John 18:38 Pilate said to him, What is truth? And when hehad said this, he went out again to the Jews, and said tothem, I find in him no fault at all.
How Euclid’s ‘Elements’ work• Definitions• Postulates• Axioms• Agreed Method Each step in the proof is an application of one of the above.
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
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 tape1 0 1 0 0 1 1 0 1 1 0 0
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
Turing machine is a 5-tupleCurrent Input Output Move Next 0 0 1 - 1 0 1 - R 0 1 1 - L 1 1 0 - R Stop
Current Input Output Move Next0 0 0 R 00 1 0 R 11 0 1 L 101 1 1 R 110 0 0 R 1110 1 0 R 10011 0 1 Stop 011 1 1 R 11100 0 1 L 101100 1 1 R 100101 0 1 L 10101 1 1 L 101
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.
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