We therefore add, temporarily in this subsection , two assumptions on top of the exclusiveness condition and the finite support condition so that equations like (14) become mathematically correct. The first assumption is that "clause" bodies are mutually exclusive i.e. if there are two clauses B ← W and B ← W', PDB(W A W' | 0) = 0, and the second assumption is that body atoms are in dependent, i.e. if A ← B 1 ∧ · · · ∧ B k is a rule, PDB ( B 1 ∧ · · · ∧ B k | 0 ) = PDB ( B 1 | 0 ) · · PDB ( B k | 0 ) holds.

Please note that "clause" used in this subsection has a special meaning. It is intended to mean G ← τ where G is a goal and τ is a tabled explanation for G obtained by OLDT search both of which will be explained in the next subsection. 25 In other words, these additional conditions are not imposed on a source program but on the result of OLDT search. So clauses for auxiliary computations do not need to satisfy them.

Now suppose clauses about A occur in DB like

A ← B1,1 ∧ · · · ∧ B1, i 1 A ← B L 1 ∧ ··· ∧ B L i 1 ,
where B h,j (1 ≤ h ≤ L,1 ≤ j ≤ ih ) is an atom. Theorem A.1 in Appendix A and the above assumptions ensure

$$\beta(A)=\prod_{j=1}^{i_{1}}\beta(B_{1,j})+\cdots+\prod_{j=1}^{i_{L}}\beta(B_{L,j}).\tag{1}$$
$$(15)$$

(15) suggests that β ( G t ) can be considered as a function of β ( A ) if these equations about inside probabilities are hierarchically organized in such a way that β( G t ) belongs to the top layer and any β(A) appearing on the left hand side only refers to β(B)'s which belong to the lower layers. We refer to this condition as the acyclic support condition. Under the acyclic support condition, equations of the form (15) have a unique solution, and the computation of PDB(G | 0) via inside probabilities allows us to take advantage of reusing intermediate results stored as β( A ), thereby contributing to faster computation of PDB( Gt | 0 ).

Next we tackle a more intricate problem, the computation of ESE¥pB(Gi) Paew(S  | 0)σίν(S).  Since the sum equals Σπ Σπεα(i,n,v)∈ S ∈ φ σβ(Gι) P msu(S | θ), we concentrate on the computation of

$$\mu\left(\,G_{t},{\mathfrak{m}}\,\right)\stackrel{\mathrm{def}}{=}\sum_{{\mathfrak{m}}\in S\,\in\,\psi\,_{D B}\left(\,G_{t}\,\right)}P_{{\mathfrak{m}}{\mathfrak{s}}{\mathfrak{w}}}(\,S\,\mid\,\theta\,)$$
$\beta\left(\text{m}\right)$. 
where m = msw(i,n,v). First we note that if an explanation S contains m like S = a1 A · · · A a h A m, then we have β(S) = β(α 1 ) · ·β(α h )β( m). So μ(G t , m) is expressed as µ(G t , m) = α(G t , m)β(m)
(16)
where α(Gt,m) = βμ(Gt,m) does not depend on β(m). Generalizing this observation to arbitrary ground atoms, we introduce the outside probability of ground atom A w.rt. G t by

$$\begin{array}{l}{{\alpha(G_{t},\,A)\stackrel{\mathrm{def}}{=}\frac{\partial\beta(G_{t})}{\partial\beta(A)}}}\\ {{=}}\end{array}$$

25. The logical relationship (13) corresponds to (20) where f, g and h are table atoms