Title: Turing Machines
1Turing Machines
2John 1838 Pilate said to him, What is truth?
And when he had said this, he went out again to
the Jews, and said to them, I find in him no
fault at all.
3How Euclids Elements work
- Definitions
- Postulates
- Axioms
- Agreed Method
Each step in the proof is an application of one
of the above.
4Hilberts question (1900)
- Is there, or could there possibly be, a definite
method that could decide whether a particular
mathematical expression is true?
- What exactly do we mean by a definite method?
Turings answer - mechanical algorithmic -
the Turing machine
5Turings Concept
- A machine
- With a finite set of states
- Unrestricted input and output
- Unlimited storage space
- Simplest possible operations
Read/write head
Infinite tape
6Basic operations of the machine
- Read
- Read the symbol on the current square
- Change the inner state of the machine
- Write
- Change the symbol on the current square
- Change the inner state of the machine
- Move
- Tape can move any distance left or right
7Turing-Kara
8Turing machine is a 5-tuple
Current Input Output Move Next
0 0 1 - 1
0 1 - R 0
1 1 - L 1
1 0 - R Stop
9Current Input Output Move Next
0 0 0 R 0
0 1 0 R 1
1 0 1 L 10
1 1 1 R 1
10 0 0 R 11
10 1 0 R 100
11 0 1 Stop 0
11 1 1 R 11
100 0 1 L 101
100 1 1 R 100
101 0 1 L 10
101 1 1 L 101
10Universal Turing Machines
- Each quintuple could be coded into a single
number, each number coded on tape. - We therefore have the idea of a Turing machine
which reads its own instructions. - Such a machine could mimic any possible Turing
machine.
11Turing Machines and Computability
- Is there an algorithm which will establish the
truth of mathematical proposition p? - . is equivalent to the question
- Is there a possible Turing machine which will
calculate all values of p and stop if p is
false and if so, can we know in advance if it
will stop.
The Goldbach conjecture Every even number
greater than two, is the sum of two prime numbers.
The Entscheidungsproblem
We could make a Turing machine to test each
successive number and find out if it is the sum
of two primes. If it isnt then the machine
halts having disproved the conjecture
The Halting Problem