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Statistical model checking for parameterized model t Delahaye 1 Paulin Fournier 1 Didier Lime 2 Beno Universit e de Nantes - LS2N, UMR 6004 - Nantes, France Ecole Centrale de Nantes - LS2N, UMR 6004 - Nantes, France SynCoP 2018

  1. Statistical model checking for parameterized model ıt Delahaye 1 Paulin Fournier 1 Didier Lime 2 Benoˆ Universit´ e de Nantes - LS2N, UMR 6004 - Nantes, France ´ Ecole Centrale de Nantes - LS2N, UMR 6004 - Nantes, France SynCoP 2018 Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 1 / 23

  2. Introduction Motivation Model-checking of pMC ◮ Suffers from the same problems as standard MC: state space explosion ◮ Produces large rational functions as the probability of satisfying a property ◮ Existing cannot handle large number of parameters In some cases (large models, large number of parameters), it could be beneficial to use approximation techniques such as Statistical Model Checking, but ◮ SMC is limited to models without non-determinism ◮ Attempts to extend SMC to models with non-det have limitations ◮ No precision / confidence intervals ◮ Cannot be easily adapted to parameters (dependent transition probabilities) Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 2 / 23

  3. Introduction Contribution We propose a parametric version of SMC for parameterized models (pMC). ◮ Computes an approximation of the probability of satisfying a property ◮ as a parametric function ◮ polynomial ◮ with parametric confidence intervals ◮ Allows to compute the value of the probability for all parameter values (with varying precision) Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 3 / 23

  4. Introduction Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 4 / 23

  5. Parametric Markov Chains and Properties Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 5 / 23

  6. Parametric Markov Chains and Properties Background - Properties Markov Chains Definition (Markov chain) A Markov chain (MC, for short) is a tuple M = ( S , s 0 , P ) where S is a denumerable set of states, s 0 ∈ S is the initial state and P : S × S → [0 , 1] is the transition probability function. ◮ Finite run: ρ = s 0 s 1 . . . s n s.t. P ( s i , s i +1 ) > 0 ◮ Γ( l ): set of all runs of length l in M ◮ Probability of a finite run: P M ( ρ ) = Π n i =1 P ( s i − 1 , s i ) Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 6 / 23

  7. Parametric Markov Chains and Properties Background - Properties Properties In the context of SMC, we only consider properties on bounded runs. Let r : Γ( l ) → R be a reward function. Reachability P M ( ♦ ≤ l s ). ρ | = ♦ ≤ l s , if there exists i ≤ l such that s i = s . Safety P M ( � = l E ). ρ | = � = l E , if for all i ≤ l , s i ∈ E . M ( r ) = � Expected reward E l M ( r ). E l ρ ∈ Γ( l ) P M ( ρ ) r ( ρ ) is the expected value of r on the runs of length l . Remark For any property ϕ ⊆ Γ( l ), P M ( ϕ ) = E l M ( ✶ ϕ ) ⇒ we focus on properties of the form E l M ( r ). Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 7 / 23

  8. Parametric Markov Chains and Properties Parametric Markov Chains Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 8 / 23

  9. Parametric Markov Chains and Properties Parametric Markov Chains Parametric Markov Chains (pMCs) Definition (Parametric Markov chain) A Parametric Markov chain is a tuple M = ( S , s 0 , P , X ) such that S is a finite set of states, s 0 ∈ S is the initial state, X is a finite set of parameters, and P : S × S → Poly ( X ) is a parametric transition probability function. If v ∈ R X is a valuation of the parameters, ◮ P v : transition probabilities under v : P v ( s , s ′ ) = P ( s , s ′ )( v ) ◮ v is valid if ( S , s 0 , P v ) is a MC ◮ M v = ( S , s 0 , P v ) ◮ Runs and probabilities are similar to MC, but parametric Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 9 / 23

  10. Parametric Markov Chains and Properties Parametric Markov Chains Example 1 q 1 p r 1 2 0 . 5 p 0 1 0 . 5 3 4 q r pMC M 1 Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 10 / 23

  11. Parametric Markov Chains and Properties Parametric Markov Chains Example 1 q 1 0 . 5 1 p r 0 . 5 1 2 1 2 0 . 5 0 . 5 p 0 . 5 0 0 1 1 0 . 5 0 . 5 3 4 3 4 q 0 . 5 r MC M v pMC M 1 1 for parameter valuation v such that v ( p ) = v ( q ) = 0 . 5 and v ( r ) = 0 Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 10 / 23

  12. Monte Carlo and pMCs Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 11 / 23

  13. Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 0 . 5 1 2 0 . 5 0 . 5 0 1 0 . 5 3 4 0 . 5 Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

  14. Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 0 . 5 1 2 0 . 5 0 . 5 0 1 0 . 5 3 4 0 . 5 ◮ Run n simulations ρ i of length l Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

  15. Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 ◮ Run n simulations ρ i of length l Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

  16. Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 ◮ Run n simulations ρ i of length l ◮ r ( ρ i ) = 1 if ρ i reaches 4 Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

  17. Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 r ( ρ 1 ) = 0 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 2 ) = 1 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 3 ) = 0 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 r ( ρ 4 ) = 0 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 r ( ρ 5 ) = 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 6 ) = 0 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 r ( ρ 7 ) = 0 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 8 ) = 1 ◮ Run n simulations ρ i of length l ◮ r ( ρ i ) = 1 if ρ i reaches 4 Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

  18. Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 r ( ρ 1 ) = 0 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 2 ) = 1 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 3 ) = 0 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 r ( ρ 4 ) = 0 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 r ( ρ 5 ) = 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 6 ) = 0 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 r ( ρ 7 ) = 0 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 8 ) = 1 ◮ Run n simulations ρ i of length l ◮ r ( ρ i ) = 1 if ρ i reaches 4 � r ( ρ i ) ◮ E l ⇒ Here, E 5 M ( r ) ∼ M ( r ) ∼ 0 . 375 (exact: 0.3125) n Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

  19. Monte Carlo and pMCs Intuition for pMCs q 1 p r 1 2 0 . 5 p 0 1 0 . 5 3 4 q r ◮ How to run simulations? Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

  20. Monte Carlo and pMCs Intuition for pMCs 0.33 — q 1 0.33 — p 0.33 — r 1 2 0 . 5 0.33 — p 0 1 0 . 5 3 4 0.33 — q 0.33 — r ◮ How to run simulations? Use a normalization function f (uniform?) → M f Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

  21. Monte Carlo and pMCs Intuition for pMCs 0.33 — q 1 ρ 1 = 0 · 1 · 1 · 2 · 2 · 2 0.33 — p 0.33 — r ρ 2 = 0 · 1 · 0 · 3 · 3 · 4 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 1 · 0 0.33 — p 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 4 1 ρ 6 = 0 · 3 · 3 · 3 · 4 · 4 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 3 4 0.33 — q ρ 8 = 0 · 1 · 2 · 2 · 2 · 2 0.33 — r ◮ How to run simulations? Use a normalization function f (uniform?) → M f Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

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