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

Late Weak Bisimilarity for Markov Automata Christian Eisentraut 1 Jens Chr. Godskesen 2 Holger Hermanns 1 Lei Song 3 , 1 Lijun Zhang 4 , 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 , Act τ , s ) where , ¯ , s ∈ S is the initial state, ◮ ¯ ◮ S is a finite but non-empty set of states, . ◮ Act τ = Act ∪ { τ } is a set of actions including the internal action τ , ⊂ S × Act τ × 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 , Act τ , s ) , ¯ , where s ∈ S is the initial state, ◮ ¯ ◮ S is a finite but non-empty set of states, . ◮ Act τ = Act ∪ { τ } is a set of actions including the internal action τ , ⊂ S × Act τ × 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 , Act τ , s ) , ¯ , where s ∈ S is the initial state, ◮ ¯ ◮ S is a finite but non-empty set of states, . ◮ Act τ = Act ∪ { τ } is a set of actions including the internal action τ , ⊂ S × Act τ × S is a finite set of transitions, ◮ ⊂ S × R > 0 × S is a finite set of Markovian transitions. ◮ 4 / 25

Example s 0 6 2 β α s 1 s 4 s 5 µ 1 2 3 3 s 3 s 2 5 / 25

Early Weak Bisimilarity s 1 s 0 s 2 ≈ �≈ α τ α s 3 2 1 3 3 τ s 4 s 5 2 1 3 3 1 2 3 3 α α r 1 r 2 r 1 r 2 r 1 r 2 (a) (b) (c) 6 / 25

Early Weak Bisimilarity Early Weak Bisimilarity A relation R ⊆ Dist ( S ) × Dist ( S ) is an early weak bisimulation over M iff µ R ν implies: ⇒ ν ′ such that µ ′ R ν ′ ; ◮ whenever µ θ θ → µ ′ , there exists a ν − = 0 ≤ i ≤ n p i · µ i , there exists ◮ whenever µ = � τ 0 ≤ i ≤ n p i · ν i such that µ i R ν i for each 0 ≤ i ≤ n = ⇒ � ν 0 ≤ i ≤ n p i = 1; where � ◮ symmetrically for ν . s • ≈ r iff δ s • ≈ δ r 7 / 25

Early Weak Bisimilarity Early Weak Bisimilarity A relation R ⊆ Dist ( S ) × Dist ( S ) is an early weak bisimulation over M iff µ R ν implies: ⇒ ν ′ such that µ ′ R ν ′ ; ◮ whenever µ θ θ → µ ′ , there exists a ν − = 0 ≤ i ≤ n p i · µ i , there exists ◮ whenever µ = � τ 0 ≤ i ≤ n p i · ν i such that µ i R ν i for each 0 ≤ i ≤ n = ⇒ � ν 0 ≤ i ≤ n p i = 1; where � ◮ symmetrically for ν . s • ≈ r iff δ s • ≈ δ r → µ ′ iff µ ′ = µ θ µ ( s ) · µ s � − θ s ∈ Supp ( µ ) ∧ s − → µ s 7 / 25

Early Weak Bisimilarity Early Weak Bisimilarity A relation R ⊆ Dist ( S ) × Dist ( S ) is an early weak bisimulation over M iff µ R ν implies: ◮ whenever µ θ θ ⇒ ν ′ such that µ ′ R ν ′ ; − → µ ′ , there exists a ν = 0 ≤ i ≤ n p i · µ i , there exists ◮ whenever µ = � τ 0 ≤ i ≤ n p i · ν i such that µ i R ν i for each 0 ≤ i ≤ n ν = ⇒ � 0 ≤ i ≤ n p i = 1; where � ◮ symmetrically for ν . { 1 2 : s 1 , 1 2 : s 2 } = 1 2 δ s 1 + 1 2 δ s 2 = 2 3 { 1 4 : s 1 , 3 4 : s 2 } + 1 3 δ s 1 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 s 0 i print(“I am going to toss”); r = rand () ; if r ≥ 1 2 then 1 1 print(“head”); 2 2 else s 1 s 2 print(“tail”); end h t s 3 s 4 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 s ′ 0 τ r = rand () ; if r ≥ 1 2 then 1 1 print(“I am going to toss”); 2 2 print(“head”); s 5 s 6 else print(“I am going to toss”); i i print(“tail”); s 1 s 2 end h t s 3 s 4 11 / 25

The guesser r 0 i i r 1 r 2 t h r 3 r 4 Suc Suc r 5 r 6 12 / 25

s 2 � r 1 1 2 i 1 h Suc 2 s 1 � r 1 s 3 � r 3 s 3 � r 5 s 0 � r 0 t Suc s 2 � r 2 s 2 � r 4 s 4 � r 6 1 i 2 1 s 1 � r 2 2 13 / 25

s 1 � r 2 i s 5 � r 0 Suc 1 h i s 1 � r 1 s 3 � r 3 s 3 � r 5 2 τ s ′ 0 � r 0 t Suc s 2 � r 2 s 4 � r 4 s 4 � r 6 i 1 2 s 6 � r 0 i s 2 � r 1 14 / 25

s 1 � r 2 i s 5 � r 0 Suc 1 h i s 1 � r 1 s 3 � r 3 s 3 � r 5 2 τ s ′ 0 � r 0 t Suc s 2 � r 2 s 4 � r 4 s 4 � r 6 i 1 2 s 6 � r 0 i s 2 � r 1 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 s 1 s 2 β β α α α τ 17 / 25

Distributed Schedulers s � A r 1 α µ 18 / 25

Distributed Schedulers s � A r 1 s � A r 2 α α µ µ 19 / 25

Late Weak Bisimilarity Late Weak Bisimilarity A relation R ⊆ Dist ( S ) × Dist ( S ) is a late weak bisimulation over M iff µ R ν implies: ⇒ ν ′ such that µ ′ R ν ′ ; ◮ whenever µ θ θ → µ ′ , there exists a ν − = ◮ if not − → 0 ≤ i ≤ n p i · µ i and µ , then there exists µ = � 0 ≤ i ≤ n p i · ν i such that − → τ = ⇒ � µ i and µ i R ν i for each ν 0 ≤ i ≤ n where � 0 ≤ i ≤ n p i = 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 over M iff µ R ν implies: ⇒ ν ′ such that µ ′ R ν ′ ; ◮ whenever µ θ θ → µ ′ , there exists a ν − = ◮ if not − → 0 ≤ i ≤ n p i · µ i and µ , then there exists µ = � 0 ≤ i ≤ n p i · ν i such that − → τ = ⇒ � µ i and µ i R ν i for each ν 0 ≤ i ≤ n where � 0 ≤ i ≤ n p i = 1; ◮ symmetrically for ν . where − → µ if all states in µ have the same observable actions. • r iff δ s ≈ • δ r s ≈ 20 / 25

Examples s 1 s 0 s 2 • • ≈ ≈ α τ α s 3 2 1 3 3 τ s 4 s 5 1 2 3 3 2 1 3 3 α α r 1 r 2 r 1 r 2 r 1 r 2 21 / 25

Examples s 0 s 1 • ≈ τ β α ν 1 2 r 1 µ 3 3 s 2 s 3 1 2 3 3 β α α β r 2 r 3 r 1 r 2 r 3 r 1 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