Late Weak Bisimilarity for Markov Automata Christian Eisentraut 1 - - PowerPoint PPT Presentation

Late Weak Bisimilarity for Markov Automata

Christian Eisentraut1 Jens Chr. Godskesen2 Holger Hermanns1 Lei Song3,1 Lijun Zhang4,1

Saarland University, Germany IT University of Copenhagen, Denmark Max-Planck-Institut f¨ ur Informatik, Germany Institute of Software, Chinese Academy of Sciences, China

September 27, 2013 ISCAS, Beijing

1 / 25

Markov Automata

Markov Automata An MA M is a tuple (S, Actτ, , , ¯ s) where

◮ ¯

s ∈ S is the initial state,

◮ S is a finite but non-empty set of states, ◮ Actτ = Act .

∪ {τ} is a set of actions including the internal action τ,

◮

⊂ S × Actτ × Dist(S) is a finite set of probabilistic transitions,

◮

⊂ S × R>0 × S is a finite set of Markovian transitions.

2 / 25

Probabilistic Automata

Probabilistic Automata A Probabilistic Automaton M is a tuple (S, Actτ, , , ¯ s) where

◮ ¯

s ∈ S is the initial state,

◮ S is a finite but non-empty set of states, ◮ Actτ = Act .

∪ {τ} is a set of actions including the internal action τ,

◮

⊂ S × Actτ × Dist(S) is a finite set of probabilistic transitions,

◮

= ∅.

3 / 25

Interactive Markov Chain

Interactive Markov Chain An Interactive Markov Chain M is a tuple (S, Actτ, , , ¯ s) where

◮ ¯

s ∈ S is the initial state,

◮ S is a finite but non-empty set of states, ◮ Actτ = Act .

∪ {τ} is a set of actions including the internal action τ,

◮

⊂ S × Actτ × S is a finite set of transitions,

◮

⊂ S × R>0 × S is a finite set of Markovian transitions.

4 / 25

Example

s2 s3 µ s1 s0 s4 s5 β α 2 6

1 3 2 3

5 / 25

Early Weak Bisimilarity

r1 r2 r1 r2 r1 r2 s4 s5 s3 s1 ≈ s0 ≈ s2 (a) (b) (c) α τ

1 3 2 3

α

1 3 2 3

τ

1 3 2 3

α α

6 / 25

Early Weak Bisimilarity

Early Weak Bisimilarity A relation R ⊆ Dist(S) × Dist(S) is an early weak bisimulation

• ver M iff µ R ν implies:

◮ whenever µ θ

− → µ′, there exists a ν

θ

= ⇒ ν′ such that µ′ R ν′;

◮ whenever µ = 0≤i≤n pi · µi, there exists

ν

τ

= ⇒

0≤i≤n pi · νi such that µi R νi for each 0 ≤ i ≤ n

where

0≤i≤n pi = 1; ◮ symmetrically for ν.

s • ≈ r iff δs • ≈ δr

7 / 25

Early Weak Bisimilarity

Early Weak Bisimilarity A relation R ⊆ Dist(S) × Dist(S) is an early weak bisimulation

• ver M iff µ R ν implies:

◮ whenever µ θ

− → µ′, there exists a ν

θ

= ⇒ ν′ such that µ′ R ν′;

◮ whenever µ = 0≤i≤n pi · µi, there exists

ν

τ

= ⇒

0≤i≤n pi · νi such that µi R νi for each 0 ≤ i ≤ n

where

0≤i≤n pi = 1; ◮ symmetrically for ν.

s • ≈ r iff δs • ≈ δr µ θ − → µ′ iff µ′ =

• s∈Supp(µ)∧s

θ

− →µs µ(s) · µs

7 / 25

Early Weak Bisimilarity

Early Weak Bisimilarity A relation R ⊆ Dist(S) × Dist(S) is an early weak bisimulation

• ver M iff µ R ν implies:

◮ whenever µ θ

− → µ′, there exists a ν

θ

= ⇒ ν′ such that µ′ R ν′;

◮ whenever µ = 0≤i≤n pi · µi, there exists

ν

τ

= ⇒

0≤i≤n pi · νi such that µi R νi for each 0 ≤ i ≤ n

where

0≤i≤n pi = 1; ◮ symmetrically for ν.

{1 2 : s1, 1 2 : s2} =1 2δs1 + 1 2δs2 =2 3{1 4 : s1, 3 4 : s2} + 1 3δs1

8 / 25

Properties of • ≈

◮ Relation on distributions. ◮ •

≈ is strictly coarser than Weak Probabilistic Bisimulation by Segala.

◮ •

≈ is compositional.

◮ •

≈ is the coarsest compositional equivalence preserving trace distribution equivalence.

9 / 25

A piece of probabilistic program

print(“I am going to toss”); r = rand(); if r ≥ 1

2 then

print(“head”); else print(“tail”); end

10 / 25

A piece of probabilistic program

print(“I am going to toss”); r = rand(); if r ≥ 1

2 then

print(“head”); else print(“tail”); end s3 s4 s1 s2 s0 i

1 2 1 2

h t

10 / 25

Another piece of probabilistic program

r = rand(); if r ≥ 1

2 then

print(“I am going to toss”); print(“head”); else print(“I am going to toss”); print(“tail”); end

11 / 25

Another piece of probabilistic program

r = rand(); if r ≥ 1

2 then

print(“I am going to toss”); print(“head”); else print(“I am going to toss”); print(“tail”); end s3 s4 s1 s2 s5 s6 s′ τ

1 2 1 2

i i h t

11 / 25

The guesser

r5 r6 r3 r4 r1 r2 r0 i i h t Suc Suc

12 / 25

s0 r0 s2 r1 s1 r1 s3 r3 s3 r5 s2 r2 s1 r2 s2 r4 s4 r6 i i

1 2 1 2

h Suc

1 2 1 2

t Suc

13 / 25

s′

0 r0

s6 r0 s5 r0 s1 r2 s2 r2 s2 r1 s4 r4 s4 r6 s1 r1 s3 r3 s3 r5 τ

1 2 1 2

i i h Suc i i t Suc

14 / 25

s′

0 r0

s6 r0 s5 r0 s1 r2 s2 r2 s2 r1 s4 r4 s4 r6 s1 r1 s3 r3 s3 r5 τ

1 2 1 2

i i h Suc i i t Suc

15 / 25

Two Less Powerful Schedulers

Partial Information Schedulers

• L. De Alfaro. The verification of probabilistic systems under

memoryless partial-information policies is hard. Technical report, DTIC Document, 1999.

16 / 25

Two Less Powerful Schedulers

Partial Information Schedulers

• L. De Alfaro. The verification of probabilistic systems under

memoryless partial-information policies is hard. Technical report, DTIC Document, 1999. Distributed Schedulers Sergio Giro and Pedro R. D’Argenio. Quantitative model checking revisited: neither decidable nor approximable. In FORMATS, pages 179–194, Berlin, Heidelberg, 2007. Springer-Verlag.

16 / 25

Partial Information Schedulers

s1 s2 α α β α τ β

17 / 25

Distributed Schedulers

s A r1 µ α

18 / 25

Distributed Schedulers

s A r1 µ s A r2 µ α α

19 / 25

Late Weak Bisimilarity

Late Weak Bisimilarity A relation R ⊆ Dist(S) × Dist(S) is a late weak bisimulation

• ver M iff µ R ν implies:

◮ whenever µ θ

− → µ′, there exists a ν

θ

= ⇒ ν′ such that µ′ R ν′;

◮ if not −

→ µ , then there exists µ =

0≤i≤n pi · µi and

ν

τ

= ⇒

0≤i≤n pi · νi such that −

→ µi and µi R νi for each 0 ≤ i ≤ n where

0≤i≤n pi = 1; ◮ symmetrically for ν.

where − → µ if all states in µ have the same observable actions.

20 / 25

Late Weak Bisimilarity

Late Weak Bisimilarity A relation R ⊆ Dist(S) × Dist(S) is a late weak bisimulation

• ver M iff µ R ν implies:

◮ whenever µ θ

− → µ′, there exists a ν

θ

= ⇒ ν′ such that µ′ R ν′;

◮ if not −

→ µ , then there exists µ =

0≤i≤n pi · µi and

ν

τ

= ⇒

0≤i≤n pi · νi such that −

→ µi and µi R νi for each 0 ≤ i ≤ n where

0≤i≤n pi = 1; ◮ symmetrically for ν.

where − → µ if all states in µ have the same observable actions. s ≈

• r iff δs ≈
• δr

20 / 25

Examples

r1 r2 r1 r2 r1 r2 s4 s5 s3 s1 ≈

• s0

≈

• s2

α τ

1 3 2 3

α

1 3 2 3

τ

1 3 2 3

α α

21 / 25

Examples

r2 r3 r1 r2 r3 r1 µ r1 s0 ≈

• s2

s3 ν s1 β α

1 3 2 3

τ

1 3 2 3

β α α β

22 / 25

Properties of ≈

• ◮ Relation on distributions.

◮ ≈

• is strictly coarser than •

≈.

◮ •

≈ is compositional w.r.t. to partial information and distributed schedulers.

◮ •

≈ is the coarsest compositional equivalence preserving trace distribution equivalence w.r.t. partial information and distributed schedulers.

23 / 25

Conclusion and Future Work

◮ Efficient Decision Algorithm (Currently exponential). ◮ Logical Characterization. ◮ . . ..

24 / 25

Thank You Q& A

25 / 25