Turing Machines – Decidability

Lecture 25Section 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 C1 yields 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 machine M 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 M decides 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.