The collapse of ≤_m and ≤_T would entail that every Turing-complete set is also many-one complete, yet Post's construction of simple sets demonstrates Turing-complete sets lacking many-one completeness.

?Rate how convincing each reason is below to see the overall strength.

1 reason for

1 reason against

Reasons For

1 perspective

Reason for

1.Post's simple sets are Turing-complete but lack many-one completeness, establishing that ≤_T and ≤_m are genuinely distinct notions.
?
Think about whether this reason is strong or weak
2.If ≤_m and ≤_T collapsed, all sets in the same Turing degree would share many-one degree, contradicting Post's construction.
?
Think about whether this reason is strong or weak
3.The existence of counterexamples (simple sets) is logically sufficient to refute claims about universal equivalence of reduction notions.
?
Think about whether this reason is strong or weak

Reasons Against

1 perspective

Reason against

1.The claim conflates 'collapse of relations' with 'every instance follows the rule'—reducibilities can differ without contradicting Post's results.
?
Think about whether this reason is strong or weak
2.Simple sets being Turing-complete doesn't entail they lack many-one completeness; Post proved they're not m-complete, which is consistent.
?
Think about whether this reason is strong or weak
3.The argument assumes any collapse would force uniform reduction types, but local distinctions can persist even if reducibilities asymptotically align.
?
Think about whether this reason is strong or weak

Next step

Based on where you are in your exploration

Strongest counterpoint

Explore the most compelling reason on the other side.

Key Terms

Many-one complete(computability theory): A description of a problem that is both as hard as other problems in its difficulty class and can be used to solve all problems in that class through straightforward transformations.
Post's construction(mathematical logic and computability theory): A famous mathematical technique developed by logician Emil Post that creates examples of problems with specific computational properties, particularly problems that are very hard but not the hardest possible.
Simple sets(computability theory): A special type of mathematical set created by Post's method that is extremely difficult to compute but lacks certain properties that the absolute hardest problems have.
Turing-complete(computability theory): A description of a computational problem or system that is as hard as the hardest problems that computers can theoretically solve—essentially the maximum level of computational difficulty.
≤_T (Turing reducibility)(mathematical logic and computability theory): A symbol representing a more flexible way to compare problem difficulty: problem A is Turing reducible to problem B if you could solve A by using a solution to B as a tool, possibly multiple times.
≤_m (many-one reducibility)(mathematical logic and computability theory): A symbol representing a way to compare the difficulty of computational problems: problem A is many-one reducible to problem B if you can transform any instance of A into an instance of B using a single, straightforward conversion.

Connections

2 topics

Truth & Knowledge1 linked Modality & Possibility1 linked

If ≤_m and ≤_T collapsed, all sets in the same Turing degree would share many-on...Many-one reducibility implies Turing reducibility (A ≤_m B implies A ≤_T B)

Details

Type: claim
Perspectives: 2 (1 for, 1 against)
Edits: 1 edit