Prove that for each arbitrary set of natural numbers $A$ we have:  $$\{bin(n) | n\in A\} \in BPP \to \{0^n|n\in A\}\in QP$$ where $bin(n)$ is binary representation of $n$ $$QP = \bigcup_{c\in \mathbb{N}} DTIME\left(2^{(\log n)^c}\right) $$  I have no idea how to move from probablistic machines to non-probablisitic machine. Any ideas ?  EditLet us note, that $DTIME(n)\subseteq QP$, because for $c=1$ we have $2^{\log n}=n$.  Now we use the fact that $BPP\subset EXP$.  Let $X=\{bin(n)|n\in A\}$ will be decided by exponetnial turing machine $M$.   Now, we give machine $N$ which in $DTIME(n)$ (so in particular in $QP$) recognize  $Y=\{0^n | n\in A\}$.The $N$ works in following way:1. Convert unary ($n$ bits) number to binary number ($\log n$) bits.2. Launch $M$ on this converted input - it takes $2^{\log n}=n$ steps.3. $n\in DTIME(n)\subseteq QP$.  
Show that if language belongs to $BPP$ then also similar language belgons to $QP$
complexity theory;time complexity;probabilistic algorithms