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Turing Machines – DecidabilityPowerPoint Presentation

Turing Machines – Decidability

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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 qQ.
- 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.

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