How to define the languages of the implicit set system problems?
There are different notions of complexity used for various computational problems, not all tasks fit naturally to the standard model of decision/search problems. The &quot;classic&quot; analysis of complexity talks about how many resources we need to compute $f(x)$ given $x$, but this misses a lot of interesting questions we can ask. Consider communication complexity, where we want to compute $f(x,y)$ where one sides holds $x$ and the other holds $y$, while minimizing the number of bits exchanged between the parties. This requires a new definition of complexity. Same goes for query complexity, where we are given a black box access to some function $f$ and we want to know how many evaluations of $f$ are required to determine if it satisfies some property $P$.
The seemingly natural way (this of course depends on the context) to determine hardness of set system problems would be in terms of standard complexity and the number of queries to the membership oracle. As the set system is given, I don't want to involve hardness of deciding whether some set is in the system, so I'll only charge one operation for querying the membership oracle. This allows, in a sense, to isolate the hardness of evaluating the property from the hardness of the system's description.