Reconsidering the Probabilistic Bisimilarity ongoing research

1
Reconsidering the Probabilistic
Bisimilarity(ongoing research)

• Sonja Georgievska
• Joint work with Suzana Andova and Nikola Trcka
• Department of Mathematics and Computer Science,
• Eindhoven University of Technology

2
Labeled Transition Systems

• Formalism for modeling of qualitative
  (functional) behavior
• Directed graphs
• labels on edges actions that the system can
  perform
• nodes states of the system between performing
  two consecutive actions

3
The Bisimilarity Equivalence

• Equates states with the same action transition
  potential
• Two systems are bisimilar iff their starting
  states are bisimilar

a
a
a
a


b
b
b
b
b
a
a
a
Note same colour bisimilar states
?
b
b
c
c
4
Adding Probabilistic Behaviour

• To model quantitative aspects of systems
• With explicit probabilistic states alternating
  model
• Without probabilistic states probabilistic
  automata
• (action transition leads to a distribution over
  states)

a
1/3
2/3
d
b
c
1/4
3/4
c
5
Standard Probabilistic Bisimilarity
s and t are bisimilar iff if s can perform an
action and end in a distribution, then t can
perform the same action and end in an
equivalent distribution
6
Question

• Why are the following two systems not bisimilar?

How do we know/prove where exactly the
probabilistic behaviour happens?
a
a
2/3
1/3
? ?
b
b
b
2/3
1/3
c
d
c
d
7
Previous work

• Indeed, people have tried to distribute action
  over probabilistic choice
• Problems
• Compositionality issues (KwiatkowskaNorman98)
• x1 x2 should imply x1 parallel y x2
  parallel y
• Lost of the idempotency rule xxx (Morgan et
  al.96, Cazorla et al. 03)
• (Alternative choice between two equal systems x
  should yield the system x)

8
Our approach

• Each state has its distribution over classes
• Each class has a representor state
• Two states are bisimilar iff they have the same
  distribution in some class assignement

9
Current results

• A new (weaker) definition of probabilistic
  bisimilarity
• Compositionality idempotency preserved
• Proved to be equivalence for acyclic graphs (we
  still do not know how to prove for cyclic graphs
  ?)

10
Perspective?

• In a setting with silent steps t, we could
  finally equate (without compositionality lost)

• Therefore, we could add time to the model with
  silent steps without unnecessary restrictions
• The goal is a more powerful stochastic process
  algebra for quantitative analysis