1. Lecture 5 Turing Machines
Jan Maluszynski, IDA, 2007
ida.liu.se
Jan Maluszynski - HT 2007

2. Motivation/Example of TM Formal Definition Multitape Turing Machines
Outline
Motivation/Example of TM
Formal Definition
Multitape Turing Machines
Nondeterministic Turing Machines
Enumerators
Turing-computable functions
Church-Turing thesis
Other models
Jan Maluszynski - HT 2007

3. One head scanning a cell, moving right/left
Basic Turing Machine
Half-infinite tape
One head scanning a cell, moving right/left
Content of the scanned cell can be changed while head is moving
Finite number of states
Moves controlled by the content of the scanned cell and actual state
Jan Maluszynski - HT 2007

4. Example TM
Jan Maluszynski - HT 2007

5. (Q,, , , q0 , qaccept , qreject ) Q set of states
Turing Machine
(Q,, , , q0 , qaccept , qreject )
Q set of states
 input alphabet not containing  (the blank symbol)
 tape alphabet including ; 
: QQ{L,R} transition function
q0 the start state
qaccept the accepting state
qreject the rejecting state Qaccept  Qreject
No transitions out from accept and reject states
Jan Maluszynski - HT 2007

6. Example of a two-tape TM
Multitape TM
Example of a two-tape TM
Jan Maluszynski - HT 2007

7. Example of a non-deterministic TM
Jan Maluszynski - HT 2007

8. Variants of TMs are equivalent
Every multitape TM has an equivalent single-tape TM
Every non-deterministic TM has an equivalent deterministic TM
Every TM with doubly infinite tape has an equivalent TM with half-infinite tape.
A language is Turing-recognizable iff some TM enumerator enumerates it.
Jan Maluszynski - HT 2007

9. L is Turing decidable if L=L(M) for some TM that halts on every input
Uses/kinds of TMs
Language recognizer:
L is Turing recognizable if L=L(M) for some TM: for some strings M may loop
Language decider:
L is Turing decidable if L=L(M) for some TM that halts on every input
Language enumerator :
Starts with empty tape enumerates all strings of L
Function computing device:
Transforms a given input string to an output string
Jan Maluszynski - HT 2007

10. Intuitive notion of algorithm -> TM algorithm
Church-Turing thesis
A function is effectively computable iff there is a Turing machine that computes it.
Intuitive notion of algorithm -> TM algorithm
Other formal models of computations:
Lambda-calculus
Partial recursive functions
Random-Access machines
…..
have been proved equivalent to the TM model.
Jan Maluszynski - HT 2007