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Statistical probabilistic model checking

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### Statistical probabilistic model checking

Håkan L. S. Younes

Carnegie Mellon University

(now at Google Inc.)

Introduction

- Model-independent approach to probabilistic model checking
- Relies on simulation and statistical sampling
- Wrong answer possible, but can be bounded (probabilistically)
- Low memory requirements (can handle large/infinite models)
- Trivially parallelizable (distributed sampling gives linear speedup)

- Topics covered in this talk:
- Error control
- Hypothesis testing vs. estimation
- Dealing with unbounded properties/infinite trajectories

Statistical probabilistic model checking

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

- Example: tandem queuing network

arrive

route

depart

q1

q2

“The probability is at least 0.1 that both queuesbecome full within 5 minutes”

Statistical probabilistic model checking

Probabilistic temporal logic (PCTL, CSL)

- Standard logic operators: , , …
- Probabilistic operator: ≥ []
- Holds in state s iff probability is at least for paths satisfying and starting in s

- Bounded until: ≤T
- Holds over path iff becomes true along within time T, and is true until then

- Unbounded until:
- Holds over path iff becomes true eventually along , and is true until then

Statistical probabilistic model checking

Property examples

- “The probability is at least 0.1 that both queues become full within 5 minutes”
- ≥0.1[≤5full1 full2]

- “The probability is at most 0.05 that the second queue becomes full before the first queue”
- ≤0.05[ full1full2]

Statistical probabilistic model checking

The problem (in detail)

- Before we propose a solution, we need to fully define the problem
- Possible outcomes of model-checking algorithm
- Ideal vs. realistic error control

Statistical probabilistic model checking

Possible outcomes of model-checking algorithm

- Given a state s and a formula , a model-checking algorithm A can:
- Accept as true in s (s)
- Reject as false in s (s)
- Return an undecided result (sI)

- An error occurs if:
- A rejects when is true (false negative)
- A accepts when is false (false positive)

- Note: an undecided result is not an error, but still not desirable

Statistical probabilistic model checking

Ideal error control

- Bound the probability of false negatives/positives and undecided results under all circumstances
- Bound on false negatives: Pr[s|s ]
- Bound on false positives: Pr[s|s ]
- Bound on undecided results: Pr[sI]

- If , , and are all low, then model-checking algorithm A produces a correct result with high probability

Statistical probabilistic model checking

Unrealistic expectations

- Ideal error control for verifying probabilistic formula ≥ [] in state s

False negatives

1

1 –

1 – –

s≥ []

s≥ []

Probability of acceptingP≥[] as true in s

Undecided

+

p

False positives

Actual probability of holding

Statistical probabilistic model checking

Relaxing the problem

- Indifference region of width 2 centered around probability thresholds
- Probabilistic operator: ≥ []
- Holds in state s if probability is at least + for paths satisfying and starting in s
- Does not hold if probability is at most − for paths satisfying and starting in s
- “Too close to call” if probability is within distance of (indifference)

- Essentially three-valued logic, but we care only about true and false

Statistical probabilistic model checking

Error control for relaxed problem

- Option 1: bound the probability of false positives/negatives outside of the indifference region; no undecided results
- Bound on false negatives: Pr[s|s]
- Bound on false positives: Pr[s|s]
- No undecided results: = 0Pr[sI] = 0

- Option 2: bound the probability of undecided results outside of the indifference region; low error probability under all circumstances
- Bound on false negatives: Pr[s|s ]
- Bound on false positives: Pr[s|s ]
- Bound on undecided results: Pr[sI|(s) (s)]

Statistical probabilistic model checking

Realistic error control—no undecided results

- Error control for verifying probabilistic formula ≥ [] in state s

False negatives

1

1 –

s≥ []

s≥ []

Probability of acceptingP≥[] as true in s

High error probabilityin indifference region

p

−

+

False positives

Actual probability of holding

Statistical probabilistic model checking

Realistic error control—with undecided results

- Error control for verifying probabilistic formula ≥ [] in state s

Rejection probability

Acceptance probability

1

1 –

s≥ []

s≥ []

Probability of acceptingP≥[] as true in s

High undecided probability in indifference region

p

−

+

Actual probability of holding

−

Statistical probabilistic model checking

The solution

- Statistical sampling (hypothesis testing vs. estimation)
- Undecided results
- Avoiding infinite sample trajectories in simulation for unbounded until

Statistical probabilistic model checking

Verifying probabilistic properties—no undecided resultsYounes & Simmons (CAV’02, Information and Computation’06)

- Use acceptance sampling to verify ≥ [] in state s
- Test hypothesis H0: p + against hypothesis H1: p –
- Observation: verify over sample trajectories generated using simulation

Statistical probabilistic model checking

Acceptance sampling with fixed sample size

- Single sampling plan: n, c
- Generate n sample trajectories
- Accept H0: p + iff more than c paths satisfy
- Pick n and c such that:
- Probability of accepting H1 when H0 holds is at most
- Probability of accepting H0 when H1 holds is at most

- Sequential single sampling plan:
- Accept H0 after m < n observations if more than c observations are positive
- Accept H1 after m < n observations if at most k observations are positive and k + (n – m) ≤ c

Statistical probabilistic model checking

Graphical representation of sequential single sampling plan

Continue untilline is crossed

accept

c

continue

Number of positiveobservations

reject

n – c

n

Start here

Number of observations

Make observations

Statistical probabilistic model checking

Sequential probability ratio test (SPRT)Wald (Annals of Mathematical Statistics’45)

- More efficient than sequential single sampling plan
- After m observations, k positive, compute ratio:
- Accept H0: p + if ≤ ∕ (1 – )
- Accept H1: p – if ≥ (1 – )∕

- No fixed upper bound on sample size, but much smaller on average

Statistical probabilistic model checking

Graphical representation of SPRT

Continue untilline is crossed

accept

continue

Number of positiveobservations

reject

Start here

Number of observations

Make observations

Statistical probabilistic model checking

Statistical estimationHérault et al. (VMCAI’04)

- Estimate p with confidence interval of width 2
- Accept H0: p + iff center of confidence interval is at least
- Choosing sample size:

- Same as single sampling plan n, n + 1; never more efficient!

Statistical probabilistic model checking

Acceptance sampling with undecided resultsYounes (VMCAI’06)

- Simultaneous acceptance sampling plans
- H0: p against H1: p –
- H0: p + against H1: p

- Combining the results
- Accept ≥ [] if H0 and H0 are accepted
- Reject ≥ [] if H1 and H1 are accepted
- Undecided result otherwise

Statistical probabilistic model checking

Graphical representation of SPRT with undecided results

Continue untilline is crossed

accept

undecided

Number of positiveobservations

reject

continue

Start here

Number of observations

Make observations

Statistical probabilistic model checking

Unbounded until—avoiding infinite sample trajectoriesYounes (unpublished manuscript)

- Premature termination with probability pt after each state transition
- Ensures finite sample trajectories
- Change value of positive sample trajectory from 1 to (1 – pt)–||
- Inspired by Monte Carlo method for matrix inversion by Forsythe & Leibler (1950)

- Observations no longer 0 or 1: previous methods do not apply
- Use sequential estimation by Chow & Robbins (1965)
- Lower pt means fewer samples, by longer trajectories

- Note: Sen et al. (CAV’05) tried to handle unbounded until with termination probability, but flawed because observations are still 0 or 1

Statistical probabilistic model checking

Empirical evaluation

Statistical probabilistic model checking

Numerical vs. statistical (tandem queuing network)Younes et al. (TACAS’04)

P≥0.5(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

Statistical probabilistic model checking

Numerical vs. statistical (symmetric polling system)Younes et al. (TACAS’04)

serv1 P≥0.5(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

Statistical probabilistic model checking

= 10–2

= 0

15

Verification time (seconds)

10

5

0

14

14.1

14.2

14.3

14.4

14.5

Formula time bound (T)

Undecided results (symmetric polling system)Younes (VMCAI’06)serv1 P≥0.5[U ≤Tpoll1]

Statistical probabilistic model checking

Undecided results (symmetric polling system)

= = = 10–2

Statistical probabilistic model checking

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