D.3.3 ALGORITHMS FOR INCREMENTAL ... - SecureChange

Dealing with Known Unknowns: A Goal-based Approach 15
– EM is the original enterprise model,
– R o is set of observable evolution rules.
– R c is a set of controllable rules applying to the original enterprise model and
other evolved ones.
– Dep ⊆ R o×R o×N is a set of evolution-dependent relations between observable
rules.
R o = [ j r oα = [ j ffff
SM α pα i
−−→ SMi
α
R c = [ n
r cαi = [ nSM α i
Dep = [ n
r oα
o
i
r oβ
where SM α ⊆ EM ∨ ∃SM α′ p α′
i
−−→ SMi α′ .SM α ⊆ SMi
α′
oo
∗
−→ SMij
α
The calculation of the max belief and residual risk for a given configuration
C on a given evolutionary enterprise model eEM is complicated by the need to
consider both multi-part and multi-step evolutions.
However, once we have defined a semantics for the basic set-up of probabilities,
we can now leverage on the mathematics behind the theory or probabilities.Thus
we can use the mechanisms of conditional probabilities for multi-step
evolution and of independent combination of events for multi-part evolution.
The max belief and residual risk of a given configuration C and an evolutionary
enterprise model eEM〈EM, R o, R c, Dep〉 are defined as follows.
(3)
MaxB(C)
max
P r
∀( V α SMα −−→SM pα i
i α). S α SMα i ∈SDA(C)
RRisk(C) 1 −
X
∀( V α SMα pα i
−−→SM α i ). S α SMα i ∈SDA(C) P r
^
SM α pα i
−−→ SMi
α
α
!
^
SM α pα i
−−→ SMi
α
α
! (4)
8 An Incremental Algorithm to Calculate Max Belief and Residual Risk
Developing an algorithm to calculate max belief and residual risk, as in Formula 4,
for an evolutionary goal model is not practical. With the heuristic assumption that
observable rules in different parts are independent we can efficiently eliminate
recursion by transforming the model into a suitable hypegraph and use efficient
hyperpath algorithms.
An evolutionary goal model is converted in to an evolutionary hypergraph as
follows.
Definition 7 (Evolutionary hypergraph) The hypergraph G〈V, E〉 of an evolutionary
goal model 〈EM, R o〉 is constructed as follows:
• For each goal g, create a goal node g.