Journal of Software - Academy Publisher
Journal of Software - Academy Publisher
Journal of Software - Academy Publisher
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JOURNAL OF SOFTWARE, VOL. 6, NO. 5, MAY 2011 777<br />
S ′ 0<br />
ν0 ��<br />
��<br />
��<br />
�<br />
��<br />
ν1,Readcrd ��<br />
′ S 1<br />
executions. Let S ⊆ Act. Then<br />
P||SQ ∼ �<br />
[λi].Pi + �<br />
(bj,pj).Pj +<br />
I<br />
J<br />
�<br />
[µk].Qk + �<br />
(bl,ql).Ql +<br />
K<br />
�<br />
A∩B∩S<br />
(b j ,p j )=(b l ,q l )<br />
L<br />
(bj,pj).(Pj||SQl)<br />
where A = {(bj,pj)|j ∈ J}, B = {(bl,ql)|l ∈ L}<br />
and<br />
Example 5.6 Revisit Example 5.3, we assume that<br />
P = (true).([ν4].Msg PASS )+<br />
(¬true).([ν5].Msg WrongCard ))<br />
Q = (true).([ν4].Msg OK )+<br />
(¬true).([ν5].Msg Again ))<br />
ν2,Identifycrd ��<br />
as time delays characterized by µ4, µ5, ν4 and ν5 have no<br />
influences with boolean expression true and ¬true. We<br />
can exchange the position between delays and boolean<br />
expressions by shorten true to bp and bq, Msg PASS<br />
to MP , Msg WrongCard to MW , Msg OK to MO and<br />
Msg Again to MA. Then, we have<br />
and<br />
P = [ν4].(bp,MP)+[ν5].(¬bp,MW)<br />
Q = [µ4].(bq,MO)+[µ5].(¬bq,MA)<br />
as there is no action for P and Q to synchronize, then it<br />
can be expanded as<br />
P|| ∅Q = ([ν4].(bp,MP)+[ν5].(¬bp,MW))|| ∅<br />
([µ4].(bq,MO)+[µ5].(¬bq,MA))<br />
∼ [ν4].(bp,MP)+[ν5].(¬bp,MW)+<br />
[µ4].(bq,MO)+[µ5].(¬bq,MA)<br />
We know that ν4, ν5, µ4 and µ5 are variables <strong>of</strong> exponential<br />
distributions, by the “memoryless” property <strong>of</strong><br />
Markovian process, at any time point, the time passed cannot<br />
influence them. So it is reasonable for[ν4].(bp,MP)+<br />
[ν5].(¬bp,MW)+[µ4].(bq,MO)+[µ5].(¬bq,MA) to simulate<br />
the execution <strong>of</strong> P|| ∅Q.<br />
© 2011 ACADEMY PUBLISHER<br />
S ′ 2<br />
ν3,Log crd<br />
��<br />
true ��<br />
��<br />
��<br />
�<br />
�����<br />
� ¬true ���<br />
ν7<br />
Figure 3. Illustration <strong>of</strong> Syscrd<br />
S ′ 3<br />
S ′ 4<br />
�����<br />
����<br />
�<br />
ν 4 ,Msg PASS<br />
�������<br />
��<br />
�����<br />
��<br />
ν 5 ,Msg Wrong Card<br />
C. Weak Markovian Congruence<br />
S ′ 5<br />
ν6,Done<br />
��<br />
′ S 6<br />
The weak Markovian congruence abstracts away internal<br />
actions. To treat internal transitions properly, we need<br />
the following definition.<br />
Definition 5.7 (Weak Markovian Bisimulation) An<br />
equivalence relation R with R = LYAWN × LYAWN<br />
is called weak Markovian bisimulation is a family <strong>of</strong><br />
symmetric relationsR = {R b | b ∈ BExp}, and satisfies:<br />
iff PR b Q implies for all a ∈ Act and all equivalence<br />
classes C <strong>of</strong> R b :<br />
1) If P b1,av<br />
−−−→ P ′ with bv(a) ∩ fv(b,P,Q) = ∅,<br />
then there is a b ∧ b1-partition B with<br />
fv(B) ⊆ fv(b,P,Q) such that for each b ′ ∈ B<br />
there exist b2, a ′ and u ′ with b ′ |= b2, a = b′ a ′ ,<br />
Q b2,â′<br />
−−−→ Q ′′ and such that<br />
• If a ≡ c?x then there is a b ′ -partition B ′ such<br />
that for each b ′′ ∈ B there are b ′ 2 and Q′′ with<br />
b ′′ |= b ′ b′<br />
2 , Q′ −−→ Q ′′ and P ′ Rb′′ Q ′′ ;<br />
• otherwise P ′ Rb′ Q ′ .<br />
2 ,i<br />
2) P i −→ P ′ and P ′ � i −→ imply γM(p,C i ) = γM(q,C i )<br />
P andQare called weakly Markovian bisimulation equivalent<br />
(P ≈ Q) if there is a weak Markovian bisimulation<br />
R such that PRQ.<br />
Example 5.8 Revisit Example 5.3, we can abstract<br />
away some internal actions to form a system as which can<br />
be observed by the outside observers. First we simplify<br />
Sysfp in Figure 4.<br />
Based on Definition 5.7, we know that system Sysfp<br />
and Syscrd in the relation <strong>of</strong> weakly Markovian bisimulation<br />
in Fig.4 iff they satisfy: µi = νi for i = 1,2,3,4.<br />
In [14], Hermanns proved that ≈ is a congruence for<br />
all IMC operators (and hence also for YAMN ) except<br />
choice operator. The reasons for this are well known due<br />
to Milner [23], and the deficiency is fixed with the follow<br />
definition:<br />
Definition 5.9 (Weak Markovian Congruence)P and<br />
Q are said to be weakly Markovian congruent(P ≃ Q) is<br />
a family <strong>of</strong> symmetric relations R = {R b | b ∈ BExp},<br />
if and only if for all a ∈ Act, all C ∈ YAMN/ ≈: