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Branching Bisimulation Congruence for Probabilistic Transition SystemsNikola Trčka

and

Sonja Georgievska

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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: a

b

c

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Branching Bisimulation Equivalence

Equates states with the same action potential Preserves branching structure Abstracts from internal (tau labeled) transitions

a

a

τ

b

a b

Same colour - equivalent states

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Adding Probabilities To model quantitative aspects of systems Several existing models

Further refinements and extensions: reactive, generative strictly alternating, non-strictly alternating stratified models

a bc

1/3 1/6

1/2

Fully probabilistic

a1/3

2/3

b1/4

3/4

Simple Segala

a b d

c1/4 3/4 1

Alternating

f

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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

a c

1/2

1/21/3 2/3

1/6 5/6 a b

τ

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Parallel composition Probabilistic choice resolved first

Parallel probabilistic choices are combined

b c

1/3 2/3

|| a =b||a

1/3 2/3

c||a

a b

1/3 2/3

|| =a||d

1/6 1/3

b||cc d

1/2 1/21/6

a||c

1/3

b||d

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Branching bisimulation for probabilistic systems

1. Fully probabilistic systems [Baier and Hermanns, 1997]2. Strictly alternating model [Andova and Willemse, 2005]3. 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.

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Our goal

Define branching bisimulation for probabilistic transition systems that: is a congruence relationdoes not use the notion of schedulers is a conservative extension of branching

bisimulation for transition systems

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Congruence problem Direct adaptation of branching bisimulation of [Andova and Willemse]

does not work

τ

1/2 1/2

1/2 1/2

a

a b

b

c

c

τ

1/21/2

a b

τ

1/2 1/2

a||c b||c

c1/2 1/2

a||c b||c

|| =

|| =

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What we want…

τ

1

a

“invisible” transition

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What we want…

τ τ

1/21/2

1/21/2

1/21/2

1/4

1/41/4

1/4

A B C D A B C D

A, B, C and D – nondeterministic states

τ

1

a

“invisible” transition

Light blue states have same

“probabilistic potential”

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What we don’t want…

τ

1/2 1/2

b c

a

because of the priority of the probabilistic

choice in parallel composition

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What we don’t want…

τ

1/2 1/2

b c

a

τ

1/2 1/2

b c

a

because of the priority of the probabilistic

choice in parallel composition

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sa

t(a)

…...

then

If

Our branching bisimulationR – equivalence relation, (s, t) in R

R is branching bisimulation iff

First condition (statement):

tau or probabilistic step

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The first condition… Preserves “branching potential” for action

transitions

Ensures that all three states are equivalent

τ

1

a

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The first condition… Preserves “branching potential” for action

transitions

Ensures that all three states are equivalent

But still…

τ

1

a

a b

π 1-π

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The first condition… Preserves “branching potential” for action

transitions

Ensures that all three states are equivalent

But still…

τ

1

a

a b

π 1-πτ

1/2 1/2

b c

a

...and still

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Second condition (preliminaries):

Define

“all probabilities are left unchanged, except that a nondeterministic state reaches itself with probability one”

Nondeterministic states

prob. trans.

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Second condition - preliminaries

Cesaro limes of P:

Π(s,t) – “probability that s will ‘end up’ in t (without performing actions!)”

A, B, C and D - nondeterministic

“Cesaro” probabilities

1/21/2

1/21/2 1/2

A B C D

1/4

1/4

1/4

A B C D

1/2 1/2 1/2

1/21/4

1/2

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Second condition (statement)

s

||V

Cesaro prob. = π

t

Cesaro prob. = π

Extra requirement:

A nondeterministic state can be related only to a

state that eventually reaches a

nondeterministic one

Note: it should also hold when the blue states are grey

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The second condition…

τ

1/2 1/2

b c

a

This state reaches its class with Cesaro probability 1…

…which is not true for this state

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The second condition…

τ

1/2 1/2

b c

a

a b

π 1-π

τ1

“A nondeterministic state can be related only to a state that

eventually reaches a nondeterministic one”

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What else is equivalent…The light blue

states reach same classes with same

Cesaro probabilities

τ τ

1/21/2

1/21/2

1/21/2

1/4

1/41/4

1/4

A B C D A B C D

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What else is equivalent or not…

τ

1/2 1/2τ

a b

τ

τ

1/2 1/2τ

a b

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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

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Questions?