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The universal halting problem, also known (in recursion theory) as totality, is the problem of determining whether a given computer program will halt for every input (the name totality comes from the equivalent question of whether the computed function is total). This problem is not only undecidable, as the halting problem is, but highly ...
In computability theory, the halting problem is a decision problem which can be stated as follows: . Given the description of an arbitrary program and a finite input, decide whether the program finishes running or will run forever.
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).
We can now show that H decides the halting problem: Assume that the algorithm represented by a halts on input i. In this case F t = F b and, because P(b) = "yes" and the output of P(x) depends only on F x, it follows that P(t) = "yes" and, therefore H(a, i) = "yes". Assume that the algorithm represented by a does not halt on input i.
Because many outstanding problems in number theory, such as Goldbach's conjecture, are equivalent to solving the halting problem for special programs (which would basically search for counter-examples and halt if one is found), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems. But as the halting ...
To show that a decision problem P is undecidable we must find a reduction from a decision problem which is already known to be undecidable to P. That reduction function must be a computable function. In particular, we often show that a problem P is undecidable by showing that the halting problem reduces to P.
Turing completeness, as just defined above, corresponds only partially to Turing completeness in the sense of computational universality. Specifically, a Turing machine is a universal Turing machine if its halting problem (i.e., the set of inputs for which it eventually halts) is many-one complete for the set of recursively enumerable sets.
"The Halting Problem of Alan Turing - A Most Merry and Illustrated Explanation." The Halting Problem of Alan Turing - A Most Merry and Illustrated Explanation. N.p., n.d. Web. 10 April 2017. Turner, Raymond, and Nicola Angius. "The Philosophy of Computer Science." Stanford Encyclopedia of Philosophy. Stanford University, 20 August 2013.