# Whether there exists a probabilistic automaton satisfying $\Pr \{ x \in L\}=\frac{\Pr \{ x \in L_1\}}{\Pr \{ x \in L_1\}+\Pr \{ x \in L_2\}}$

Suppose that there are two probabilistic automata $$A_1$$ and $$A_2$$ with a same finite alphabet $$\Sigma$$. The languages of them are $$\mathcal{L}_{1} \subseteq \Sigma^*$$ and $$\mathcal{L}_{2} \subseteq \Sigma^*$$ respectively.

For every input $$x \in \Sigma^*$$, $$\Pr \{ x \in \mathcal{L}_{1}\}=p_1(x)$$ and $$\Pr \{ x \in \mathcal{L}_{2}\}=p_2(x)$$. Whether there exists a probabilistic automaton $$A$$ satisfying $$\Pr \{ x \in \mathcal{L}\}=\frac{p_1(x)}{p_1(x)+p_2(x)}$$ where $$\mathcal{L} \subseteq \Sigma^*$$ is the language of $$A$$?

• What do you think? Have you tried constructing such an automaton $A$? Have you tried finding $A_1,A_2$ such that no such $A$ exists? – Yuval Filmus Feb 5 at 9:32
• What about the trivial answer $\mathcal{L}_1 = \{\}$? – orlp Feb 5 at 9:38