Turing machines are abstract computing devices that can be used to model algorithmic processes and assess computability. A Turing machine operates on an infinite tape divided into discrete cells, each of which can contain a symbol. The machine has a read/write head and a finite set of internal states, and follows a set of rules to manipulate symbols on the tape and transition between states. This allows Turing machines to systematically process inputs to provide outputs. Alan Turing showed that a universal Turing machine could simulate any other Turing machine, laying the foundation for the modern computer. Turing machines are useful tools for investigating questions around computability and the limits of mechanical computation.

1. Turing Machines

2. John 18:38 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 Euclid’s ‘Elements’ work
• Definitions
• Postulates
• Axioms
• Agreed Method Each step in the proof is an
application of one of the above.

4. Hilbert’s 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? – Turing’s answer -
mechanical – algorithmic - the Turing
machine

5. Turing’s Concept
• A machine
– With a finite set of states
– Unrestricted input and output
– Unlimited storage space
– Simplest possible operations
Read/write head
Infinite tape
1 0 1 0 0 1 1 0 1 1 0 0

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 The Goldbach conjecture
will establish the truth of
Every even number
mathematical proposition ‘p’? greater than two, is the
The Entscheidungsproblem sum of two prime
numbers.
…. is equivalent to the question
• Is there a possible Turing We could make a Turing
machine which will calculate machine to test each
all values of ‘p’ and stop if ‘p’ successive number and find
out if it is the sum of two
is false … and if so, can we primes. If it isn’t then the
know in advance if it will stop. machine halts – having
The Halting Problem disproved the conjecture