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Turing Machines – Decidability. Lecture 25 Section 3.1 Fri, Oct 19, 2007. Turing Machine as Calculator. Design a Turing Machine that will compare (<) two integers. Input: 0110#11100 Output: 1 (true) Input: 11100#0110 Output: 0 (false). Turing Machine as Calculator.

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Turing machines decidability

Turing Machines – Decidability

Lecture 25

Section 3.1

Fri, Oct 19, 2007


Turing machine as calculator
Turing Machine as Calculator

  • Design a Turing Machine that will compare (<) two integers.

    • Input: 0110#11100

    • Output: 1 (true)

    • Input: 11100#0110

    • Output: 0 (false)


Turing machine as calculator1
Turing Machine as Calculator

  • Design a Turing Machine that will add two integers.

    • Input: 0110#11100

    • Output: 100010


Turing machine as calculator2
Turing Machine as Calculator

  • Design a Turing Machine that will multiply two integers.

    • Input: 0110#11100

    • Output: 10101000


Turing machine as calculator3
Turing Machine as Calculator

  • Design a Turing Machine that will find the square root of an integer.

    • Input: 11100

    • Output: 101


Configurations
Configurations

  • The current “state” of a Turing machine is fully described by specifying

    • The current state,

    • The current tape position,

    • The current tape contents.


Configurations1
Configurations

  • This can be summarized in a triple uqv, called a configuration, where u, v * and qQ.

  • The interpretation is

    • The current state is q.

    • The current tape content is uv.

    • The current tape position is at the first symbol in v.


Computations
Computations

  • We say that a configuration C1yields a configuration C2 if there is a transition that takes the Turing machine from C1 to C2.

  • A computation is a sequence of configurations C1, …, Cn, where Ci yields Ci + 1 for i = 1, …, n – 1.


Example
Example

  • Our machine that accepts {w#w} will perform the following computation on input 101#101:

    • q0101#101

    • $q301#101

    • $0q31#101

    • $01q3#101

    • $01#q4101


Example1
Example

  • $01q5#$01

  • $0q61#$01

  • $q601#$01

  • q6$01#$01

  • $q001#$01

  • $$q11#$01

  • etc.


Accepting and rejecting configurations
Accepting and Rejecting Configurations

  • The start configuration on input w is q0w.

  • An accepting configuration is one where the state is qaccept.

  • A rejecting configuration is one where the state is qreject.


Accepting input
Accepting Input

  • A Turing Machine accepts input w if there is a computation C1, …, Cn, where

    • C1 is the start configuration on w.

    • Cn is an accepting configuration.


Rejecting input
Rejecting Input

  • A Turing Machine rejects input w if there is a computation C1, …, Cn, where

    • C1 is the start configuration on w.

    • Cn is a rejecting configuration.


The third possibility
The Third Possibility

  • It is possible that a Turing Machine neither accepts nor rejects an input w.


Turing recognizable languages
Turing-Recognizable Languages

  • The language of a Turing machineM is the set of input strings that are accepted by M.

    L(M) = {w | M accepts w}.

  • A language is Turing-recognizable if it is accepted by some Turing machine.


Turing decidable languages
Turing-Decidable Languages

  • A Turing Machine is a decider if it halts on all inputs.

  • A Turing machine Mdecides a language L if M accepts every string in L and rejects every string not in L.

  • A language is Turing-decidable if it is decided by some Turing machine.


Example2
Example

  • The language {w#w | w *} is a Turing-decidable language.

  • Every Turing-decidable language is Turing-recognizable, but not every Turing-recognizable language is Turing-decidable.


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