Turing Machines  Decidability

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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 (&lt;) 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

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
• Design a Turing Machine that will add two integers.
• Input: 0110#11100
• Output: 100010
Turing Machine as Calculator
• Design a Turing Machine that will multiply two integers.
• Input: 0110#11100
• Output: 10101000
Turing Machine as Calculator
• Design a Turing Machine that will find the square root of an integer.
• Input: 11100
• Output: 101
Configurations
• The current “state” of a Turing machine is fully described by specifying
• The current state,
• The current tape position,
• The current tape contents.
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
• 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
• 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
Example
• \$01q5#\$01
• \$0q61#\$01
• \$q601#\$01
• q6\$01#\$01
• \$q001#\$01
• \$\$q11#\$01
• etc.
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
• 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
• 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
• It is possible that a Turing Machine neither accepts nor rejects an input w.
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
• 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.
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.