Turing Machines

A more powerful computation model than a PDA ?

[Section 9.1]

Turing Machines

Some history:

- introduced by Alan Turing in 1936

- models a “human computer” (human writes/rewrites symbols on a sheet of paper, the human’s state of mind changes based on what s/he has seen)

- a reasonable model for real computers

Church-Turing Thesis:

For any problem L (given by a language) there exists an algorithm iff there exists a Turing machine which terminates on every input.

[Section 9.1]

Real-world problems vs. languages

Example : the airline problem - given are airports and available flights, is it possible to get from a place A to a place B ?

[Section 9.1]

[Section 9.1]

Verbal explanation:

- The tape is infinite to the right and it initially contains the input string (the rest of the tape contains blanks - M).

- The TM starts in an initial state, reading the first symbol on the tape.

- The head can move left, right, or stay at its current position.

- The TM has two special states, ha (accept) and hr (reject).

- If the head moves to the left of the first symbol, this automatically means the change of state to hr (reject).

- The transition is specified by a state and a tape symbol (to which the head points). It returns a new state, new tape symbol (to rewrite the original), and a head-move (L/R/S).

Turing Machines cont.

Turing Machines cont.[Section 9.1]

Example : As a warm-up, give a Turing machine for a*b*c*

Simplified transition diagram : we do not have to draw transitions leading to hr.

[Section 9.1]

A Turing machine (TM) is a 5-tuple (Q,,,q0,) where

- Q is a finite set of states not containing ha, hr (the two halting states)

- is a finite alphabet (input symbols)

- is a finite alphabet (tape symbols) such that µ and does not contain M (the blank symbol)

- qo 2 Q is the initial state

- : ___________ _______________________ is a partial function defining the transitions

Turing Machines cont.

[Section 9.1]

Let T = (Q,,,q0,) be a TM.

A configuration of T is ________________.

The initial configuration is ________________.

We use `T to say that T can get from one configuration to another configuration using a single transition.

We use `T* to say that T can get from one configuration to

another configuration using a sequence of transitions.

Turing Machines cont.

[Section 9.1]

Let T = (Q,,,q0,) be a TM and x 2 *. We say that x is accepted by T if

_____________________ .

The language accepted by T, denoted L(T), is the set of all strings in * that are accepted by T.

A string x can be rejected in two ways : either the computation of T on x ends in the state hr, or the computation of T on x gets into an infinite loop.

A language accepted by a TM is called recursively enumerable. A language for which there is a TM which never goes to an infinite loop is called recursive.

Turing Machines cont.

[Section 9.1]

Example: Give a TM accepting { akbkck | k ¸ 0 }.

Turing Machines cont.

[Section 9.2]

Let T = (Q,,,q0,) be a TM and let f be a total function from * to *. We say that T computes f if for every x 2 *,

(q0, Mx) `T* (ha, Mf(x)).

Example : give a TM that computes the function f(1n) = 12n

Turing Machines and functions

[Section 9.4]

Possible attempts to make Turing machines stronger :

- 2-way infinite tape

- several heads, several tapes

- random access (the head can jump to any position)

- nondeterminism

- etc.

Note: All of the above changes can be simulated by a TM.

A stronger machine than a TM ?

[Section 9.4]

Example : How to simulate a 2-way infinite tape using a regular TM ?

A stronger machine than a TM ?

[Section 9.5]

The definition is the same as Turing machines, except that the transition function goes from Q £ ( [ {M}) to subsets of (Q [ {ha,hr}) £ ( [ {M}) £ {R,L,S}.

Thm : Let T1 be an NTM. Then there exist a TM T2 such that L(T1)=L(T2).

Nondeterministic TM (NTM)