How can I go from $R(s)$ to $R(s,a)$ in this specific MDP? I'm trying to implement a research paper, as explained in this other post, here the author of the paper assumed R as a function of both states and actions, while the code (and the MDP) I'm using to test this algorithm assumes R as a function of only states.
My question is:
Given $\mathcal{X}$ as the set of states of an MDP and $\mathcal{A}$ as the set of actions of an MDP. Supposing I have four states ($1$,$2$,$3$,$4$), two actions $a$ and $b$ and a reward function $R: \mathcal{X}\to\mathbb{R}$ s.t.
$R(1) = 0$
$R(2) = 0$
$R(3) = 0$
$R(4) = 1$
If I need to change the current reward function to a new reward function $R:\mathcal{X}\ \times \mathcal{A} \to\mathbb{R}$ is it ok to compute it as $\forall a,R(s,a) = R(s)$?
$R(1,a) = 0$
$R(1,b) = 0$
$R(2,a) = 0$
$R(2,b) = 0$
$R(3,a) = 0$
$R(3,b) = 0$
$R(4,a) = 1$
$R(4,b) = 1$
More generally, what's the correct way of generalising a reward function
$R: \mathcal{X}\to\mathbb{R}$ to a reward function $R:\mathcal{X}\ \times \mathcal{A} \to\mathbb{R}$?
As explained here, I can write $R(s,a) = R(s)\  \forall a$ since the reward of my specific MDP is dependent exclusively to the state $s$.