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Cf. Chapter 2, "Algorithms and Turing Machines". An over-complicated presentation (see Davis's paper for a better model), but a thorough presentation of Turing machines and the halting problem, and Church's Lambda Calculus. Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed ...
Alan Turing proved in 1936 that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist. Hence, the halting problem is undecidable for Turing machines.
The halting problem (determining whether a Turing machine halts on a given input) and the mortality problem (determining whether it halts for every starting configuration). Determining whether a Turing machine is a busy beaver champion (i.e., is the longest-running among halting Turing machines with the same number of states and symbols).
The termination analysis is even more difficult than the Halting problem: the termination analysis in the model of Turing machines as the model of programs implementing computable functions would have the goal of deciding whether a given Turing machine is a total Turing machine, and this problem is at level of the arithmetical hierarchy and ...
The classic example is the halting problem: create an algorithm that takes as input a program in some Turing-complete language and some data to be fed to that program, and determines whether the program, operating on the input, will eventually stop or will continue forever.
A machine with an oracle for the halting problem can determine whether particular Turing machines will halt on particular inputs, but it cannot determine, in general, whether machines equivalent to itself will halt. This creates a hierarchy of machines, each with a more powerful halting oracle and an even harder halting problem.
If we have an algorithm that decides a non-trivial property, we can construct a Turing machine that decides the halting problem. For the formal proof, algorithms are presumed to define partial functions over strings and are themselves represented by strings. The partial function computed by the algorithm represented by a string a is denoted F a.
No halting probability is computable. The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω cannot be computed. The algorithm proceeds as follows.