with 
Although the set K is recursively enumerable, the halting problem is not solvable by a computable function.
There are many equivalent formulations of the Halting problem; any set whose Turing degree is the same as that of the Halting problem can be thought of as such a formulation. Examples of such sets include:
Importance and consequences
The historical importance of the halting problem lies in the fact that it was one of the first problems to be proved undecidable. (Turing's proof went to press in May 1936, whereas Church's proof of the undecidability of a problem in the lambda calculus had already been published in April 1936.) Subsequently, many other such problems have been described; the typical method of proving a problem to be undecidable is with the technique of reduction. To do this, the computer scientist shows that if a solution to the new problem was found, it could be used to decide an undecidable problem (by transforming instances of the undecidable problem into instances of the new problem). Since we already know that no method can decide the old problem, no method can decide the new problem either.
One such consequence of the halting problem's undecidability is that there cannot be a general algorithm that decides whether a given statement about natural numbers is true or not. The reason for this is that the proposition that states that a certain algorithm will halt given a certain input can be converted into an equivalent statement about natural numbers. If we had an algorithm that could solve every statement about natural numbers, it could certainly solve this one; but that would determine whether the original program halts, which is impossible, since the halting problem is undecidable.
Read more at Wikipedia.org
Home
Accessories
8mm, Hi8
JVC
