  In the Group Testing problem, the objective is to learn a subset K of some
much larger domain N, using the shortest-possible sequence of queries Q. A
feedback to a query provides some information about the intersection between
the query and subset K. Several specific feedbacks have been studied in the
literature, often proving different formulas for the estimate of the query
complexity of the problem, defined as the shortest length of queries' sequence
solving Group Testing problem with specific feedback. In this paper we study
what are the properties of the feedback that influence the query complexity of
Group Testing and what is their measurable impact. We propose a generic
framework that covers a vast majority of relevant settings considered in the
literature, which depends on two fundamental parameters of the feedback: input
capacity $\alpha$ and output expressiveness $\beta$. They upper bound the
logarithm of the size of the feedback function domain and image, respectively.
To justify the value of the framework, we prove upper bounds on query
complexity of non adaptive, deterministic Group Testing under some "efficient"
feedbacks, for minimum, maximum and general expressiveness, and complement them
with a lower bound on all feedbacks with given parameters $\alpha,\beta$. Our
upper bounds also hold if the feedback function could get an input twisted by a
malicious adversary, in case the intersection of a query and the hidden set is
bigger than the feedback capacity $\alpha$. We also show that slight change in
the feedback function may result in substantial worsening of the query
complexity. Additionally, we analyze explicitly constructed randomized
counterparts of the deterministic results. Our results provide some insights to
what are the most useful bits of information an output-restricted feedback
could provide, and open a number of challenging research directions.
