A state is:$A_{q}=(A_{q}^{0},...,A_{q}^{E})$ where $A_{i}^{j}$ is interval, $q$ and $E$ are positive integer
The initial state is $A_{m}=((0,1),\emptyset...,\emptyset)$ , $m>E$
Procedure： Every time before asking a question, randomly select a number x between 0 and 1, interval $T=(0,x)$
(here the original problem is ask question and then answer yes or no, i think we can skip understand original problem)
A 'Yes' answer will result in a new state $A_{q-1}$: $^{y}A_{q-1}=(^{y}A_{q-1}^{0},...,^{y}A_{q-1}^{E})$ with
$^{y}A_{q-1}^{e}=(A^{e}_{q}\cap T)\cup (A^{e-1}_{q}-T)$ for $1 \le e \le E$
and
$^{y}A_{q-1}^{0} = A_{q}^{0} \cap T$
A 'No' answer will result in a new state $A_{q-1}$: $^{n}A_{q-1}=(^{n}A_{q-1}^{0},...,^{n}A_{q-1}^{E})$ with
$^{n}A_{q-1}^{e}=(A^{e}_{q}- T)\cup (A^{e-1}_{q}\cap T)$ for $1 \le e \le E$
and
$^{n}A_{q-1}^{0} = A_{q}^{0} - T$
Ask m questions (repeat the procedure m times) in total, and finally get the state $A_{0}$
The question is that one paper says that :
"The set $\cup_{e=0}^{E}A_{0}^{e}$ consists of a finite number of intervals $I_{1}, I_{2},.. I_{r}$ (where $r\leq E+1$). Define $depth(I_{j})$ to be the number of sets $A^{e}_{0}$, $0\le e\le E$, that intersect $I_{j}$, and define $depth(A_{0})=\sum^{r}_{j=1} depth(I_{j})$. Using the fact that $A_{0}=(A_{0}^{0},...,A_{0}^{E})$ can be described completely by no more than $E+1$ previous 'Yes' answers and $E+1$ previous 'No' answers, it's not hard to see that $depth(A_{0})\le E+1$."
I cannot understand :
Why r must be $\le E+1$
Why $A_{0}$ can be described completely by no more than $E+1$ previous 'Yes' answers and $E+1$ previous 'No' answers, it's not hard to see that $depth(A_{0})\le E+1$
Hint: $A_q^0$ can only be $(0,1), T, (0,1)-T, \emptyset$.
More specifically: $$A_m^0=(0,1)$$ If the answer to the first question is 'yes', then: $$A_{m-1}^0=T$$ else it's: $$A_{m-1}^0=(0,1)-T$$ Note that $T\cap ((0,1)-T)=((0,1)-T)\cap T=\emptyset$, as after a 'yes' the set only contains elements of $T$ and after a 'no' it doesn't contain any.
So if the answers given oppose, then for $k\gt1$:
$$A_{m-k}^0=\emptyset$$