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Monday, August 16, 2010

Non-undecidability of singular questions

The many memorious among youse may remember a tiny problem set I offered as a challenge aeons ago, which included, among others, a question about the decidability of a couple of representative singular questions (singular in the sense of "Is John homicidal?" vs. "Which people named 'John' are homicidal?"). The answer given in my hearing by Prof. Burton Dreben when asked by a student was echoed almost verbatim by Shonk in answer to my challenge.

The same answer may be found in Marvin Minsky, Computation, Finite and Infinite Machines (Prentice-Hall, 1967) regarding the halting problem for one Turing machine; but, interestingly, Minsky goes further: "For a given (T0,t0) it could well be that no one will ever find out whether it halts or not. It could conceivably be that there is no way to find out, in some obscure sense. But it could not be that someone could prove that there is no way to find out...Suppose that it had allegedly been proven that there is no way to find out if (T0,t0) halts or not. Suppose also that a great experimental project were launched, involving the construction and operation of (T0,t0)...Now there are two cases in fact--either (T0,t0) does eventually halt, or it never halts. In the first case, the experimental approach will ultimately succeed, and the question will be settled. This would certainly contradict any proof that the question could never be settled! Hence it must happen that (T0,t0) never stops. In other words, the proof that there is no way to find out can be used to prove that (T0,t0) never halts. Hence we would be able to find out. Contradiction!"

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