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Turing Machines

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Title: Turing Machines


1
Turing Machines
2
John 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.
3
How Euclids Elements work
  • Definitions
  • Postulates
  • Axioms
  • Agreed Method

Each step in the proof is an application of one
of the above.
4
Hilberts 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

5
Turings Concept
  • A machine
  • With a finite set of states
  • Unrestricted input and output
  • Unlimited storage space
  • Simplest possible operations

Read/write head
Infinite tape
6
Basic 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

7
Turing-Kara
8
Turing 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
9
Current 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
10
Universal 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.

11
Turing 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
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