Probabilistic Model Checking MEFISTO- 11/2003 Igor Melatti, Finite - PowerPoint PPT Presentation

MEFISTO- 11/2003 Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier Giuseppe Della Penna Benedetto Intrigila Igor Melatti Dip. di Informatica, Universit` a di L ’Aquila Enrico Tronci Marisa Venturini Zilli Dip. di Informatica, Universit` a di Roma “La Sapienza” Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –1–

Probabilistic Model Checking MEFISTO- 11/2003 Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-a–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis • Given the description of a Markov Chain, it verifies a PCTL property Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-b–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis • Given the description of a Markov Chain, it verifies a PCTL property • PCTL: Probabilistic CTL – A → P ≥ α [ true UB ] – A → P >α [ true U ≤ k B ] Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-c–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis • Given the description of a Markov Chain, it verifies a PCTL property • PCTL: Probabilistic CTL – A → P ≥ α [ true UB ] – A → P >α [ true U ≤ k B ] • Very few available probabilistic model checkers Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-d–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis • Given the description of a Markov Chain, it verifies a PCTL property • PCTL: Probabilistic CTL – A → P ≥ α [ true UB ] – A → P >α [ true U ≤ k B ] • Very few available probabilistic model checkers – PRISM Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-e–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis • Given the description of a Markov Chain, it verifies a PCTL property • PCTL: Probabilistic CTL – A → P ≥ α [ true UB ] – A → P >α [ true U ≤ k B ] • Very few available probabilistic model checkers – PRISM – Two Towers Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-f–

Probabilistic Model Checking MEFISTO- 11/2003 • Markov Chain analysis • Given the description of a Markov Chain, it verifies a PCTL property • PCTL: Probabilistic CTL – A → P ≥ α [ true UB ] – A → P >α [ true U ≤ k B ] • Very few available probabilistic model checkers – PRISM – Two Towers – FHP-Mur ϕ (new) Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –2-g–

PRISM MEFISTO- 11/2003 PRISM Probabilistic Symbolic Model Checker Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –3–

PRISM MEFISTO- 11/2003 PRISM Probabilistic Symbolic Model Checker • State-of-the-art probabilistic model checker Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –3-a–

PRISM MEFISTO- 11/2003 PRISM Probabilistic Symbolic Model Checker • State-of-the-art probabilistic model checker • Implicit verification algorithm (MTBDD-based) Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –3-b–

PRISM MEFISTO- 11/2003 PRISM Probabilistic Symbolic Model Checker • State-of-the-art probabilistic model checker • Implicit verification algorithm (MTBDD-based) • It allows to verify three types of Markov Chains: DTMC, with PCTL are the “classic” ones, here we will deal with these only MDP, with PCTL non-determinism added CTMC, with CSL continuous time managed Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –3-c–

PRISM MEFISTO- 11/2003 PRISM Probabilistic Symbolic Model Checker • State-of-the-art probabilistic model checker • Implicit verification algorithm (MTBDD-based) • It allows to verify three types of Markov Chains: DTMC, with PCTL are the “classic” ones, here we will deal with these only MDP, with PCTL non-determinism added CTMC, with CSL continuous time managed • Three verification modalities: – totally MTBDD-based (calculating fix points) – algebraic (on the Markov Chain transition matrix) – an hybrid modality between the two previous ones Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –3-d–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ • Explicit probabilistic model checker Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4-a–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ • Explicit probabilistic model checker – symbolic and explicit verification are not comparable in non-probabilistic model checking Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4-b–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ • Explicit probabilistic model checker – symbolic and explicit verification are not comparable in non-probabilistic model checking – we will show that this holds also for probabilistic model checking Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4-c–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ • Explicit probabilistic model checker – symbolic and explicit verification are not comparable in non-probabilistic model checking – we will show that this holds also for probabilistic model checking • Mur ϕ modified in the input language and in the verification algorithm Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4-d–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ • Explicit probabilistic model checker – symbolic and explicit verification are not comparable in non-probabilistic model checking – we will show that this holds also for probabilistic model checking • Mur ϕ modified in the input language and in the verification algorithm • Specialized in verifying a particular type of PCTL properties – P ≤ α [ true U ≤ k φ ] ≡ Pr (( ∃ i ≤ k φ ( π ( i ))) | π ∈ Path ( M )) ≤ α – φ is a boolean function defined on states Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4-e–

FHP-Mur ϕ MEFISTO- 11/2003 • FiniteHorizonProbabilistic-Mur ϕ • Explicit probabilistic model checker – symbolic and explicit verification are not comparable in non-probabilistic model checking – we will show that this holds also for probabilistic model checking • Mur ϕ modified in the input language and in the verification algorithm • Specialized in verifying a particular type of PCTL properties – P ≤ α [ true U ≤ k φ ] ≡ Pr (( ∃ i ≤ k φ ( π ( i ))) | π ∈ Path ( M )) ≤ α – φ is a boolean function defined on states – If φ models an error, we are asking if the error probability is acceptable Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –4-f–

FHP-Mur ϕ ’s input language MEFISTO- 11/2003 • We added finite precision real numbers and probabilities: Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –5–

FHP-Mur ϕ ’s input language MEFISTO- 11/2003 • We added finite precision real numbers and probabilities: – on the initial states (initial probability distribution) ∗ n initial states with probability p 1 , . . . , p n ∗ � n i =1 p i = 1 has always to hold Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –5-a–

FHP-Mur ϕ ’s input language MEFISTO- 11/2003 • We added finite precision real numbers and probabilities: – on the initial states (initial probability distribution) ∗ n initial states with probability p 1 , . . . , p n ∗ � n i =1 p i = 1 has always to hold – on the rules (they now define a Markov Chain transition function) ∗ s 1 , . . . , s n successor states of s with probability p 1 , . . . , p n � n ∗ i =1 p i = 1 has always to hold Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –5-b–

FHP-Mur ϕ ’s input language MEFISTO- 11/2003 • We added finite precision real numbers and probabilities: – on the initial states (initial probability distribution) ∗ n initial states with probability p 1 , . . . , p n ∗ � n i =1 p i = 1 has always to hold – on the rules (they now define a Markov Chain transition function) ∗ s 1 , . . . , s n successor states of s with probability p 1 , . . . , p n � n ∗ i =1 p i = 1 has always to hold – on the invariant to be verified ∗ property to be verified: is the probability of the event “an error state (i.e., not satisfying the invariant) is reachable within a given number of steps” less than a given α ? ∗ i.e., does Pr ( ∃ i ≤ k : φ ( π ( i )) | π is a Markov Chain path ) ≤ α hold? ∗ equivalent to the PCTL formula P ≤ α [ true U ≤ k φ ] Igor Melatti, Finite Horizon Analysis of Markov Chains with the Mur ϕ Verifier –5-c–