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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


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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.