Representation As A Set
The conventional representation of decision problems is the set of objects possessing the property in question. The halting set
- K := { (i, x) | program i will eventually halt if run with input x}
represents the halting problem.
This set is recursively enumerable, which means there is a computable function that lists all of the pairs (i, x) it contains. However, the complement of this set is not recursively enumerable.
There are many equivalent formulations of the halting problem; any set whose Turing degree equals that of the halting problem is such a formulation. Examples of such sets include:
- { i | program i eventually halts when run with input 0 }
- { i | there is an input x such that program i eventually halts when run with input x }.
Read more about this topic: Halting Problem
Famous quotes containing the word set:
“Was it an intellectual consequence of this ‘rebirth,’ of this new dignity and rigor, that, at about the same time, his sense of beauty was observed to undergo an almost excessive resurgence, that his style took on the noble purity, simplicity and symmetry that were to set upon all his subsequent works that so evident and evidently intentional stamp of the classical master.”
—Thomas Mann (1875–1955)