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

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:

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

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

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

Parallel composition Probabilistic choice resolved first Parallel probabilistic choices are combined

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.

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

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

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

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

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

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

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

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

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…

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…

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

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

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

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

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

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

What else is equivalent or not…

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

Questions?