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Statistical Probabilistic Model Checking

Statistical Probabilistic Model Checking. Håkan L. S. Younes Carnegie Mellon University. Introduction. Model checking for stochastic processes Stochastic discrete event systems Probabilistic time-bounded properties Model independent approach Discrete event simulation

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Statistical Probabilistic Model Checking

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  1. StatisticalProbabilistic Model Checking Håkan L. S. Younes Carnegie Mellon University

  2. Introduction • Model checking for stochastic processes • Stochastic discrete event systems • Probabilistic time-bounded properties • Model independent approach • Discrete event simulation • Statistical hypothesis testing

  3. Example:Tandem Queuing Network arrive route depart q1 q2 q1 = 0q2 = 0 q1 = 0q2 = 0 q1 = 1q2 = 0 q1 = 1q2 = 0 q1 = 2q2 = 0 q1 = 2q2 = 0 q1 = 1q2 = 1 q1 = 1q2 = 1 q1 = 1q2 = 0 q1 = 1q2 = 0 t = 0 t = 1.2 t = 3.7 t = 3.9 t = 5.5 With both queues empty, is the probability less than 0.5that both queues become full within 5 seconds?

  4. Probabilistic Model Checking • Given a model M, a state s, and a property , does  hold in s for M? • Model: stochastic discrete event system • Property: probabilistic temporal logic formula

  5. Continuous Stochastic Logic (CSL) • State formulas • Truth value is determined in a single state • Path formulas • Truth value is determined over a path Discrete-time analogue: PCTL

  6. State Formulas • Standard logic operators: , 1  2, … • Probabilistic operator: P≥ () • Holds in state s iff probability is at least  that  holds over paths starting in s • P< ()  P≥1– ()

  7. Path Formulas • Until: 1U≤T 2 • Holds over path iff 2 becomes true in some state along  before time T, and 1 is true in all prior states

  8. CSL Example • With both queues empty, is the probability less than 0.5 that both queues become full within 5 seconds? • State: q1 = 0  q2 = 0 • Property: P<0.5(true U≤5 q1 = 2  q2 = 2)

  9. Model Checking Probabilistic Time-Bounded Properties • Numerical Methods • Provide highly accurate results • Expensive for systems with many states • Statistical Methods • Low memory requirements • Adapt to difficulty of problem (sequential) • Expensive if high accuracy is required

  10. Statistical Solution Method [Younes & Simmons 2002] • Use discrete event simulation to generate sample paths • Use acceptance sampling to verify probabilistic properties • Hypothesis: P≥ () • Observation: verify  over a sample path Not estimation!

  11. Error Bounds • Probability of false negative: ≤  • We say that  is false when it is true • Probability of false positive: ≤  • We say that  is true when it is false

  12. Performance of Test 1 –  Probability of acceptingP≥() as true   Actual probability of  holding

  13. False negatives False positives Ideal Performance of Test 1 –  Unrealistic! Probability of acceptingP≥() as true   Actual probability of  holding

  14. False negatives Indifference region p1 p0 False positives Realistic Performance of Test 2 1 –  Probability of acceptingP≥() as true   Actual probability of  holding

  15. SequentialAcceptance Sampling [Wald 1945] True, false, or another observation?

  16. Number of positiveobservations Number of observations Graphical Representation of Sequential Test

  17. Number of positiveobservations Number of observations Graphical Representation of Sequential Test • We can find an acceptance line and a rejection line given , , , and  acceptance line Continue untilline is crossed accept continue Verify  oversample paths rejection line Start here reject

  18. Special Case • p0 = 1 and p1 = 1 – 2 • Reject at first negative observation • Accept at stage m if p1m ≤  • Sample size at most dlog  / log p1e • “Five nines”: p1 = 1 – 10–5

  19.  a 1 2 … … 1 − a Case Study:Tandem Queuing Network • M/Cox2/1 queue sequentially composed with M/M/1 queue • Each queue has capacity n • State space of size O(n2)

  20. Tandem Queuing Network (results) [Younes et al. 2004] P≥0.5(true U≤T full) 106 105 104  = 10−6 = = 10−2 = 0.5·10−2 103 Verification time (seconds) 102 101 100 10−1 10−2 101 102 103 104 105 106 107 108 109 1010 1011 Size of state space

  21. Tandem Queuing Network (results) [Younes et al. 2004] P≥0.5(true U≤T full) 106 105 104  = 10−6 = = 10−2 = 0.5·10−2 103 Verification time (seconds) 102 101 100 10−1 10−2 101 102 103 104 T

  22. … Server Case Study:Symmetric Polling System • Single server, n polling stations • Stations are attended in cyclic order • Each station can hold one message • State space of size O(n·2n)     Polling stations

  23. Symmetric Polling System (results) [Younes et al. 2004] serv1  P≥0.5(true U≤T poll1) 106 105 104  = 10−6 = = 10−2 = 0.5·10−2 103 Verification time (seconds) 102 101 100 10−1 10−2 102 104 106 108 1010 1012 1014 Size of state space

  24. Symmetric Polling System (results) [Younes et al. 2004] serv1  P≥0.5(true U≤T poll1) 106 105 104  = 10−6 = = 10−2 = 0.5·10−2 103 Verification time (seconds) 102 101 100 10−1 10−2 101 102 103 T

  25. Symmetric Polling System (results) [Younes et al. 2004] serv1  P≥0.5(true U≤T poll1) 102 n = 10 T = 40 101 Verification time (seconds) ==10−10 100 ==10−8 ==10−6 ==10−4 10−1 ==10−2 (=10−6) 10−4 10−3 10−2 

  26. Tandem Queuing Network: Distributed Sampling • Use multiple machines to generate samples • m1: Pentium IV 3GHz • m2: Pentium III 733MHz • m3: Pentium III 500MHz

  27. Summary • Acceptance sampling can be used to verify probabilistic properties of systems • Sequential acceptance sampling adapts to the difficulty of the problem • Statistical methods are easy to parallelize

  28. Other Research • Failure trace analysis • “failure scenario” [Younes & Simmons 2004a] • Planning/Controller synthesis • CSL goals [Younes & Simmons 2004a] • Rewards (GSMDPs) [Younes & Simmons 2004b]

  29. Tools • Ymer • Statistical probabilistic model checking • Tempastic-DTP • Decision theoretic planning with asynchronous events

  30. References Wald, A. 1945. Sequential tests of statistical hypotheses. Ann. Math. Statist. 16: 117-186. Younes, H. L. S., M. Kwiatkowska, G. Norman, and D. Parker. 2004. Numerical vs. statistical probabilistic model checking: An empirical study. In Proc. TACAS-2004. Younes, H. L. S., R. G. Simmons. 2002. Probabilistic verification of discrete event systems using acceptance sampling. In Proc. CAV-2002. Younes, H. L. S., R. G. Simmons. 2004a. Policy generation for continuous-time stochastic domains with concurrency. In Proc. ICAPS-2004. Younes, H. L. S., R. G. Simmons. 2004b. Solving generalized semi-Markov decision processes using continuous phase-type distributions. In Proc. AAAI-2004.

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