Between my textbook and various online sources (namely wikipedia), I'm very confused... can somebody clear up which words are synonymous and which mean different things?Many-to-one reductionMapping reductionTuring reductionCook reductionKarp reductionPolynomial-time many-to-one reductionPolynomial time turing reductionI've also seen others, but I can't recall them currently.
What is the difference between these terms?
complexity theory;computability;terminology;reductions
Let $A,B\subseteq \Sigma^*$ be languages.Many-to-one: A (computable) function $f:\Sigma^*\to \Sigma^*$  such that $\forall x\in \Sigma^*$, $x\in A\iff f(x)\in B$.The names Mapping reduction and Karp reductions, to my knowledge, refer to Many to one.The Many to one means that $f$ may not be injective.Turing reduction: we say that $A\le_T B$ if, given an oracle to the language $B$, we can use it to solve $A$. The word solve here should be in the context of a specific complexity/computability class.Turing reductions are weaker than many-to-one reductions. The latter can be viewed as Turing reductions where we are only allowed to call the oracle once - at the very end of the run.polynomial time many-to-one reductions - simply adding a constraint that the reduction $f$ is computable in polynomial time.polynomial time Turing reduction (= Cook reduction) - add the constraint that the oracle machine runs in polynomial time, counting each oracle call as $O(1)$.