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:

Turing Machine as Calculator Design a Turing Machine that will multiply two integers. Input: 0110#11100 Output:

Turing Machine as Calculator Design a Turing Machine that will find the square root of an integer. Input: 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 C 1 yields a configuration C 2 if there is a transition that takes the Turing machine from C 1 to C 2. A computation is a sequence of configurations C 1, …, C n, where C i yields C i + 1 for i = 1, …, n – 1.

Example Our machine that accepts {w#w} will perform the following computation on input 101#101: q 0 101#101 $q 3 01#101 $0q 3 1#101 $01q 3 #101 $01#q 4 101

Example $01q 5 #$01 $0q 6 1#$01 $q 6 01#$01 q 6 $01#$01 $q 0 01#$01 $$q 1 1#$01 etc.

Accepting and Rejecting Configurations The start configuration on input w is q 0 w. An accepting configuration is one where the state is q accept. A rejecting configuration is one where the state is q reject.

Accepting Input A Turing Machine accepts input w if there is a computation C 1, …, C n, where C 1 is the start configuration on w. C n is an accepting configuration.

Rejecting Input A Turing Machine rejects input w if there is a computation C 1, …, C n, where C 1 is the start configuration on w. C n 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.