1. Turing Machines A more powerful computation model than a PDA ?
[Section 9.1]
Turing Machines
A more powerful computation model than a PDA ?

2. Turing Machines Some history: introduced by Alan Turing in 1936
[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.

3. Real-world problems vs. languages
[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 ?

4. Turing Machines cont. Verbal explanation:
[Section 9.1]
Turing Machines cont.
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).

5. [Section 9.1]
Turing Machines cont.
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.

6. [Section 9.1]
Turing Machines cont.
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

7. Turing Machines cont. Let T = (Q,,,q0,) be a TM.
[Section 9.1]
Turing Machines cont.
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.

8. [Section 9.1]
Turing Machines cont.
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.

9. Turing Machines cont. Example: Give a TM accepting { akbkck | k ¸ 0 }.
[Section 9.1]
Turing Machines cont.
Example: Give a TM accepting { akbkck | k ¸ 0 }.

10. Turing Machines and functions
[Section 9.2]
Turing Machines and functions
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

11. A stronger machine than a TM ?
[Section 9.4]
A stronger machine than a TM ?
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.

12. A stronger machine than a TM ?
[Section 9.4]
A stronger machine than a TM ?
Example : How to simulate a 2-way infinite tape using a regular TM ?

13. Nondeterministic TM (NTM)
[Section 9.5]
Nondeterministic TM (NTM)
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).