Properties
- The relations of many-one reducibility and 1 reducibility are transitive and reflexive and thus induce a preorder on the powerset of the natural numbers.
- if and only if
- A set is many-one reducible to the halting problem if and only if it is recursively enumerable. This says that with regards to many-one reducibility, the halting problem is the most complicated of all computer programs. Thus the halting problem is many-one complete.
- The specialized halting problem for an individual Turing machine T (i.e., the set of inputs for which T eventually halts) is many-one complete iff T is a universal Turing machine. Emil Post showed that there exist recursively enumerable sets that are neither decidable nor m-complete, and hence that there exist nonuniversal Turing machines whose individual halting problems are nevertheless undecidable.
Read more about this topic: Many-one Reduction
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—John Locke (1632–1704)
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