Fractals/Mathematics/group/Binary adding machine

< Fractals‎ | Mathematics/group

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

Contents

StructureEdit

Alphabet   is a set consisting of two symbols so it is called binary alphabet:

 


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

  • a little-endian (least-significant-bit-first) :  
  • a big-endian :  

When :

  •   is on the right side it is easier to treat it as a binary number
  •   is on the leftt side it is easier for machine


Space of words   denote the set of all infinite strings over the alphabet.

 


The strings ( , 0, 1, 00, 01, 10, 11, 000, etc.) would all be in this space.

  represents the empty string

ActionEdit

 
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 :

 

where :

  •   is a lsb[3] , first symbol ( here at the beginnig, but for binary notationit will be at the end of sequence)
  •   is a rest of the word c


Transformation is defined by 2 recursion formulae :

  • if the first symbol   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:

 

or in other notation :

 
 

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

 


More explicitly :[6]

 

if and only if

 

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


ExamplesEdit

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

 


where   is an element of binary alphabet X ={0,1}

without carry because lsb  :

  0 
+ 1 
---
  1

Here lsb   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

NucleusEdit

The nucleus of group G is :[7]

 

where a is an action of group G

TreeEdit

A point of the binary tree c = c0 c1 c2 . . .where corresponds to the diadic integer  

 


which translates to the n-ary addition

 

Visual representationEdit

DiagramEdit

 


Here alphabet   is a set consisting of two symbols so it is called binary alphabet:

 

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]

TableEdit

Table [9]

GAP/FREdit

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.

These functions are respectively the same as AddingGroup(2), AddingMachine(2) and AddingElement(2).


gap>LoadPackage("fr"); 
gap> Draw(NucleusMachine(BinaryAddingGroup));
gap>Draw(BinaryAddingMachine);

ReferencesEdit