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# turing machines decidability - PowerPoint PPT Presentation

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

Lecture 25

Section 3.1

Fri, Oct 19, 2007

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

• Input: 0110#11100

• Output: 1 (true)

• Input: 11100#0110

• Output: 0 (false)

• Design a Turing Machine that will add two integers.

• Input: 0110#11100

• Output: 100010

• Design a Turing Machine that will multiply two integers.

• Input: 0110#11100

• Output: 10101000

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

• Input: 11100

• Output: 101

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

• The current state,

• The current tape position,

• The current tape contents.

• 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.

• 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.

• 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

• \$01q5#\$01

• \$0q61#\$01

• \$q601#\$01

• q6\$01#\$01

• \$q001#\$01

• \$\$q11#\$01

• etc.

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.