Let $A$ and $B$ be two decision problems in $NP$. Consider three cases:(1) For any instance of problem $A$, one can produce, in polynomial time, an instance of problem $B$ having exactly the same number of solutions as $A$.(2) For any instance of problem $A$, one can produce, in polynomial time, an instance of problem $B$ having a fixed number of solutions for every solution of problem $A$. We can, in polynomial time, calculate the number of solutions for $A$ provided an oracle that gives the number of solutions to $B$, and vice versa. What if we can guarantee that the number of solutions for an instance of $A$ will be greater than the number of solutions for the corresponding instance of $B$ or vice versa?(3) For any instance of problem $A$, one can produce, in polynomial time, an instance of problem $B$ having a solution if and only if $A$ has a solution. However, we do not have a polynomial time algorithm to compute the number of solutions for $A$ provided some number of solutions for $B$.  We also cannot guarantee that such a polynomial time algorithm does not exist.What formal names and notation for the above sort of reductions? When is a Levin reduction called parsimonious? Instances one and two only? Is the third example necessarily worth anything?
Reductions where the number of certificates from one problem can be computed for another to varying degrees
complexity theory;reductions;np
Parsimonious reduction.Polynomial-time Turing reduction (a.k.a. Cook reduction).Polynomial-time many-one reduction (a.k.a. Karp reduction).I've not come across cardinality restrictions of the form you mention in (2) so I don't know any name for them. If you find there isn't a name already, monotone would be a good one to choose.  In the case where the number of solutions to $B$ is less than $A$, you need less than or equal to since, if $A$ has exactly one solution, you still need to reduct it to a yes instance of $B$.