1. Branching Bisimulation Congruence for Probabilistic Transition Systems
Nikola Trčka
and
Sonja Georgievska

2. Labeled Transition Systems
Formalism for modeling of qualitative (functional) behavior
Directed graphs:
nodes = states of the system
labels on arrows = actions that the system can perform
Example:

3. Branching Bisimulation Equivalence
Equates states with the same action potential
Preserves branching structure
Abstracts from internal (tau labeled) transitions
Same colour - equivalent states

4. Adding Probabilities To model quantitative aspects of systems
Several existing models
Further refinements and extensions:
reactive, generative
strictly alternating, non-strictly alternating
stratified models

5. Our model: Probabilistic Transition System
Generalization of the alternating model: also allows consecutive probabilistic states
Orthogonal extension of both labeled transition systems and Markov chains

6. Parallel composition Probabilistic choice resolved first
Parallel probabilistic choices are combined

7. Branching bisimulation for probabilistic systems
Fully probabilistic systems [Baier and Hermanns, 1997]
Strictly alternating model [Andova and Willemse, 2005]
Non-alternating model [Segala and Lynch, 1994]
Main idea in all three definitions:
if s~t and then the probability of the set of all scheduled internal computations from t not leaving the class of s and ending in the class of s’ by doing the action a is 1.

8. Our goal
Define branching bisimulation for probabilistic transition systems that:
is a congruence relation
does not use the notion of schedulers
is a conservative extension of branching bisimulation for transition systems

9. Congruence problem
Direct adaptation of branching bisimulation of [Andova and Willemse] does not work

10. “invisible” transition
What we want…
“invisible” transition

11. What we want… Light blue states have same “probabilistic potential”
“invisible” transition
A, B, C and D – nondeterministic states

12. What we don’t want…
because of the priority of the probabilistic choice in parallel composition

13. What we don’t want…
because of the priority of the probabilistic choice in parallel composition

14. Our branching bisimulation
R – equivalence relation, (s, t) in R
R is branching bisimulation iff
First condition (statement):
tau or probabilistic step

15. Ensures that all three states are equivalent
The first condition…
Preserves “branching potential” for action transitions
Ensures that all three states are equivalent

16. Ensures that all three states are equivalent
The first condition…
Preserves “branching potential” for action transitions
Ensures that all three states are equivalent
But still…

17. Ensures that all three states are equivalent
The first condition…
Preserves “branching potential” for action transitions
...and still
Ensures that all three states are equivalent
But still…

18. Second condition (preliminaries):
Define
prob. trans.
Nondeterministic states
“all probabilities are left unchanged, except that a nondeterministic state reaches itself with probability one”

19. Second condition - preliminaries
Cesaro limes of P:
Π(s,t) – “probability that s will ‘end up’ in t (without performing actions!)”
“Cesaro” probabilities
1/2
A, B, C and D - nondeterministic

20. Second condition (statement)
Note: it should also hold when the blue states are grey
Extra requirement:
A nondeterministic state can be related only to a state that eventually reaches a nondeterministic one

21. The second condition…
This state reaches its class with Cesaro probability 1…
…which is not true for this state

22. The second condition…
“A nondeterministic state can be related only to a state that eventually reaches a nondeterministic one”

23. What else is equivalent…
The light blue states reach same classes with same Cesaro probabilities

24. What else is equivalent or not…

25. Main results
We defined a branching bisimulation for a general model that includes probabilistic and nondeterministic states
It is congruence
It is stronger than [Andova and Willemse, 2005] when applied to the strictly alternating model

26. Questions?