"Adding machines have played an important role in dynamical systems, and in the theory of groups acting on trees. "[1]

# Structure

Alphabet ${\displaystyle X}$ is a set consisting of two symbols so it is called binary alphabet:

${\displaystyle X=\left\{0,1\right\}}$

Word c is a sequence of symbols ( string). It can be dsiplayed in two ways :[2]

• a little-endian (least-significant-bit-first) : ${\displaystyle c=c_{0}c_{1}..c_{n-1}}$
• a big-endian : ${\displaystyle c=c_{n-1}..c_{1}c_{0}}$

When :

• ${\displaystyle c_{0}}$ is on the right side it is easier to treat it as a binary number
• ${\displaystyle c_{0}}$ is on the leftt side it is easier for machine

Space of words ${\displaystyle X^{\omega }}$ denote the set of all infinite strings over the alphabet.

${\displaystyle X^{\omega }=\left\{0,1\right\}^{\omega }}$

The strings (${\displaystyle \epsilon }$, 0, 1, 00, 01, 10, 11, 000, etc.) would all be in this space.

${\displaystyle \epsilon }$ represents the empty string

## Action

The binary representation of decimal 149, with the lsb highlighted. The msb in an 8-bit binary number represents a value of 128 decimal. The lsb represents a value of 1.

Here is only one transformation ( action ) a ona a input word c :

${\displaystyle c=c_{0}w}$

where :

• ${\displaystyle c_{0}\,}$ is a lsb,[3] first symbol ( here at the beginnig, but for binary notationit will be at the end of sequence)
• ${\displaystyle w\,}$ is a rest of the word ${\displaystyle c\,}$

Transformation is defined by 2 recursion formulae :

• if the first symbol ${\displaystyle c_{0}\,}$ is zero then we change it to one and the rest of the word remains unchanged
• if it is one :
• we change it to zero
• carry 1 to next column.[4]
• aply action to the next column (symbol) until last column

Formally:

${\displaystyle (c)^{a}={\begin{cases}(0w)^{a}=1w\\(1w)^{a}=0w^{a}\end{cases}}}$

or in other notation :

${\displaystyle a(0w)=1w\,}$
${\displaystyle a(1w)=0a(w)\,}$

"This transformation is known as the adding machine, or odometer, since it describes the process of adding one to a binary integer." [5]

${\displaystyle (c)^{a}=c+1\,}$

More explicitly :[6]

${\displaystyle a(x_{1}x_{2}...x_{n})=y_{1}y_{2}...y_{n}\,}$

if and only if

${\displaystyle (x_{1}+2*x_{2}+4*x_{3}+...+x_{n}*2^{n-1})+1=y_{1}+y_{2}*2+y_{3}*4+...+y_{n}*2^{n-1}(mod2^{n})}$

Both input and output are binary numbers least-significant bit first.

### Examples

Word c is a sequence of n symbols ( from 0 to n-1) representing binary integer :

${\displaystyle c=c_{n-1}..c_{1}c_{0}=\sum _{i=0}^{n-1}c_{i}2^{i}}$

where ${\displaystyle c_{n}}$ is an element of binary alphabet X ={0,1}

without carry because lsb ${\displaystyle c_{0}=0}$:

  0
+ 1
---
1


Here lsb ${\displaystyle c_{0}=1}$ then c_0+1>1 so one has to carry 1 to next column

  1
+ 1
---
10

  10
+ 01
-----
11



Carry in second column ( from right to left)

  011
+ 001
-----
100

  100
+ 001
-----
101

  101
+ 001
-----
110

  0111
+ 0001
-----
1000


## Nucleus

The nucleus of group G is :[7]

${\displaystyle N=\{1,a,a^{-1}\}\,}$

where a is an action of group G

## Tree

A point of the binary tree c = c0 c1 c2 . . .where corresponds to the diadic integer ${\displaystyle \quad \xi \,}$

${\displaystyle \xi =\sum {c_{i}2^{i}|i\leq 0}}$

which translates to the n-ary addition

${\displaystyle \xi =\xi +1\,}$

# Visual representation

## Diagram

Here alphabet ${\displaystyle X}$ is a set consisting of two symbols so it is called binary alphabet:

${\displaystyle X=\left\{0,1\right\}}$

Labels shows pairs of symbols : input/output

There are 2 vertices ( nodes, states of machine) : 1 and 0.

Vertices correspond to the states .

"The states of this machine will represent the value that is "carried" to the next bit position.

Initially 1 is "carried".

The carry is "propagated" as long as the input bits are 1.

When an input bit of 0 is encountered, the carry is "absorbed" and 1 is output.

After that point, the input is just replicated." [8]

Table [9]

# GAP/FR

BinaryAddingGroup  ( global variable )


This function constructs the state-closed group generated by the adding machine on [1,2]. This group is isomorphic to the Integers.

BinaryAddingMachine        ( global variable )


This function constructs the adding machine on the alphabet [1,2]. This machine has a trivial state 1, and a non-trivial state 2. It implements the operation "add 1 with carry" on sequences.

BinaryAddingElement        ( global variable )


This function constructs the Mealy element on the adding machine, with initial state 2.

gap>LoadPackage("fr");