D.3.3 ALGORITHMS FOR INCREMENTAL ... - SecureChange

Dealing with Known Unknowns: A Goal-based Approach 17
{g 5 ← {g 9 , g 12 } , g 5 ← {g 9 , g 13 }} or {g 5 ← {g 9 , g 12 , g 13 }} with a probability of 46%
and 42%, respectively. The original part might stay unchanged with a probability
of 12%. Similarly, in r o2 the original subpart {g 2 ← {. . .}} might also evolves to
two other possibilities with probabilities of 45% and 40%. The remain unchanged
probability of r o2 is 15%.
Once an evolutionary goal model was transformed into a corresponding evolutionary
hypergraph G〈V, E〉, every node x in G is tagged with a data structure
called design alternative table (DAT), denoted as DAT(x), holding information
for further calculating the max belief and residual risk. DAT(x) is a set of tuples
S {〈Si , mb i , rr i , T i 〉}, where:
– S i ⊆ G is a set of leaf goals necessary to fulfil this node.
– mb i , rr i are the values of max belief and residual risk, respectively.
– T i is a set of strings 〈r oα, i〉 which is the i th possibility of a rule r oα. This is
used to keep track of the path of evolutions.
Obviously, two tuples in a DAT which have a same T i are two design alternatives
of an observable evolution possibility. Initially, DAT of each node is initialized
as follows.
8
< {〈{x} , 1, 0, ∅〉} if x is a leaf goal,
DAT (x) ←
(5)
: ∅
otherwise.
The DATs of leaf nodes are then propagated upward to predecessor nodes (ancestors).
This propagation is done by two operators join(⊗) and concat(⊕).
Also via these operations, DAT of an predecessor node is generated using their
successor’s DATs.
DAT(x 1 ) ⊕ DAT(x 2 ) ← DAT(x 1 ) ∪ DAT(x 2 )
DAT(x 1 ) ⊗ DAT(x 2 ) ← [ i,j
˘
〈Si ∪ S j , mb i · mb j , 1 − rr i · rr j , T i ∪ T j 〉 ˛˛
〈S i , mb i , rr i , T i 〉 ∈ DAT(x 1 ), 〈S j , mb j , rr j , T j 〉 ∈ DAT(x 2 )¯
(6)
Lemma 1 The operator join and concat are commutative and associative.
Proof (Lemma 1) By definition,
DAT(x 1 ) ⊕ DAT(x 2 ) ← DAT(x 1 ) ∪ DAT(x 2 )
= DAT(x 2 ) ∪ DAT(x 1 ) → DAT(x 2 ) ⊕ DAT(x 1 )
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`DAT(x1 ) ⊕ DAT(x 2 )´ ⊕ DAT(x 2 ) ← (DAT(x 1 ) ∪ DAT(x 1 )) ∪ DAT(x 2 )
= DAT(x 1 ) ∪ (DAT(x 1 ) ∪ DAT(x 2 ))
→ DAT(x 1 ) ⊕ `DAT(x 2 ) ⊕ DAT(x 2 )´
From (7),(8), we conclude that concat(⊕) is commutative and associative. Similarly,
we also have join(⊗) commutative and associative.
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