In detail, let H  B ; C  H B ; C  and H B ; C  (note that   and  will coincide with the least upper bound of Pr BL C )) Pr B), Pr B L C )) Pr B), and Pr L C )) Pr B), respectively, sub ject to Pr KB and Pr B 0), where   and  in Leaf C B), Chaining C B ), and Fusion C B1 B2 : : : Bk with k 1)
are given as follows:

Leaf



Chaining

 NG $$\begin{array}{cccccccccc},\,Pr&C\ B&\xrightarrow{H^\intercal\ C,\,C^\intercal}&\\ &&\hspace*{1.0cm}Pr&C\ B&&\hspace*{1.0cm}+\ \frac{H^\beta\ C,\,C^\intercal}{Pr\ B|C}&&\\ &&\hspace*{1.0cm}Pr&C\ B&&\hspace*{1.0cm}+\ \frac{H^\beta\ C,\,C^\intercal}{Pr\ B|C}&&\\ &&\hspace*{1.0cm}Pr&C\ B&&\hspace*{1.0cm}\frac{H^\beta\ C,\,C^\intercal}{Pr\ B|C}&&\hspace*{1.0cm},\ Pr&C\ B&&\hspace*{1.0cm}\frac{H^\intercal\ C,\,C^\intercal}{Pr\ B|C}&&\\ &&\hspace*{1.0cm}Pr&C\ B&&\hspace*{1.0cm}\frac{H^\intercal\ C,\,C^\intercal}{Pr\ B|C}&&\hspace*{1.0cm}\end{array}$$
$$\beta$$

Fusion min i2[1:k H  B ; Bi

$$\beta$$

 min i2[1:k H B ; Bi

 min( min
$$\;_{i\in\ :k}H^{\gamma}\ B,B_{i}\ ,\ _{i,j\in\ :k\ ,i\ /j}H^{\alpha}\ B,B_{i}\ +H^{\beta}\ B,B_{j}$$
The functions H  H  and H  and C i H  B ; C  H B ; C  and H B ; C  are the least upper bounds of Pr BL C )) Pr B), Pr B L C )) Pr B), and Pr L C )) Pr B), respectively, sub ject to Pr Mo B ; C and Pr B 0. Hence, the following theorem shows soundness and global completeness of H  H  and H  induction on the recursive denition of H  H  and H  ).
are sound and globally complete with respect to B

(actually, it shows even more to enable proof by

## Theorem 4.2

a) For al l probabilistic interpretations Pr Mo B ; C , it holds Pr BL C))  Pr B ,
Pr BL C))  Pr B , and Pr L C))  Pr B .

b) There exists a probabilistic interpretation Pr Mo B ; C with Pr B , Pr BL C))
 Pr B , and Pr L C))  Pr B .

c) There exists a probabilistic interpretation Pr Mo B ; C with Pr B , Pr BL C))
 Pr B , and Pr L C))  Pr B .