ of $\mathbf{Q}^{n}$.
Let $J^{-}$ for $J = \{ s,t,..,w\} \subseteq \mathbf{N}$
denote the complementation of the $s$th, $t$th,..,$w$th
coordinates defined by
$$
J^{-}x = (x_{1},..,\neg x_{s},..,\neg x_{t},..,\neg
x_{w},..,x_{n}).
$$
If $J$ is a one element-set $\{ s\}$, $J^{-}$ is
denoted by $s^{-}$. Also $\mathbf{N}^-$ is denoted by
$\bar{\neg}$. A product $\tau J^{-}$, where
$\tau \in \mathrm{SYM}(\mathbf{N})$ and $J^{-}$ is a
complementation is an isometry, and the set
$O(\mathbf{Q}^{n})$ of all isometries is a transformation
group of $\mathbf{Q}^{n}$. For the product of isometries,
$$
(\sigma J^{-})(\tau K^{-}) = \sigma \tau
(\tau ^{-1}J \dot{+} K)^{-},
\eqno (1.1)
$$
where $\dot{+}$ is the symmetric difference. Further, the
Polya action of $O(\mathbf{Q}^{n})$ on the set of Boolean
functions defines $Tf = fT^{-1}$ for every Boolean function
$f$ and every isometry $T$.
We refer to the set $f$ for a Boolean function meaning
the set $f^{-1}1$, ie. the inverse image of $1$. Therefore,
$x \in f$ means $fx = 1$. Then $\neg f$ is the set $f^{-1}0$,
and $\bar{\neg}f$ is the set of complements of points of
$f^{-1}1$. We call the set of all non-fixed points of a
transformation $F$ of $\mathbf{Q}^{n}$ the \emph{carrier}
of $F$ and write $\mathrm{Car}F$. If $\mathrm{Car}F$ and
$\mathrm{Car}G$ are disjoint, then the \emph{sum} $F+G$
can be defined by
$$
(F+G)x = \left\{
\begin{array}{ll}
Fx &\mbox{if $x \in \mathrm{Car}F$,}\\
Gx &\mbox{if $x \in \mathrm{Car}G$,}\\
x &\mbox{if $x \in (\mathrm{Car}F )\cup
\mathrm{Car}G)^{c}$.}
\end{array}
\right.
$$
A transformation $F$ is called \emph{self-dual}, if
$\bar{\neg}F = F\bar{\neg}$. A function
$f: \mathbf{Q}^{n} \rightarrow \mathbf{Q}$ is called a threshold
function, if the set $f$ and $\neg f$ are separated by a
hyperplane in the real $n$-space $R^{n}$. A frequently used
property of a threshold function is that it is not
$2$-\emph{summable}.
Here, a function is called $2$-\emph{summable}, if there exist
some $a, b, c, d$ such that $a \in
f$, $b \in f$, $c \in \neg f$, $d \in \neg f$, and $a + b
= c + d$ where $+$ is the addition in $R^{n}$. A transformation
$F$ of $\mathbf{Q}^{n}$ is called a \emph{threshold
transformation}
if $p_{i}F$ is a threshold function for every $i$.
Assume that the transformation $F = (F_{1}, . . . , F_{n})$
of
$\mathbf{Q}^{n}$, where $F_{i} = p_{i}F$, is self-dual. For
a point $x \in \mathbf{Q}^{n}$, $x_{i} = 1$ and $(Fx)_{i} = 0$
iff $x \in p_{i}\cdot \neg F_{i}$. Let $f_{i}$ be defined by
$$
f_{i} = p_{i}\cdot \neg F_{i}
\eqno (1.2)
$$
for every $i$. Then
$$
F_{i} = p_{i}\cdot \neg f_{i} \vee
\bar{\neg}f_{i}.
\eqno (1.3)
$$
Conversely, for any Boolean function $f_{i}$ such that
$f_{i} = p_{i}\cdot f_{i}$ for every $i$, let $F_{i}$ be defined
by (1.3). Then $F = (F_{1},..., F_{n})$ is a self-dual
transformation,
and (1.2) is satisfied. Consequently, any self-dual
transformation
$F$ such that $F = (F_{1},..., F_{n})$ can be represented by
$$
F = [f_{1}, . . . , f_{n}].
$$
Here $x =(x_{1}, . . . , x_{n}) \in f_{i}$ if and only if $x_{i}
= 1$
and $(Fx)_{i} = 0$. Therefore,
$$
\mathrm{Car}F = \bigcup_{i=1}^n(f_{i}\cup \bar{\neg}f_{i}).
$$
If $F$ is self-dual and represented by $[f_{1}, . . . ,
f_{n}]$, then $F$ is a threshold transformation if and only if
$f_{i}$ is a
threshold function for every $i$ [4].
A transformation $F$ of $\mathbf{Q}^{n}$ is called
\emph{circular}, if $F\rho = \rho F$, where $\rho $ is the
cyclic permutation $(1, 2, . . , n)$ of $\mathbf{N}$. $F$ is
called \emph{skew-circular}, if $F(\rho n^{-}) =
( \rho n^{-})F$. $F = [f_{1}, . . . , f_{n}]$ is circular if and
only if $f_{i} = \rho ^{i-1}f_{1}$, and $F$ is denoted by
$F = \langle f_{1}\rangle $. $F$ is skew-circular, if and only
if $f_{i} = (\rho n^{-})^{i-1}f_{1}$, and $F$ is denoted
by $F = \langle\langle f_{1}\rangle\rangle $.
Let $\langle \bar{\neg},\rho \rangle $ and $\langle
\rho n^{-}\rangle $ respectively denote the subgroup of
$O(\mathbf{Q}^{n})$ generated by $\bar{\neg}$ and
$\rho $, and the subgroup generated by $\rho n^{-}$.
In this article, $[a]$ for an element $a$ in $\mathbf{Q}^{n}$
denotes the orbit of $\langle \bar{\neg},\rho \rangle $
containing $a$ if the operating transformation is self-dual
and circular; $[A]$ for a subset $A$ of $\mathbf{Q}^{n}$
denotes the union of the orbits of $\langle \bar{\neg},
\rho \rangle $containing $a \in A$. Similarly, $[a]$
and $[A]$ respectively denote the orbit of
$\langle \rho n^{-}\rangle $ containing $a$ and
the union of orbits of $\langle \rho n-\rangle $ for
$a \in A$, if the operating transformation is skew-circular.
Note that $\langle \bar{\neg},\rho n^{-}\rangle = \langle
\rho n^{-}\rangle $, since $(\rho n^{-})^{n} = \bar{\neg}$.
Then, a transformation $F^{\sim }$ of the orbit set
$ \{ [x] \mid x \in \mathbf{Q}^{n}\}$ is naturally defined
by $F^{\sim }[x] = [Fx]$. \\
\section*{2. Single-cycle one-to-one transformations}
The simplest skew-circular one-to-one threshold
transformation is\\
\textbf{Example 2.1.} $F = \langle\langle f\rangle\rangle $,
where $f = p_{1}\cdot \cdot p_{i}\cdot \cdot p_{n}$.
The graph of $F$ consists of one $2n$-cycle and loops. \\
As a generalization of Example 2.1, we now determine
a condition for the transformation $F = \langle\langle
f\rangle\rangle $ of $\mathbf{Q}^{n}$ such that
$f = p_{1}\cdot q_{2}\cdot ...\cdot q_{n}$, where
$q_{i} = p_{i}$ or $\neg p_{i}$ for every $i$,
to be one-to-one.
Let $f = \{ x\}$ , a one-element set, where
$x = (1,x_{2},..,x_{n})$. We assume $Fx = (0,x_{2},..,x_{n})$.
In order to determine a condition for
$F(\mathrm{Car}F) = \mathrm{Car}F$, suppose
$Fx = (\rho n^{-})^{h-1}x$ for $0 < h-1 < 2n$. Then
$$
p_{1}x = 1, 1^{-}x = (\rho n^{-})^{h-1}x.
\eqno (2.1)
$$
If $0 < h-1 < n$, then
$$
(0,x_{2},..,x_{n}) = (\neg x_{n+2-h},..,\neg
x_{n},1,x_{2},..,x_{n+2-(h+1)}),
$$
that is,
$$
\begin{array}{r}
\neg x_{1} = 0 = \neg x_{n+2-h}, x_{2} = \neg
x_{n+2-(h-1)},..., x_{h-1} = \neg x_{n},\\
x_{h} = 1, x_{h+1} = x_{2},..., x_{n} = x_{n+2-(h+1)},
\end{array}
$$
that is,
$$
\begin{array}{r}
x_{h} = 1, x_{h+(h-1)} = \alpha _{h}x_{h},\;
x_{h+2(h-1)} = \alpha _{h+(h-1)}x_{h+(h-1)},..,\\
x_{n+2-h} = \alpha _{n+2-(h+(h-1))}x_{n+2-(h+(h-1))},\;
\neg x_{1} = 0 = \neg x_{n+2-h},
\end{array}
\eqno (2.2)
$$
where
$$
\alpha _{i } = \left \{
\begin{array}{ll}
\neg & \mbox{for $n+2-(h-1) \leq i \leq n$,}\\
I_{\mathbf{Q}} \quad \mbox{(identity)} &\mbox{for $2
\leq i \leq n+2-(h+1)$}.
\end{array}
\right.
$$
In particular, the number of $i$ such that $\alpha _{i} =
\neg $ is $h-2$.
If $n < h-1 < 2n$, then for $h' = h - n$,
$$
(0,x_{2},...,x_{n}) = (x_{n+2-h'},..,x_{n},0,\neg
x_{2},..,\neg x_{n+2-(h'+1)}),
$$
that is,
$$
\begin{array}{r}
x_{h'} = 0,\; x_{h'+(h'-1)} = \alpha '_{h'}x_{h'},\;
x_{h'+2(h'-1)} = \alpha '_{h'+(h'-1)}x_{h'+(h'-1)},..,\\
x_{n+2-h'} = \alpha '_{n+2-(h'+(h'-1))}x_{n+2-(h'+(h'-1))},\;
\neg x_{1} = 0 = x_{n'+2-h'},
\end{array}
\eqno (2.2)'
$$
where
$$
\alpha '_{i } = \left \{
\begin{array}{ll}
I_{\mathbf{Q}} & \quad \mbox{for $n+2-(h'-1)\leq i \leq
n$,}\\
\neg & \quad \mbox{for $2 \leq i \leq n+2-(h'+1)$,}
\end{array}
\right.
$$
particularly the number of $i$ such that $\alpha _{i} =
\neg $ is $n-h'$.
If $0 < h-1 < n$ is relatively prime with $n$, the
system of equations (2.2) sequentially and uniquely determines
$x_{i}$ from $x_{h} = 1$ to $\neg x_{1} = 0$ by step $h-1$ of
their subscripts. Further, the values of $x_{i}$ change
$h-1$ times as the values of $x_{i}$ are determined from
$x_{h}$ to $x_{1}$. Therefore, if $h-1$ is relatively prime
with $2n$, then $x_{i}$ is consistently determined for every $i$.
Similarly, if $0 < h'-1 < n$, $h'-1$ is relatively prime with
$n$, and $n-h'$ is even (i.e. $n < h-1 < 2n$ and $h-1$ is
relatively prime with $2n$), then (2.2)' is uniquely solved.
Consequently (2.1) is uniquely solved if $0 < h < 2n$ and $h$
is relatively prime with $2n$.\\
\textbf{Example 2.2.} Let $n = 7$ and $h-1 = 9$. The equation
$1^{-}x = (\rho 7^{-})^{9} = \bar{\neg}(\rho 7^{-})^{2}x$
for $x = (1,x_{2},..,x_{7})$ is
$$
\begin{array}{r}
\neg x_{1} = 0 = x_{6}, x_{2} = x_{7}, x_{3} = 0, x_{4}
= \neg
x_{2},\\
x_{5} = \neg x_{3}, x_{6} = \neg x_{4}, x_{7} = \neg
x_{5},
\end{array}
$$
that is,
$$
\begin{array}{r}
x_{3} = 0, x_{5} = \neg x_{3}, x_{7} = \neg x_{5},
x_{2} = x_{7},\\
x_{4} = \neg x_{2}, x_{6} = \neg x_{4}, \neg x_{1} = 0
= x_{6}.
\end{array}
$$
The solution is $x = 1001100$.\\
Now assume $0 < h-1 < 2n$ and $h-1$ is relatively prime with
$2n$, and let $x$ be the solution of (2.1). Then
\begin{eqnarray*}
((\rho n^{-})^{h-1})^{2}x &=& (\rho
n^{-})^{h-1}(1^{-}x)\\
&=& (\neg x_{n-h+2},..,\neg
x_{n},0,x_{2},..,x_{n-h+1})\\
&=& h^{-}((\rho n^{-})^{h-1}x)\\
&=& h^{-}1^{-}x.
\end{eqnarray*}
In general,
$$
\begin{array}{lll}
(\rho n^{-})^{i(h-1)}x &=& (1+(i-1)(h-1))^{-}
(1+(i-2)(h-1))^{-}...(1+(h-1))^{-}1^{-}x\\
& & \qquad \mbox{for every positive integer $i$}.
\end{array}
\eqno (2.3)
$$
Therefore, $(\rho n^{-})^{i(h-1)}x$ for
$i = 1,..,2n-1$ are all different from $x$ and
$(\rho n^{-})^{n(h-1)}x = \bar{\neg}x$. Let $f = \{ x\}$
and $F = \langle\langle f\rangle\rangle $. Then
$$
Fx = 1^{-}x = (\rho n^{-})^{h-1}x,
$$
and $F$ is one-to-one with one $2n$-cycle. Thus we obtained
the following theorem.\\
\textbf{Theorem 2.3.} Assume $0 < h-1 < 2n$ is relatively
prime with $2n$. Then there exists a one-to-one transformation
$F =
\langle\langle f\rangle\rangle $ of $\mathbf{Q}^{n}$ such that
\begin{eqnarray*}
f &=& p_{1}\cdot \alpha _{2}q_{2}\cdot ...\cdot \alpha
_{n}q_{n}, \quad \mbox{where $\alpha _{i} = I_{\mathbf{Q}}$ or
$\neg $ for every $i$,}\\
F &=& (\rho n^{-})^{h-1} \quad \mbox{on $\mathrm{Car}F$.}
\end{eqnarray*}
In this case, $f$ is uniquely determined by (2.2) or (2.2)'.
$F$ has one $2n$-cycle.\\
Any transformation $F$ described in Theorem 2.3 is
reflective. That is, there exists an isometry of order $1$ such
that
$F^{-1} = T^{-1}FT$. See [8] for the proof. This result is
consistent with the author's conjecture given in [6] that any
minimal non-compressible one-to-one threshold transformation
is reflective.\\
\textbf{Example 2.4.} For Example 2.2 we have $F =
\langle\langle f\rangle\rangle $,
$$
f = p_1\cdot \neg p_2\cdot \neg p_3\cdot p_4\cdot
p_5\cdot \neg p_6\cdot \neg p_7.
$$
The $14$-cycle of $F$ is
$$
\begin{array}{ccccccccccccc}
1001100&\rightarrow &0001100&\rightarrow &0011100&\rightarrow
&0011000&\rightarrow &0011001&\rightarrow
&0111001&\rightarrow &0110001\\
\uparrow &&&&&&&&&&&& \downarrow\\
1001110&\leftarrow &1000110&\leftarrow &1100110&\leftarrow
&1100111&\leftarrow &1100011&\leftarrow &1110011&\leftarrow
&0110011
\end{array}
$$
\section*{3. Extended representations and neighborhood
functions}
Before we go further, we need a tool for systematic
analysis of attractiveness for various transformations. This
tool is
the \emph{extended representation} of a Boolean transformation.
The \emph{Hamming distance} $d_H$ is defined on
$\mathbf{Q}^{n}$ by
$$
d_H(x,y) = |\{ i \mid x_i \ne y_i\} |.
$$
Let $d_{SH}(x,S)$ be the \emph{signed Hamming distance} between a
point $x$ and a non-empty proper subset $S$ of $\mathbf{Q}^{n}$
defined by
$$
d_{SH}(x,S) = \left \{
\begin{array}{ll}
d_{H}(x,S) & \mbox{if $x \notin S$,}\\
1-d_{H}(x,S^{c}) & \mbox{if $x \in S$.}
\end{array}
\right.
$$
\textbf{Definition 3.1.} Let $x$ be an element of
$\mathbf{Q}^{n}$ and $F = [f_{1},...,f_{n}]$. Then
the extended representation $F^{\#}$ of $F$ is a
function from $\mathbf{Q}^{n}$ to $\mathbf{Z}^{n}$ defined by
$$
(F^{\# }x)_{i} = \left \{
\begin{array}{ll}
d_{SH}(P_{\mathbf{N}\backslash i}x,
P_{\mathbf{N}\backslash i}f_{i})&
\mbox{if $x_{i} = 1$}\\
d_{SH}(P_{\mathbf{N}\backslash i}x,
P_{\mathbf{N}\backslash i}\bar{\neg}f_{i}))&
\mbox{if $x_{i} = 0$.}
\end{array}
\right.
$$
Clearly $|(F^{\# }x)_{i}| \leq n - 1$ for every $i$ for
every $x$. In general,
$x \in f_{i}$ or $x \in \bar{\neg}f_{i}$, if and only if
$(F^{\# }x)_{i} \leq 0$. That is,
$$
x_{i} \ne (Fx)_{i} \quad \mbox{iff $(F^{\# }x)_{i}
\leq 0$.}
\eqno (3.1)
$$
For example, let
$$
f = p_{1}\cdot S_{4}\{ p_{2},p_{3},p_{4},\neg
p_{6},\neg p_{7},\neg p_{8}\} ,
$$
and $F = \langle f\rangle $ be a transformation of
$\mathbf{Q}^{8}$. Let $c =
11110000$. Then $F^{\# }c = (-2,0,2,4,-2,0,2,4)$.
Therefore, $Fc = 00111100$.
For $F = [f_{1},...,f_{n}]$, let $f_{i}|1$ be the Boolean
function defined on $\mathbf{Q}^{\mathbf{N}\backslash i}$ by
$$
(f_{i}|1)(x_{1},..,x_{i-1},x_{i+1},..,x_{n}) =
f_{i}(x_{1},..,x_{i-1},1,x_{i+1},..,x_{n}).
$$
\textbf{Proposition 3.2.} If $F = [f_{1},...,f_{n}]$ then
$$
Fk^{-} = [k^{-}f_{1},..,p_{k}\cdot (\neg
\bar{\neg}(f_{k}|1)),..,k^{-}f_{n}].
\eqno (3.2)
$$
\begin{proof} Let $Fk- = [g_{1}, ..,g_{k},..,g_{n}]$. If
$i \ne k$, then
$$
\begin{array}{llll}
g_{i} &=& p_{i}\cdot \neg (Fk^{-})_{i} &\\
&=& p_{i}\cdot \neg p_{i}(Fk^{-})\\
&=& (p_{i}\cdot \neg (p_{i}F))k^{-} \\
& =& f_{i}k^{-}& \qquad \mbox{by (1.2)}\\
&=& k^{-}f_{i}.& \qquad \mbox{Polya action}\\
g_{k} &=& p_{k}\cdot \neg (Fk^{-})_{k}&\\
&=& p_{k}\cdot (\neg p_{k}(Fk^{-}))&\\
&=& p_{k}\cdot (\neg (p_{k}\cdot \neg f_{k}\vee
\bar{\neg}f_{k})k^{-}) & \qquad \mbox{by (1.3)}\\
&=& p_{k}\cdot ((\neg (p_{k}\cdot \neg f_{k})\cdot
\neg
\bar{\neg}f_{k})k^{-})&\\
&=& p_{k}\cdot (((\neg p_{k}\vee f_{k})\cdot \neg
\bar{\neg}f_{k})k^{-})&\\
&=& p_{k}\cdot ((\neg p_{k}\cdot \neg
\bar{\neg}f_{k}\vee
f_{k}\cdot \neg \bar{\neg}f_{k})k^{-}).&
\end{array}
$$
Since
$$
p_{k}\cdot (f_{k}k^{-}) = p_{k}\cdot ((p_{k}\cdot
f_{k})k^{-}) = p_{k}\cdot
(p_{k}k^{-})\cdot (f_{k}k^{-}) =p_{k}\cdot (\neg p_{k})\cdot
(f_{k}k^{-}) =
0,
$$
$$
\begin{array}{llll}
g_{k} &=& p_{k}\cdot ((\neg p_{k}\cdot \neg
\bar{\neg}f_{k})k^{-})&\\
&=& p_{k}\cdot p_{k}\cdot ((\neg
\bar{\neg}f_{k})k^{-})&\\
&=& p_{k}\cdot (\neg f_{k}\bar{\neg})k^{-})&
\qquad \mbox{Polya action}\\
& =&p_{k}\cdot (\neg f_{k}(\mathbf{N}\backslash
k)^{-})&\\
&=& p_{k}\cdot (\neg (p_{k}\cdot
(f_{k}|1))(\mathbf{N}\backslash k)^{-})&\\
&=& p_{k}\cdot ((\neg p_{k}\vee \cdot (\neg
(f_{k}|1)))(\mathbf{N}\backslash k)^{-})&\\
&=& p_{k}\cdot (\neg p_{k}\vee \neg
(f_{k}|1)\bar{\neg})&\\
& =& p_{k}\cdot (\neg (f_{k}|1)\bar{\neg})&\\
&=& p_{k}\cdot (\neg \bar{\neg}(f_{k}|1)).& \qquad
\mbox{Polya action}
\end{array}
$$
\end{proof}
\textbf{Proposition 3.3.} If $|f_{i}| \leq 2^{n-2}$ for a
threshold transformation $F = [f_{1},...,f_{n}]$, then
$$
(f_{i}|1) \subseteq \neg \bar{\neg}(f_{i}|1).
$$
\begin{proof} Assume that $(f_{k}|1) \subseteq \neg
\bar{\neg}(f_{k}|1)$, i.e.
$\neg (f_{k}|1) \supseteq \bar{\neg}(f_{k}|1)$ does not
hold. Then there exists
some x such that $P_{\mathbf{N}\backslash k}x \in
\bar{\neg}(f_{k}|1)$ i.e.
$\bar{\neg}P_{\mathbf{N}\backslash k}x \in (f_{k}|1)$ and
$P_{\mathbf{N}\backslash k}x \in (f_{k}|1)$. Then the
2-asummability
condition for $(f_{k}|1)f$ requires that for any $r \in
\mathbf{Q}^{\mathbf{N}\backslash k}$, either $r$ or
$\bar{\neg}r$ or both
belong to $(f_{k}|1)f$. Therefore
$$
|f_{k}| = |(f_{k}|1)| \geq 2^{n-2} +1.
$$
\end{proof}
The above two propositions imply the following two
inequalities for $((Fk^{-})^{\# }x)_{i}$.
$$
(F^{\# }x)_{i} - 1 \leq ((Fk^{-})^{\# }x)_{i} \leq
(F^{\# }x)_{i} + 1,
\qquad \mbox{if $i \ne k$.}
\eqno (3.3)
$$
$$
((Fk^{-})^{\# }x)_{k} \leq (F^{\# }x)_{k}, \qquad
\mbox{if $|f_{k}| \leq 2^{n-2}$},
\eqno (3.4)
$$
since $(f_{k}|1) \subseteq \neg \bar{\neg}(f_{k}|1)$.
The following example suggests our construction of
attractors.\\
\textbf{Example 3.4.} Let $2 \leq k \leq [n/2]$,
$$
f = p_{1}\cdot S_{n-k}\{ p_{2},p_{3},..,p_{n}\}
$$
and $F = \langle f\rangle $ be a transformation of
$\mathbf{Q}^{n}$. The $2$-cycle $C = (1...1,0...0)$ is the
unique attractor, and $U_{k-1}\mathbf{C}$ is the basin of
attraction.\\
Analogously, we can construct transformations having
attractors by modifying some of the one-to-one transformations.
First we give the following general definition. Let $f$ be a
function from $\mathbf{Q}^{n}$ to $\mathbf{Q}$. Then we
identify the $\epsilon$-neighborhood $U_{\epsilon }f$
of the set $f$ with the function under which the inverse
image of $1$ is the set $U_{\epsilon }f$. That is,
$$
(U_{\epsilon }f)^{-1}1 =
U_{\epsilon }f = U_{\epsilon }(f^{-1}1).
$$
The function $U_{\epsilon }f$ is called a \emph{neighborhood
function} of $f$. Then clearly
$$
U_{\epsilon }(f\vee g) = (U_{\epsilon }f)
\vee (U_{\epsilon }g).
$$
For example, if $f$ is a one-term function,
$f = q_{k1}\cdot ...\cdot q_{km}$, where $(k1,..,km)$ is a
subsequence of $(1,2,..,n)$ and $q_{ki} = p_{ki}$ or
$\neg p_{ki}$, then
$$
U_{\epsilon }f = S_{m-\epsilon }\{q_{k1},..,q_{km}\} .
$$
We consider hereafter only the case $\epsilon = 1$ for
simplicity. Let $F = [f_{1},...,f_{n}]$ be a self-dual
transformation of $\mathbf{Q}^{n}$. Then let
$G = [g_{1},...,g_{n}]$ be the transformation defined by
$$
g_{i} = p_{i}\cdot U_{1}(f_{i}|1).
\eqno (3.5)
$$
Then
$$
(G^{\# }x)_{i} = (F^{\# }x)_{i} - 1 \quad \mbox{for
every $i$ for every $x$,}
\eqno (3.6)
$$
The following proposition immediately follows from (3.6).\\
\textbf{Proposition 3.5.} A cycle of $F = [f_{1},...,f_{n}]$
is also a cycle of $G = [g_{1},...,g_{n}]$ defined by (3.5),
if and only if $(F^{\# }x)_{i} \ne 1$ for every $i$ for any
point $x$ on the cycle.
\begin{proof} $x_{i} \ne (Gx)_{i}$ iff $(G^{\# }x)_{i}
\leq 0$ by (3.1). $(G^{\# }x)_{i} = (F^{\# }x)_{i} - 1$
by (3.6). Therefore, $Gx = Fx$ iff $(F^{\# }x)_{i} \ne 1$
for every $i$.
\end{proof}
\section*{4. Construction through expansion}
In Section 2 we determined a class of one-to-one
skew-circular
threshold transformations having single cycles. Although
neighborhood
functions are useful for constructing attractors, transformations
generated by $p_{1}\cdot U_{1}(f|1)$ usually do not preserve
the cycles of original transformations generated by $f$.
In our case, let $0 < h-1 < 2n$ and $h-1$ be relatively
prime with $2n$, and let $F = \langle\langle f\rangle\rangle $,
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot
\alpha _{n}p_{n}
$$
be a transformation determined by Theorem 2.3, and let $f =
\{ c\}$ . Let $G = \langle\langle g\rangle\rangle $,
$$
g = p_{1}\cdot S_{n-2}\{ \alpha _{2}p_{2},\alpha
_{3}p_{3},...,\alpha _{n}p_{n}\} .
\eqno (4.1)
$$
Referring to (2.3), we have
\begin{eqnarray*}
(F^{\# }c)_{i} &=& 0 \quad \mbox{iff $i = 1$;}\\
(F^{\# }c)_{i} &=& 1 \quad \mbox{iff $i = h$, \quad
since $(\rho n^{-})^{h-1}c = 1^{-}c$;}\\
(F^{\# }c)_{i} &=& 2 \quad \mbox{iff $i = 1+2(h-1) = 2h-1$;}\\
(F^{\# }c)_{i} &\geq & 3 \quad \mbox{for every other $i$.}
\end{eqnarray*}
Therefore, by (3.6),
$$
\begin{array}{lll}
(G^{\# }c)_{1} = -1, & (G^{\# }c)_{h} = 0,
& (G^{\#}c)_{2h-1} = 1,\\
(G^{\# }c)_{i} \geq 2 \quad \mbox{for every other $i$.}
\end{array}
$$
Therefore,
$$
Gc = \{ 1,h\} ^{-}c = (\rho n^{-})^{2(h-1)}c,
\eqno (4.2)
$$
and $G$ has two cycles
$\mathrm{Orb}_{(\rho n-)^{2(h-1)}}\{c,(\rho n^{-})^{h-1}c\}$ .
This set of two cycles is an attractor. See [8] for the proof.
However, we are here concerned with attractors consisting of
single cycles.
One method of constructing such attractors is through
expansion, which was introduced in [4]. Corresponding to any
function in the class described in Theorem 2.3, we have its
circular expansion $E$ of $\mathbf{Q}^{2n}$.
Specifically, for any $h$ such that $0 < h-1 < 2n$ and
$h-1$ is relatively prime with $2n$, let $E = \langle e\rangle $,
$$
e = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n}\cdot \neg
p_{n+1}\cdot \neg \alpha _{2}p_{n+2}\cdot ..\cdot \neg
\alpha _{n}p_{2n,}
\eqno (4.3)
$$
corresponding to $f$ in Theorem 2.3. Then
$$
E = \rho ^{h-1} \qquad \mbox{on $\mathrm{Car}E$.}
$$
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{2n}$ defined by
$$
g = p_{1}\cdot S_{2n-2}\{ \alpha _{2}p_{2},..,\alpha
_{n}p_{n},\neg p_{n+1},\neg \alpha _{2}p_{n+2},..,
\neg \alpha _{n}p_{2n}\}.
\eqno (4.4)
$$
Let $e = \{ c\}$. Then
$$
C = \mathrm{Orb}_{\rho ^{h-1}}c
$$
is a cycle of $E$. $(E^{\# }c)_{i}$ is even for every $i$, since
$\rho ^{n}x = \bar{\neg}x$ for every $x \in \mathrm{Car}E$.
Therefore, $C$ is a cycle of $G$ by Proposition 3.5.
We have
$$
(E^{\# }c)_{i} \leq 0 \qquad \mbox{iff $i = 1$ or $n+1$,}
$$
since $Ec = \{ 1,n+1\} ^{-}c$. Since $(\rho n^-)^{h-1}c
= Ec = \{ 1,n+1 \}^{-}c$,
$$
c = \{ 1,n+1\}^{-}(\rho n^-)^{h-1}c.
$$
Therefore, referring to (2.3),
$$
(E^{\#}c)_{i} = 2 \quad \mbox{iff $i = h$ or $n+h$.}
$$
Then, by (3.6),
$$
\begin{array}{lll}
(G^{\#}c)_{1} &\leq & -1, \qquad (G^{\#}c)_{n+1}\leq -1,\\
(G^{\#}c)_{i} &\geq & 1 \quad \mbox{for every other $i$,}\\
(G^{\#}c)_{i} &=& 1 \quad \mbox{only if $i = h$ and $n+h$.}
\end{array}
\eqno (4.5)
$$
Let $k \notin \{ 1,n+1\}$ . Then, referring to (2.3),
$$
\begin{array}{llll}
((Gk^{-})^{\# }c)_{h} &=& 2 \quad \mbox{if $k \ne h$,}
&\qquad \mbox{by (3.2)}\\
((Gk^{-})^{\# }c)_{n+h} &=& 2 \quad \mbox{if $k \ne n+h$,}
&\qquad \mbox{by (3.2)}\\
((Gk^{-})^{\# }c)_{1} &\leq & 0, &\qquad \mbox{by (3.3)}\\
((Gk^{-})^{\# }c)_{n+1} &\leq & 0, &\qquad \mbox{by (3.3)}\\
((Gk^{-})^{\# }c)_{i} &\geq & 1 \quad \mbox{for every other $i
\ne k$.} & \qquad \mbox{by (3.3)}
\end{array}
$$
Therefore,
$$
G(k^{-}c) = \rho ^{h-1}c,
$$
or
$$
G(k^{-}c) = \rho ^{h-1}((\rho ^{-(h-1)}k)^{-}c).
$$
Therefore, under $G^{\sim }$,
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [c],
$$
or
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [1^{-}c].
$$
On the other hand, referring to (4.5), we have
$$
\begin{array}{llll}
((G1^{-})^{\# }c)_{1} &\leq & (G^{\# }c)_{1} = -1,
& \qquad \mbox{by (3.4)}\\
((G1^{-})^{\# }c)_{n+1} &\leq & 0.
& \qquad \mbox{by (3.3)}
\end{array}
$$
Further, since $1^{-}(\rho ^{h-1}c)$ and $c$ differ only at
their $1$st coordinates, (3.2) implies
$$
((G1^{-})^{\# }c)_{h} = 0 \quad \mbox{and} \quad
((G1^{-})^{\#}c)_{n+h} = 0.
$$
Also $((G1^{-})^{\# }c)_{i} \ge 1$ for every other $i$ by
(3.3) and (4.5). Therefore,
$$
G(1^{-}c) = \rho ^{2(h-1)}c.
$$
Therefore, if $x \in U_{1}\mathbf{C}$ then $x \in
U_{1}\mathbf{C}$ and $\omega _{G}x = \mathbf{C}$.
Therefore, $C$ is an attractor of $G$. Thus we obtained\\
\textbf{Theorem 4.1.} Let $G = \langle g\rangle $ of
$\mathbf{Q}^{2n}$ be the transformation defined by (4.4) through
the expansion (4.3) of a transformation determined by Theorem
2.3. Then $G$ has a $2n$-cycle attractor.\\
\section*{5. Through partial neighborhood functions}
Another method of constructing single-cycle attractors
is to apply partial neighborhood functions.
Let $0 < h-1 < 2n$ and $h-1$ be relatively prime with
$2n$, and let $G = \langle\langle g\rangle\rangle $ be defined
by (4.1). Then $Gc = \{ 1,h\} ^{-}c = (\rho n^{-})^{2(h-1)}c$
by (4.2), and G has two cycles $\mathrm{Orb}_{(\rho
n-)^{2(h-1)}}\{ c,(\rho n^{-})^{h-1}c\}$ . In order to preserve
the original one cycle such that $Gc = 1^{-}c$ we remove
$c$ from $(\rho n^{-})^{h-1}g$, i.e. remove
$(\rho n^{-})^{-(h-1)}c$ from $g$. Note
$$
(\rho n^{-})^{-(h-1)}c = (1-(h-1))^{-}c = (2-h)^{-}c,
$$
referring to (2.3). $(2-h)^{-}c$ is the only element $x \ne
c$ in $g$ such that $x \in [c]$, otherwise $G^{\#}$ would
have more than two non-positive coordinates.
Further we consider the set
$$
\{ 2^{-}c,,,n^{-}c\} \backslash (2-h)^{-}c
$$
and find out some elements $i^{-}c \ne j^{-}c$ in the set
such that
$$
[i^{-}c] = [j^{-}c].
$$
We have
$$
(\rho n^{-})^{k(h-1)}i^{-}c = (\rho
^{k(h-1)}i)^{-}(\rho n^{-})^{k(h-1)}c
$$
by (1.1), and
$$
(\rho n^{-})^{k(h-1)}c = (1 +
(k-1)(h-1))^{-}(1+(k-2)(h-1))^{-}...(1+(h-
1))^{-}1^{-}c
$$
by (2.3). Therefore, $(\rho n^{-})^{k(h-1)}i^{-}c = j^{-}c$
implies
$$
(i+k(h-1))^{-}(1 +(k-1)(h-1))^{-}(1+(k-2)(h-1))^{-}
...(1+(h-1))^{-}1^{-} = j^{-},
$$
so that $k = 2$, and $(i+k(h-1))^{-}h^{-}1^{-} = j^{-}$.
Therefore,
$$
(i+k(h-1)) = 1, j = h
$$
or
$$
(i+k(h-1)) = h, j = 1.
$$
Since $j > 1$, $j = h$ and $i = 3-2h$. We want to remove
one of $(3-2h)^-c$ and $h^{-}c$ from $g$, but we must
decide which to remove. If we remove $(3-2h)^-c$ then
$Gh^{-}c^{ }= G^{2}c$. If we remove $h^{-}c$, then
$G(3-2h)^-c = (3-2h)^{-}Gc$. The former is better in view
of continuity.
Thus we obtained $V = \langle\langle v\rangle\rangle $
defined by
$$
v = p_{1}\cdot S_{n-4}(\{ \alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}
\backslash\{ \alpha _{2-h}p_{2-h},\alpha _{3-2h}p_{3-2h}\} )
\cdot \alpha _{2-h}p_{2-h}\cdot \alpha _{3-2h}p_{3-2h}.
\eqno (5.1)
$$
By the above removal, if $x \in v$, then $x \notin (\rho
n^{-})^{i}v$ for every $i \ne 0 \; \mathrm{mod} \; 2n$.
We prove the attractiveness of the cycle by
decomposition of the transformations instead of calculating
the extended representations of the transformations. For
the notion of the sum of transformations, refer to Section 1.
Let
\begin{eqnarray*}
v^{(1)} &=& p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n},\\
v^{(i)} &=& v\cdot \neg \alpha _{i}p_{i} \quad
\mbox{for $i \in \mathbf{N}\backslash \{1,2-h,3-2h\}$.}
\end{eqnarray*}
Then
$$
\begin{array}{l}
v = v^{(1)} \vee . . . \vee v^{(n)},\\
v^{(i)}\cdot v^{(j)} = 0 \quad \mbox{for every $i \ne j$},
\end{array}
$$
as clear from the above process of removing $(2-h)^{-}c$
and $(3-2h)^{-}c$ from $g$.
Let $V^{(i)} = \langle\langle v^{(i)}\rangle\rangle $.
Then, if $x \in v^{(1)}$ then $x = c$ and
$$
Vc = 1^{-}c \in [v^{(1)}].
$$
If $x \in v^{(i)}$ for $i \in \mathbf{N}\backslash
\{1,2-h,3-2h\}$, then $x = i^{-}c$, and
$$
V(i^{-}c) = V^{(i)}(i^{-}c) = \{ 1,i\} ^{-}c,
$$
since $i^{-}c \in (\rho n-)^{j}v^{(i)}$ for only $j = 0 \;
\mathrm{mod}\; 2n$. Therefore,
$$
V(i^{-}c) = (\rho n^{-})^{h-1}((\rho ^{-(h-1)}i)^{-}c
\in [v^{(i-h+1)}].
$$
Therefore,
$$
V = V^{(1)} + V^{(2)} + ... + V^{(n)},
$$
and under $V^{\sim }$,
$$
[v^{(4-3h)}] \rightarrow [v^{(5-4h)}] \rightarrow
... \rightarrow [v^{(1)}] \rightarrow [v^{(1)}] = [c].
$$
Let $C = \mathrm{Orb}_{(\rho n-)^{h-1}}c$. Then $C$ is a
cycle of $V$ and $C = [c]$. Further, $1^{-}c \in [v^{(1)}]$,
$(2-h)^{-}c \in v^{(1)}$, and $(3-2h)^{-}c \in [v^{(h)}]$.
Therefore, $U_{1}C = [v]$. Therefore, $C$ is an attractor of
$V$.
Thus we obtained the following theorem.\\
\textbf{Theorem 5.1} Let $V = \langle\langle v\rangle\rangle $
of $\mathbf{Q}^{n}$ be the transformation defined by (5.1)
from a transformation determined by Theorem 2.3. Then
$G$ has a $2n$-cycle attractor.\\
\section*{6. Another class}
Assume $0 < h-1 < n$ is relatively prime with odd $n$. Then
there exists a one-to-one reflective transformation $F = \langle
f\rangle $
of $\mathbf{Q}^{n}$ such that
\begin{eqnarray*}
f &=& p_{1}\cdot q_{2}\cdot ...\cdot q_{n}, \qquad
\mbox{where $q_{i} = p_{i}$ or $\neg p_{i}$ for every $i$,}\\
F &=& \bar{\neg}\rho ^{h-1} \qquad \mbox{on $\mathrm{Car}F$}.
\end{eqnarray*}
$F$ has one $2n$-cycle. Starting with the class of such
transformations, we can construct a class of threshold
transformations having single-cycle attractors in the same
methods developed here. See [8] for details.\\
\section*{Appendix: Finite-state dynamical system (FSDS)}
Let $X$ be a finite-element metric space with an
integer-valued distance $d$. The distance between a point $x$
and a non-empty subset $S$ of $X$ is defined by
$$
d(x,S) = \mathrm{min}\{ d(x,y) \mid y \in S\} ,
$$
where $\mathrm{min}$ denotes the minimum element. If $S$ is
a non-empty subset of $X$ then the $\epsilon $-neighborhood of
$S$, $U_{\epsilon }S$ for a positive integer $\epsilon $ is
defined by
$$
U_{\epsilon }S = \{ x \mid d(x,S) \leq \epsilon \} .
$$
Let $\varphi $ be a function from $X\times \mathbf{Z}_{+}$
to $X$. For each $t \in \mathbf{Z}_{+}$ (set of non-negative
integers) a transformation $\varphi _{t}: X \rightarrow X$ is
defined by $\varphi _{t }x = \varphi (x,t)$ for every $x \in X$.
If $\varphi _{t}$ satisfies
\begin{enumerate}
\item $\varphi _{s}\circ \varphi _{t} = \varphi _{s+t}$
for all $s,t \in \mathbf{Z}_{+}$,
\item $\varphi _{0} = I_{X}$ (the identity transformation of
$X$),
\end{enumerate}
then $\varphi $ is called a \emph{finite-state dynamical
system} (FSDS) on the \emph{statespace} $X$. If $F$ is a
transformation of $X$, $F$ defines a function $\varphi : X\times
\mathbf{Z}_{+}\rightarrow X$ by
$$
\varphi (x,t) = F^{t}x, x \in X, t \in \mathbf{Z}_{+}.
$$
Then $\varphi $ is an FSDS on $X$ such that $\varphi _{t} =
F^{t}$ and called the FSDS \emph{generated by} $F$.
A sequence $V = (v_{0},v_{1},...)$ in $X$ is a function
$V: \mathbf{Z}_{+} \rightarrow X$. The image of $V$, that is,
the set $\{ x\mid x = v_{i} \;\mbox{for some $i$} \}$ is denoted
by $\mathbf{V}$. If $\Psi $ is a set of sequences in $X$, then
$\mathrm{Im}\Psi$, the image of $\Psi $, is
$\bigcup_{V\in \Psi}\mathbf{V}$, that is, the union of the
images of the sequences belonging to $\Psi $. A sequence
$A = (a_{0},a_{1},...)$ is called \emph{cyclic}, if there exists
some $k$ such that $a_{i} = a_{j}$ for every $i$ and $j$ such
that $i = j \; \mathrm{mod} \; k$ and $a_{i} \ne a_{j}$ for every
$i$ and $j$ such that $i \ne j \; \mathrm{mod} \; k$.
The sequence $(x,Fx,F^{2}x,...)$ is called the \emph{orbit}
starting at $x$ and denoted by $\mathrm{Orb}_{F}x$.
A cyclic orbit is identified with an element of $\mathrm{CY}(F)$,
ie. the set of all cycles in the graph of $F$ (a loop is a
$1$-cycle). That is one cyclic orbit obtained from another by
shifting the starting point is regarded as the same. For a
subset
$S$ of $X$, $\mathrm{Orb}_{F}S$ is the set of all orbits
$\mathrm{Orb}_{F}x$ such that $x \in S$. The limit set
$\omega _{F}x$ $x$ is defined by
\begin{eqnarray*}
\omega _{F}x = \{ y &\mid & \mbox{For any}\; k \in
\mathbf{\mathbf{Z}}_+, \; \mbox{there exists}\\
&& \qquad \mbox{some}\; t > k \; \mbox{such that}\; y =
F^{t}x \}.
\end{eqnarray*}
$\omega _{F}S$ is the union of all limit sets $\omega _{F}x$
such that $x \in S$. \\
\textbf{Definition} A subset $\Phi $ of $\mathrm{CY}(F)$ is
called \emph{attractive} or an \emph{attractor} in the FSDS
generated by $F$, if there exists some $\epsilon $-neighborhood
$U_{\epsilon }(\mathrm{Im}\Phi )$ satisfying
\begin{enumerate}
\item $F(U_{\epsilon }(\mathrm{Im}\Phi )) \subseteq
U_{\epsilon}(\mathrm{Im}\Phi )$;
\item $\omega _{F}(U_{\epsilon }(\mathrm{Im}\Phi ))
= \mathrm{Im}\Phi $.
\end{enumerate}
In particular, if $\Phi $ consists of one cycle, the cycle
is called attractive. The \emph{basin of attraction} for
an attractor $\Phi $ is the set of all points $x$ such that
$F^{ k }x \in \mathrm{Im}\Phi $ for some $k$.
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\end{document}