Turing Machine

M. S. Ramaiah School of Advanced Studies 1
CSN2505 AFM Presentation
Turing Machine
Anshuman Biswal
PT 2012 Batch, Reg. No.: CJB0412001
M. Sc. (Engg.) in Computer Science and Networking
Module Leader: Dr. Hariharan Ramasangu
Module Name: Applied Formal Methods
Module Code : CSN2505

Marking
Head Maximum Score
Technical Content 10
Grasp and Understanding 10
Delivery – Technical and
General Aspects
10
Handling Questions 10
Total 40

Presentation Outline
• Introduction and Historical background
• Historical Background-TM in WW-II
• A Thinking machine-Successor Program Example
• Turing Machine
• Turing Machine Formal Definition and Description
• Turing Machine Transition Functions
• Turing Machine Instantaneous Description and Move
• Turing Machine Example
• Turing Machine Variants
• Conclusion
• Referenses

Introduction and Historical Background
• The desire to capture the complexities of mathematical reasoning and
computation in a simple and powerful manner has been a major theme
of 20th Century mathematics.
• The 1930’s saw two of these great negative results in the foundations
of mathematics, the first of which was Godel’s very unexpected proof
of 1931 (Godel’s Incompleteness Theorem).
• 1936 saw the independent development of three influential models of
computation :lambda calculus, recursive functions and Turing
machines.
• Alonzo Church’s lambda calculus and Steven Kleene’s recursive
functions were arguably more elegant in this respect, but it was the
mechanical action of Turing’s machines that most agreed with
intuitions about how people go about computing mathematical
functions.

• 1936-Alan Turing a British cryptologist invented a mechanical device,
which from that day forth was called the “Turing machine”.
• Based on Turing’s prior thoughts and ideas, his invention was able to
compute every given mathematical expression .
• It only consists of three main components: A tape (e.g. paper) that can
hold data and serves as some kind of memory,a head that can read data
from or write data to the tape and finally an application, which controls
the whole process.
• The Turing machine addresses two major problems of theoretical
computer science, namely the decision problem and the halting
problem.
Introduction and Historical Background

• In 1936 Alan Turing was member of the Royal Army
and worked in the British codebreaking centre, located at
Bletchley Park in South East England.
• His major task during World War II was breaking the
Enigma, a device that was used by the Germans, in order
to encode military messages. At least two different
versions of the Enigma were commonly used at wartime:
On the one hand the so-called “Wehrmachts-Enigma”
and on the other hand a naval edition, which was primary
used by German warships and submarines
Historical Background-TM in WW-II

• Alan Turing made use of terms and phrases that were
frequently used in German military so-called “cribs”.
• He searched certain patterns and tried to understand the
encoding algorithm. However a single message could be
interpreted in more than 150 quadrillion ways (>
150,000,000,000,000 or 150 * 10¹²).
• British scientists made use of technical support. They used a
cluster of four Turing machines which could successfully
decode a secret message within several hours and thus was a
huge advantage for the Royal British Army in their battle
against Nazi Germany. This machine is called the “Turing
Bombe”.
Historical Background-TM in WW-II

A Thinking Machine-Successor Program
Imagine a super-organized, obsessive-compulsive human computer.
The computer wants to avoid mistakes so everything written down is
completely specified one letter/number at a time. The computer
follows a finite set of rules which are referred to every time another
symbol is written down. Rules are such that at any given time, only
one rule is active so no ambiguity can arise. Each rule activates
another rule depending on what letter/number is currently read, EG:
Sample Rules:
If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read •, write 1, HALT!
Let’s see how they are carried out on a piece of paper that contains
the reverse binary representation of 47:

1 1 1 1 0 1

0 1 1 1 0 1

0 0 1 1 0 1

0 0 0 1 0 1

0 0 0 0 0 1

0 0 0 0 1 1

So the successor’s output on 111101 was
000011 which is the reverse binary
representation of 48.
Similarly, the successor of 127 should be 128:

1 1 1 1 1 1 1

0 1 1 1 1 1 1

0 0 1 1 1 1 1

0 0 0 1 1 1 1

0 0 0 0 1 1 1

0 0 0 0 0 1 1

0 0 0 0 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

It was hard for the ancients to believe that any
algorithm could be carried out on such a device.
For us, it’s much easier to believe, especially if
we’ve programmed in assembly!
However, ancients did finally believe Turing when
Church’s lambda-calculus paradigm (on which
lisp programming is based) proved equivalent!
A Thinking Machine

Turing Machines
A Turing Machine (TM) is a device with a finite
amount of read-only “hard” memory (states), and
an unbounded amount of read/write tape-memory.
There is no separate input. Rather, the input is
assumed to reside on the tape at the time when the
TM starts running.

TM Formal Definition
• DEF: A Turing machine (TM) consists of a
the 8-tuple: M = (Q, , Γ, δ, q0, B, qa, qr)
G is the tape alphabet which necessarily contains the
blank symbol •, as well as the input alphabet S. d is
as follows:
Therefore given a non-halt state p, and a tape symbol
x, d(p,x) = (q,y,D) means that TM goes into state q,
replaces x by y, and the tape head moves in
direction D.
}RL,{}),{-(:δ rejacc GG QQ qq

Turing Machine Description
• Q = finite states of the control , set of states
•  = finite set of input symbols, which is a subset of Γ below,
input alphabet
• Γ = finite set of tape symbols, tape alphabet
• δ = partial transition function. δ(q,X) are a state and tape
symbol X.
– The output is the triple, (p, Y, D)
– Where p = next state, Y = new symbol written on the tape, D =
direction to move the tape head
• q0= start state for finite control, initial state
• B = blank symbol. This symbol is in Γ but not in .
• qaccept = set of final or accepting states of Q.
• qreject = set of rejecting states of Q

TM Transition Functions
The transition function for Turing machines is
given by
}RL,{}),{-(:δ rejacc GG QQ qq
This means: When the machine is in a given state (Q) and reads a given
symbol (Γ) from the tape, it replaces the symbol on the tape with some
other symbol (Γ), goes to some other state (Q), and moves the tape head
one square left (L) or right (R)

TM Instantaneous description and Move
• An instantaneous description or configuration of a Turing machine
requires (1) the state the Turing machine is in, (2) the contents of the
tape, and (3) the position of the tape head on the tape. This can be
summarized in a string of the form
xi...xjqmxk...xl
where the x's are the symbols on the tape, qm is the current state, and the
tape head is on the square containing xk (the symbol immediately
following qm).
• A move of a Turing machine can therefore be represented as a pair of
instaneous descriptions, separated by the symbol " ├”.
For example, if (q5, b) = (q8, c, R)then a possible move might be
abbabq5babb ├ abbabcq8abb

TM Example
• Make a TM that recognizes the language L = { w#w | w  (0,1)* }. That
is, we have a language separated by a # symbol with the same string on
both sides.
• Here is a strategy we can employ to create the Turing machine:
– read the first symbol, then cross it out, and then, remember what that
symbol was, move until it reaches the #
– Then check to see if the symbols match, and if they do cross it off
and go back to the symbol after the one we crossed out,
– and then continue in this same manner, except going to the first non-
crossed off symbol in the transversals.
– Finally we need to make sure that we accept if this is all that the
input consists of.
• Here is a formal description of a Turing machine which does this
1. Q = {q1; q2; : : : q8; qa; qr}
2.  = {0,1,#} and G = {0, 1,x,#,_}
3. The start state is q1. The accept state is qa. The reject state is qr.

TM Example
And now we need to specify the transition function. We do this
with a state diagram.

• Now there are some points about this transition function which we
should point out. First of all where is the reject state qr? Well we have
obmitted it. Indeed we have followed a convention where every
allowed transition which is not labeled is a transition to qr.
•All rules should be of the form s → a, d where s, a ϵ G and d ϵ {L,R}
But some of these don't have an a, i.e. they are of the form
s → d. This is shorthand for the fact that the same symbol which is
on the tape is being written onto the tape, i.e. that symbol is not
changed. Finally we often write multiple elements which lead to the
transition separated via commas.
TM Example

TM Example- Instantaneous Description
• Sometimes it is useful to describe what a TM does in terms of its ID
(instantaneous description.)
• The ID shows all non-blank cells in the tape, pointer to the cell the head is
over with the name of the current state
– use the turnstile symbol ├ to denote the move.
– As before, to denote zero or many moves, we can use ├*.
• Let's run through how this machine acts on an input. Lets do 010#010.

TM Example
Inputs i.e Symbols on Tape is 010#010 • q1010#010 ├ xq210#010
• ├ x1q20#010├ x10q2#010
• ├ x10#q4010 ├ x10q6#x10
• ├ x1q70#x10 ├ xq710#x10
• ├ q7x10#x10 ├ xq110#x10
• ├ xxq30#x10 ├ xx0q3#x10
• ├ xx0#q5x10 ├ xx0#xq510
• ├ xx0#q6xx0 ├ xx0q6#xx0
• ├ xxq70#xx0 ├ xq7x0#xx0
• ├ xxq10#xx0 ├ xxxq2#xx0
• ├ xxx#q4xx0 ├ xxx#xq4x0
• ├ xxx#xq6xx ├ xxx#q6xxx
• ├xxxq6#xxx├xxq7x#xxx
• ├xxxq1#xxx├xxx#q8xxx
• ├ xxx#xq8xx ├ xxx#xxq8x
• ├ xxx#xxxq8 ├ xxx#xxx_qa

Turing Machine Variants
• There are a number of other types of Turing machines in addition to
the one we have seen.
• Turing Machines with Two Dimensional Tapes : It has one finite
control, one read-write head and one two dimensional tape. The tape
has the top end and the left end but extends indefinitely to the right and
down.
•
Here the numbers indicate the correspondence
of squares in the two tapes: square i of the two
dimensional tape is mapped to square i of the
one dimensional tape. h and v are symbols
which are not in the tape alphabet and they
are used to mark the left and the top end
of the tape, respectively.

• Turing Machines with Multiple Tapes : This is a kind of Turing machines
that have one finite control and more than one tapes each with its own read-
write head. It is denoted by a 5-tuple < Q , , q0, d >
• Its transition function is a partial function
d : Q X ( G  {} )n -> ( Q  { h } ) X ( G  {} )n X{R ,L,S }n .
• A configuration for this kind of Turing machine must show the current state
the machine is in and the state of each tape.
• Turing Machines with Multiple Heads : This is a kind of Turing machines
that have one finite control and one tape but more than one read-write heads.
In each state only one of the heads is allowed to read and write. It is denoted
by a 5-tuple < Q , , G, q0, d >. The transition function is a partial function
d : Q X { H1 , H2 ... , Hn } X( G  {} ) -> (Q  { h } )X( G  {} )X {
R , L , S } ,
where H1 , H2 ... , Hn denote the tape heads.

• Turing Machines with Infinite Tape : This is a kind of Turing
machines that have one finite control and one tape which extends
infinitely in both directions.
• Nondeterministic Turing Machines :A nondeterministic Turing
machine is a Turing machine which, at any state it is in and for the tape
symbol it is reading, can take any action selecting from a set of
specified actions rather than taking one definite predetermined action.
Even in the same situation it may take different actions at different
times. Here an action means the combination of writing a symbol on
the tape, moving the tape head and going to a next state.
• Formally a nondeterministic Turing machine is a Turing machine
whose transition function takes values that are subsets of
(Q  { h } ) X (G  {}X{ R , L , S }

Conclusion
• It is often said that Turing machines, are as powerful as
real machines, and are able to execute any operation that a
real program can.
• According to Church-Turing thesis any possible
calculation can be performed by an algorithm, running on a
computer that provides sufficient time and storage space.
Admittedly it is just pure theory, when we say that a
Turing machine can solve every given mathematical
problem. In practice this idea is very limited like they do
not model the concurrency well i.e. There is a bound on the
size of the integer

References
[1] H,Willkommen. (June 25 2006) “The Turing Machine
Artikel ” [Online]. Available
from:http://www.saschaseidel.de/html/programmierung/pr
ogrammierungindex.html (Accessed: 08 June 2013).
[2] Wikipedia. “Turing machine”. [Online].Available from:
http://en.wikipedia.org/wiki/Turing_machine (Accessed:18
June 2013)

Turing Machine

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Turing Machine