# How to obtain the policy in the form of a finite-state controller from the value function vectors over the belief space of the POMDP?

I was reading this paper by Hansen.

It says the following:

A correspondence between vectors and one-step policy choices plays an important role in this interpretation of a policy. Each vector in $$\mathcal{V}'$$ corresponds to the choice of an action, and for each possible observation, choice of a vector in $$\mathcal{V}$$. Among all possible one-step policy choices, the vectors in $$\mathcal{V}'$$ correspond to those that optimize the value of some belief state. To describe this correspondence between vectors and one-step policy choices, we introduce the following notation. For each vector $$\mathcal{v}_i$$ in $$\mathcal{V}'$$, let $$a(i)$$ denote the choice of action and, for each possible observation $$z$$, let $$l(i,z)$$ denote the index of the successor vector in $$\mathcal{V}$$. Given this correspondence between vectors and one-step policy choices, Kaelbling et al. (1996) point out that an optimal policy for a finite-horizon POMDP can be represented by an acyclic finite-state controller in which each machine state corresponds to a vector in a nonstationary value function. I am unable to guess how the left-side finite-state controller is formed from the right side belief space diagram. Does the above text provide enough explanation for the conversion? If yes, I am not really able to fully get it. Can someone please explain?