Given set $S = \{0,1\}^n$, consider a subset $S' \subseteq S$. By determine we mean there is a deterministic TM $A_{S'}$ which always halts and defines a mapping $\{0,1\}^n \rightarrow {0,1}$ such that $A_{S'}(x) = 1$ iff $x \in S'$.
If $S_1'$ contains all even numbers in $S$, $\forall x \in S$, we can determine whether $x \in S_1'$ in polynomial time.
If $S_2'$ contains all numbers in $S$ which are smaller than $2^{n-1}$, $\forall x \in S$, we can determine whether $x \in S_2'$ in polynomial time.
If $S_3'$ contains $2^{n-1}$ randomly chosen numbers in $S$, $\forall x \in S$, intuitively, it may not be possible to determine whether $x \in S_3'$ in polynomial time using a polynomial-size TM.
Note that the set $S'$ itself is not part of the input. In this way the running time of $A_{S'}$ doesn't depend on the size of $S'$.
Intuitively, the size and running time (which have some trade-off) of the most efficient TM $A_{S'}$ (that can determine whether $x \in S'$ correctly) can be viewed as the indistinguishability complexity (defined by me) of the subset $S'$.
Question:
Is there any formal theory that exactly defines this idea?
How to measure this trade-off between the size and the running time?
For $S = \{0,1\}^{n}$, what is the subset $S'$ that has the largest indistinguishability complexity?