I have two decision problems (function problems, actually) that are closely linked. The first one consists in finding a variable assignment to $x$ Boolean variables so that a specification $A$ is met.
The whole problem can be reduced to FSAT and vice versa, making it FNP-Complete.
Now there's a second problem related to the first, which is to determine all variable assignments to the $x$ variables, for which $A$ is met.
What is the computational complexity of this problem?
It can be solved by applying SAT $2^x$ (i.e., an exponential number of) times; so that is not a valid reduction.
This is like finding all satisfying assignments to a SAT problem. Strangely I did not find much on this in Papadimitriou's book.