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Chapter 3 Section 1 contains a quality description of the halting problem, a proof by contradiction, and a helpful graphic representation of the Halting Problem. Taylor Booth, Sequential Machines and Automata Theory, Wiley, New York, 1967. Cf. Chapter 9, Turing Machines. Difficult book, meant for electrical engineers and technical specialists.
This proof proceeds by reductio ad absurdum: we assume that there is a non-trivial property that is decided by an algorithm, and then show that it follows that we can decide the halting problem, which is not possible, and therefore a contradiction. Let us now assume that P(a) is an algorithm that decides some non-trivial property of F a.
If proof by contradiction were intuitionistically valid, we would obtain an algorithm for deciding whether an arbitrary Turing machine M halts, thereby violating the (intuitionistically valid) proof of non-solvability of the Halting problem.
In which case, if P 1 (S) is the set of one-element subsets of S and f is a proposed bijection from P 1 (S) to P(S), one is able to use proof by contradiction to prove that |P 1 (S)| < |P(S)|. The proof follows by the fact that if f were indeed a map onto P(S), then we could find r in S, such that f({r}) coincides with the modified diagonal set ...
Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. [9] This method of proof has also been presented by Shoenfield (1967); Charlesworth (1981); and Hopcroft & Ullman (1979). [10] Franzén (2005) explains how ...
Turing's proof is a proof by Alan Turing, first published in November 1936 [1] ... His first theorem is most relevant to the halting problem, ...
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One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.