Introduction to Turing Machines

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Introduction toTuring Machines
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The Turing Machine
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A TM consists of an infinite length tape, on
which input is provided as a finite sequence of
symbols. A head reads the input tape. The TM
starts at start state s0. On reading an input
symbol it optionally replaces it with another
symbol, changes its internal state and moves
one cell to the right or left.
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The Turing Machine
A TM is defined as TM ltS, T, s0, d, Hgt
where, S is a set of TM states
T is a set of tape symbols s0
is the start state H ? S is a
set of halting states d S x T ?S x T x L,R
is the transition function
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Simple TM Examples
Turing Machine U1 Given a string of 1s on a
tape (followed by an infinite number of 0s), add
one more 1 at the end of the string.
111100000000. ? 1111100000000.
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Simple TM Examples
TM U1 d(s0, 1) -- (s0, 1, R) d(s0, 0)
-- (h, 1, STOP)
s0111100000.. ? 1s011100000.. ?
11s01100000.. ? 111s0100000.. ?
1111s000000.. ?
11111h0000.. STOP
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Exercice
state symbol ?(state, symbol)
S0 b (halt, a, stop)
S0 a (S1 , a, right )
S1 b (halt, b, stop)
S1 a (S0 , a, right )
Input aaaabb What is the output for this
input?
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Solution

• s0 aaaabb
• s1 aaaabb
• s0 aaaabb
• s1 aaaabb
• s1 aaaabb
• halt aaaaab

• Input a finite sequence of a symbol, followed
  by an infinite sequence of b.
• Describe what the output this machine generates.

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Turings Thesis
Any mathematical problem solving that can be
described by a mechanical procedure (algorithm)
can be modeled by a Turing machine. All
computers today perform only mechanical problem
solving. They are no more expressive than a
Turing machine.
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Turings Thesis

• Turings thesis is not a theorem there is no
  proof for the thesis.
• The theorem may be refuted by showing at least
  one task that is performed by a digital computer
  which cannot be performed by a Turing machine.
• Many contentions have been made to this end.
  However, till date there have not been any
  conclusive evidence to refute Turings thesis.

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Conclusions

• Turing machines are a minimal extension over PDAs
  which provide greater expressiveness.
• TMs are at a level that is much below the
  assembly language of any typical microprocessor.
• So in the practical world, TMs are more useful in
  what they cannot do rather than in what they can.