Turing machines: an Introduction
Contents
What is a Turing machine?
What is a busy beaver?
What are busy beavers used for?
Last update: July 2008
What is a Turing machine?

A Turing machine has a twoway infinite tape, made of cells.
In each cell, there is a symbol.
There is a finite number of symbols, denoted by 0, 1, ...
The symbol 0 is the blank symbol.
Initially, the Turing machine holds a finite input.
This input is a string of symbols, for example: 101001.
The other cells of the tape hold blank symbols.
In the kind of machine we consider, the input can contain
the blank symbol 0.
. . . 
0 
0 
1 
0 
1 
0 
0 
1 
0 
0 
. . . 

A Turing machine has a tape head.
With this tape head, the machine can read and write on the tape.
The tape head moves one cell left or right at each step.
These directions are denoted by L and R.
In the kind of machine we consider, the tape head cannot keep
still.
Initially, the tape head scans the leftmost symbol of the input.

A Turing machine has a finite number of states.
The states are denoted by A, B, C, ...
In addition to these states, there is a special state,
the halting state, denoted by H.
Initially, the state is A, so A is called the initial state.
Below, you can see the initial configuration of a Turing machine
on the input 101001:
. . . 
0 
0 
A 1 
0 
1 
0 
0 
1 
0 
0 
. . . 
The position of the tape head is indicated by writing the
current state on the scanned cell.

A Turing machine has a next move function.
The next move function says that, when the machine is in
state q and scans symbol a on a cell, then the
machine writes a symbol b instead of a, moves
one cell in the direction left or right, and enters state
p.
We suppose that, when the machine stops, it writes a 1,
moves right, and enters state H.
The next move function is given by a table.
For example, we give below the table of a machine
from Lin and Rado, with 3 states A, B, C, and 2 symbols
0, 1, and the computation of this machine on an initially blank tape.
 0  1 
A  1RB  1RH 
B  0RC  1RB 
C  1LC  1LA 

initially 
. . . 
0 
0 
0 
A0 
0 
0 
0 
0 
0 
0 
. . .

step 1 
. . . 
0 
0 
0 
1 
B0 
0 
0 
0 
0 
0 
. . .

step 2 
. . . 
0 
0 
0 
1 
0 
C0 
0 
0 
0 
0 
. . .

step 3 
. . . 
0 
0 
0 
1 
C0 
1 
0 
0 
0 
0 
. . .

step 4 
. . . 
0 
0 
0 
C1 
1 
1 
0 
0 
0 
0 
. . .

step 5 
. . . 
0 
0 
A0 
1 
1 
1 
0 
0 
0 
0 
. . .

step 6 
. . . 
0 
0 
1 
B1 
1 
1 
0 
0 
0 
0 
. . .

step 7 
. . . 
0 
0 
1 
1 
B1 
1 
0 
0 
0 
0 
. . .

step 8 
. . . 
0 
0 
1 
1 
1 
B1 
0 
0 
0 
0 
. . .

step 9 
. . . 
0 
0 
1 
1 
1 
1 
B0 
0 
0 
0 
. . .

step 10 
. . . 
0 
0 
1 
1 
1 
1 
0 
C0 
0 
0 
. . .

step 11 
. . . 
0 
0 
1 
1 
1 
1 
C0 
1 
0 
0 
. . .

step 12 
. . . 
0 
0 
1 
1 
1 
C1 
1 
1 
0 
0 
. . .

step 13 
. . . 
0 
0 
1 
1 
A1 
1 
1 
1 
0 
0 
. . .

step 14 
. . . 
0 
0 
1 
1 
1 
H1 
1 
1 
0 
0 
. . .


The machine stops in 14 steps, leaving 6 symbols 1 on the tape.

Formal mathematical definition.
Let S = {0,1,...} be the finite set of symbols,
and let Q = {A,B,...} be the finite set of states.
Then the next move function is a mapping
d : Q × S
> S × {L,R} × Q
or {(1,R,H)}.
For example, if d(A,0) = (1,R,B), then it means
that, when the machine is in state A scanning
symbol 0 on a cell, then it replaces 0 by 1 on
this cell, moves one cell right, and enters state B.
What is a busy beaver?

Consider Turing machines with fixed numbers of states and
symbols.
If there are k states and n symbols, then the number
of possible next move functions, or possible tables, is
(2kn+1)^{kn}.
Thus, there are (2kn+1)^{kn}
Turing machines with k states and n symbols.
Each of them can be launched on a blank tape, that is
a tape with symbols 0 in all cells.
Then some of them never stop, and the other ones
eventually stop.
Those which stop are called busy beavers.

Busy beavers compete in two competitions:

to take the most time to stop,

to leave the most nonblank symbols on the tape when stopping.
We denote by s(M) the time taken by busy beaver M,
and by sigma(M) the number of nonblank symbols left
on the tape by M.
We denote by S(k,n) the time taken by the busy
beaver with k states and n symbols which takes the most time to stop:
S(k,n) = max {s(M) : M is a busy beaver with k states and n symbols}
We denote by Sigma(k,n) the number of nonblank symbols
left on the tape by the busy beaver with k states and
n symbols which leaves the most nonblank symbols on the tape when
stopping:
Sigma(k,n) = max {sigma(M) : M is a busy beaver with k states and n symbols}

Example: the Lin and Rado's Turing machine given
above, with 3 states and 2 symbols, takes 14 steps to stop,
and leaves 6 symbols 1 on the tape when stopping.
So here: s(M) = 14 and sigma(M) = 6.
This machine is the winner for the most nonblank
symbols left on the tape, so we have Sigma(3,2) = 6.
It is not the winner for the most time: another machine
takes 21 steps to stop, and we have S(3,2) = 21.
What are busy beavers used for?

In 1936, Turing defined Turing machines as a universal
model of computation on natural numbers.
This means that all computable functions you can imagine
can be computed by a Turing machine.
Other authors (Church, Kleene, Post, Markov) defined
other models of computation, but these models compute
the same functions as Turing machines.
So it is currently assumed that "computable" means
"computable by a Turing machine". This assumption
is called Church's Thesis, or, more accurately,
ChurchTuring Thesis.

Then, how to find a noncomputable function?
Such a function can be defined from a model of computation by an
abstract device called diagonalization method. But
usually this method does not give an explicit function.
Here enter busy beavers.

In 1962, Rado defined busy beaver competition. He
considered Turing machines with 2 symbols, and defined the
functions S(n,2) and Sigma(n,2). He proved that these functions
are not computable. Moreover, these functions
grow faster than any computable function (that is,
for any computable function f, there is an integer N
such that, for all n > N, we have S(n,2) > f(n)).
The definitions of functions S(n,2) and Sigma(n,2) are
explicit enough to allow their computations for small n.
The promise is to get quickly large numbers. So winning
busy beavers are simply definable machines with
complex behavior. That's why they are interesting.
Now, you know enough about Turing machines and busy beavers
to enjoy the results of the
busy beaver competition.
Link to Pascal Michel's home page