
## Finite Support Condition: For Every G E Obs( Db) Ψ Db ( G ) Is Finite.

PDB (G) is then computed from the support set ψ DB (G) = { S1,...,Sm } (0 ≤ m), with the help of the exclusiveness condition, as a finite summation L i = 1 P F ( S i ). This condition prevents an infinite summation that is hardly computable.

The fourth condition simplifies the probability computation to multiplication. Recall that an explanation S for G E obs(DB) is a conjunction a 1 ∧ · · ∧ a m of some abducibles
{a1, . . . , am} C F (1 < m). In order to reduce the computation of PF(S) = PF(a1^ . . . Aam)
to the multiplication P F ( a 1 ) . . . P F ( a m ), we assume

## Distribution Condition:

F is a set Fmsw of ground atoms with a parameterized distribution Pmsw specified below.

Here atom msw(i,n,v) is intended to simulate a multi-ary random switch whose name is i and whose outcome is v on trial n. It is a generalization of primitive probabilistic events such as coin tossing and dice rolling.

1. F msw consists of probabilistic atoms msw (i, n,v). The arguments i, n and v are ground terms called switch name, trial-id and a value (of the switch i), respectively. We assume that a finite set Vi of ground terms called the value set of i is associated with each i, and v E Vi holds.

2. Write Vi as {vi,v2,vi,vm} (m = |Vi |). Then,one of the ground atoms {msu(i,vi,vi),
msw(i,n,v2), ..., msw(i,n,vm)} becomes exclusively true (takes on value 1) on each trial. With each i, a parameter Bi,v E [0,1] such that EvEvi Bi,v = 1 is associated. Bi,v is the probability of msw(i, ,v) being true (v E Vi).

3. For each ground terms i, i , n, n ′ , v ∈ Vi and v ′ ∈ Vi ′ , random variable msw(i, n,v) is independent of msw( i' ,n' ,v') if n ̸ = n' or i ̸ = i'.
In other words, we introduce a family of parameterized finite distributions P(i,n) such that

$$P_{(i,n)}(\texttt{msw}\,(i,n\,,v_{1})\,=\,x_{1},\ldots,\texttt{msw}\,(i\,,n\,,v_{m}\,)\,=\,x_{m}\,\mid\,\theta_{i,v1},\ldots,\theta_{i,v\,m}\,)$$ $$\stackrel{{\text{def}}}{{=}}\left\{\begin{array}{ll}\theta_{i,v1}^{v1}\cdots\theta_{i,vm}^{vn}&\text{if}\quad\sum_{k=1}^{m}x_{k}\,=\,1\\ 0&\text{o.v.}\end{array}\right.\tag{7}$$
$n\,=\,|V_{i}|,\;x_{k}\in\{0,1\},(1\,\leq\,k)$
where m = |Vi|, x k E {0,1} (1 ≤ k ≤ m), and define P new as their infinite product

$$P_{\bf m w}\ {\stackrel{\mathrm{def}}{=}}\ \prod_{i,n}P_{(i,n)}.$$

Under this condition, we can compute Pmmw(S), the probability of an explanation S, as the product of parameters. Suppose msw(ij,n,v) and msw(ij,n',v') are different conjuncts in an explanation S = msu(ii,ni,vi) A . . A msu(ik,nk,vk). If either j = j' or n = n' holds, they are independent by construction. Else if j = j and n = n ′ but v ̸ = v ′ , they are not independent but P n m ( S ) = 0 by construction. As a result, whichever condition may hold, Pnnw(S) is computed from the parameters.