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A reduction is the transformation of one problem into another problem. A example of using a reduction would be to be to show if a problem P is undecidable. This would be achieved by transforming or performing a reduction of a decision problem $P$ into an undecidable problem. If this can be achieved then we have shown that this problem P is undecidable.

6 votes
Accepted

$\mathsf{TC^0}$-completeness and reductions

For question #1: If there is a complete problem for uniform TC$^0$ under uniform AC$^0$ m-reductions, then the counting hierarchy collapses. I don't know if this qualifies as "bad". … For question #3: MAJORITY and division are both complete for TC$^0$ under AC$^0$-Turing reductions. …
Eric Allender's user avatar
3 votes

Are the notions of #P-completeness via Turing reductions and polynomial many-one counting re...

One can construct artifical examples of functions in $FP^{\#P}$ that are complete under poly-time Turing reductions but seem unlikely to be complete under reductions that make only one query. …
Eric Allender's user avatar
9 votes
Accepted

Is Circuit Minimization $P$-hard under logspace reductions?

Not only is MCSP not known to be hard for P under logspace reductions; it's not even known to be hard for NL under logspace reductions. … If you don't like non-uniform reductions, then MKTP is also hard for $NISZK_L$ under (uniform) probabilistic $NC^0$ reductions. But no hardness under uniform deterministic reductions is known. …
Eric Allender's user avatar