D.3.3 ALGORITHMS FOR INCREMENTAL ... - SecureChange

Dealing with Known Unknowns: A Goal-based Approach 21
Algorithm 3. Calculating max belief and residual risk using DAT.
1 procedure calcMBRR(DAT : DAT, C : set of Goal, out mb, rr : number)
2 begin
3 trails ←∅ ;
4 totalBelief ←0;
5 mb ←0;
6 for each 〈 S i, mb i, rr i, T i 〉 in DAT do
7 if (C ⊇ S i and T i /∈ trails) then
8 mb ←max(mb, mb i);
9 totalBelief ←totalBelief + (1 − rr i);
10 trails ←trails ∪ {T i}
11 rr ←1 − totalBelief;
12 end
This values need to be multiplied by the complexity of the join and concat
operations which is bounded by the number of design alternatives in the DAT.
The second point of the theorem is that DATs for all nodes are properly generated
when the algorithm terminates. Since join and concat operators are
commutative and associative (see Lemma 1), the DAT calculation of a node by the
for loop at line 9 obviously follows Formula 9. Furthermore, when a node have
all of its children processed, it is immediately put in queue Q. This means that it
will be processed sometime later. Therefore, when the algorithm terminates, all
nodes have their DAT calculated as Formula 9.
Theorem 2 (Stability of DAT) Each time a node changes, its DAT as well as its
ancestors’ DATs are updated.
Proof (Theorem 2) When Algo. 1 is running, once a node x has been visited,
REACH[x] is set to -1 (line 8). After that, DAT[x] is no longer touch by Algo. 1
because of the precondition check at line 7.
Later on, if users change x e.g., adding new predecessors, Algo. 2 is invoked.
Therefore, the DATs of x and its ancestors are updated.
Algo. 3 presents the algorithm calculating max belief and residual risk of a
configuration C using DAT of the root node, which generated by Algo. 1. Basically,
this algorithm is a translation of Formula 10 in programming language. The
algorithm walks though all elements DA i of the DAT (the for loop at line 6),
checks whether C supports DA i , and there is no DA j which have the same trail
with DA i . For each matched element, the max belief and total belief values are
updated (line 8–9). When the for loop terminates, the residual risk is the complement
of calculated total belief (line 11).
Example 5 (Max Belief, Residual Risk and DAT) We apply the algorithm to the
hypergraph denoted in Example 4. Table 2 presents some entries in the DAT of the
root goal g 1 , for the full DAT (g 1 ), please refer to Table 3 in the appendix. Next, we
apply Formula 10 to calculate the max belief and residual risk of a configuration
C= {g 9 , g 12 , g 6 , g 7 , g 11 , g 15 , g 16 , g 17 , g 19 , g 20 } as follows.