Turing Machine

Theory of Computation
Topic: Definition of Turing Machine

INTRODUCING TURING MACHINES
Introduced by Alan Turing in 1936.
A simple mathematical model of a computer.
Models the computing capability of a computer.

DEFINITION
A Turing machine (TM) is a finite-state machine with an infinite tape
and a tape head that can read or write one tape cell and move left or
right.
It normally accepts the input string, or completes its computation, by
entering a final or accepting state.
Tape is use for input and working storage.

Representation of Turing Machine
•
Turing Machine is represented by-
• M=(Q,Σ, Γ,δ,q0,B,F) ,
• Where
• Q is the finite state of states
• Σ a set of τ not including B, is the set of input symbols,
• τ is the finite state of allowable tape symbols,
• δ is the next move function, a mapping from Q × Γ to
• Q × Γ ×{L,R}
• Q0 in Q is the start state,
• B a symbol of Γ is the blank,
• F is the set of final states.

THE TURING MACHINE MODELTHE TURING MACHINE MODEL

Transition function
One move (denoted by |---) in a TM does the following:
δ(q , X) = (p ,Y ,R/L)
• q is the current state
• X is the current tape symbol pointed by tape head
• State changes from q to p

Turing machine as language acceptors
A Turing machine halts when it no longer has available moves.
 If it halts in a final state, it accepts its input, otherwise it rejects its
input.
For language accepted by M ,we define
L(M)={ w ε ∑+
: q0w |– x1qfx2 for some qf ε F , x1 ,x2ε Γ *
}

Turing machine as transducers
• To use a Turing machine as a transducer, treat the entire nonblank
portion of the initial tape as input
• Treat the entire nonblank portion of the tape when the machine halts
as output.
A Turing machine defines a function y = f (x) for strings x, y ε ∑*
if
q0x |*– qf y
• A function index is “Turing computable” if there exists a Turing
machine that can perform the above task.

ID of a TMID of a TM
• Instantaneous Description or ID :
 X1 X2…Xi-1 q Xi Xi+1 …Xn
Means:
 q is the current state
 Tape head is pointing to Xi
 X1X2…Xi-1XiXi+1… Xn are the current tape symbols
 δ (q , Xi ) = (p ,Y , R ) is same as:δ (q , Xi ) = (p ,Y , R ) is same as:
X1 X2…Xi-1 q Xi Xi+1 …Xn|---- X1 X2…Xi-1 YY pp Xi+1…Xn
 δδ (q Xi) = (p Y L) same as:(q Xi) = (p Y L) same as:
X1 X2…Xi-1 q Xi Xi+1 …Xn|---- X1 X2…ppXi-1YY Xi+1 …Xn

VARIATIONS OF TURING MACHINESVARIATIONS OF TURING MACHINES
Multitape Turing Machines
Non deterministic Turing machines
Multihead Turing Machines
Off-line Turing machines
Multidimensional Turing machines

MULTITAPE TURING MACHINES
A Turing Machine with several tapes
 Every Tape’s have their Controlled own R/W Head
 For N- tape TM M=(Q,Σ, Γ,δ,q0,B,F)
• we define δ : Q X ΓN
Q X ΓN
X { L , R} N

NON DETERMINISTIC TURINGNON DETERMINISTIC TURING
MACHINESMACHINES
It is similar to DTM except that for any input symbol and current
state it has a number of choices
A string is accepted by a NDTM if there is a sequence of moves that
leads to a final state
The transaction function δ : Q X Γ 2 QXΓ X{L,R}

MULTIHEAD TURING MACHINEMULTIHEAD TURING MACHINE
Multihead TM has a number of heads instead of one.
 Each head indepently read/ write symbols and move left / right or
keep stationery.

OFF- LINE TURING MACHINEOFF- LINE TURING MACHINE
An Offline Turing Machine has two tapes
1. One tape is read-only and contains the input
2. The other is read-write and is initially blank.

MULTIDIMENSIONAL TURINGMULTIDIMENSIONAL TURING
MACHINEMACHINE
A Multidimensional TM has a multidimensional tape.
• For example, a two-dimensional Turing machine would read and
write on an infinite plane divided into squares, like a checkerboard.
 For a two- Dimensional Turing Machine transaction function define
as:
• δ : Q X Γ Q X Γ X { L , R,U,D}

Applications of TM
• Check Decidability If TM cannot solve a problem in countable time
then there could not be any algorithm which could solve that
problem (That is the problem is undecidable).For a decision problem
if its TM halt in countable time for all finite length inputs then we can
say that the problem could be solved by an algorithm in countable
time.
• Classify Problem TM helps to classify decidable problems into classes
of Polynomial Hierarchy. Suppose we found that the problem is
decidable. Then out target become how efficiently we can solve it.
The efficiency been calculated in number of steps, extra space used ,
length of the code/size of the FSM.
• Design and Implement Algorithm for Practical Machines TM helps to
propagate idea of algorithm in other practical machines. After the
successful check of 1,2 criteria we can use our practical
devices/computers to design and implement algorithm.

Turing Machine

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Turing Machine