Probabilistic ASP

Weighted Approach

1. Total Choices. $N(C = x) = \prod_{a \in x} w_a \prod_{\neg a \in x} (1 - w_a)$.
2. Stable Models. $N(S = x | C = c) = \alpha_{x,c}$, where the set of parameters $\alpha_{x,c}$ is such that: $$ \begin{cases} \alpha_{x,c} \geq 0, & \forall c, x\cr \alpha_{x,c} = 0, & \forall x \not\supseteq c \cr \sum_{x} \alpha_{x,c} = 1, & \forall c. \end{cases} $$
3. Worlds. $N(W = x)$
   1. If $x$ is a total choice: $ N(W = x) = \prod_{a \in x} w_a \prod_{\neg a \in x} (1 - w_a). $$