A Turing Machine (TM) is a mathematically idealised machine having:
If Initial state __ and reading __ then print __, move L/R and go into state __
Example of TM: UN+1 (see Penrose (1989) or Luger (1994))
A TM is a deterministic system: initial state + input gives an output with probability 1 for TM; between 0&1 for a probabilistic automaton.
Universal Turing Machine: Each instruction of an arbitrary Turing machine can be coded into a binary numeral. The TM can thus be represented by a long numeral which is then used as the initial input of another TM. This UTM now acts on the remainder of the input just like the original TM would have done. How is this done?
UTM thus takes the coded version of other Turing machines as input and then emulates their behaviour.
Turing's paper was a response to Hilbert's 10th problem: Is there an algorithm for solution of a given class of mathematical problems?
Algorithm = mechanical procedure = TM
For the procedure to be algorithmic the TM should stop. But Turing and Church showed that there is no algorithmic procedure to find out whether a given 3#3 acting on an input m will stop.
Church's thesis: The set of functions that are Turing computable is identical to the set of functions that any human or machine could compute. Turing himself identified some functions which are not Turing-computable but such functions have not been shown to be computable in any other sense either.
Apparently random behaviour can be simulated by Turing machines with transition probabilities between 0 and 1 (determinate but not predictable). Non-deterministic specifications are used to model creative processes.