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Bayesian Statistical Parameter Synthesis for Linear Temporal Properties of Stochastic Models Luca Bortolussi 1 Simone Silvetti 2,3 1DMG, University of Trieste, Trieste, Italy lbortolussi@units.it 2DIMA, University of Udine, Udine, Italy 3Esteco

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Bayesian Statistical Parameter Synthesis for Linear Temporal Properties of Stochastic Models

Luca Bortolussi 1 Simone Silvetti 2,3

1DMG, University of Trieste, Trieste, Italy lbortolussi@units.it 2DIMA, University of Udine, Udine, Italy 3Esteco SpA, Area Science Park, Trieste, Italy simone.silvetti@gmail.com

24th International Conference on Tools and Algorithms for the Construction and Analysis of Systems

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 1 / 25

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Outline

1

Introduction Parametric Chemical Reaction Networks Signal Temporal Logic Verification: a statistical approach

2

Bayesian Threshold Synthesis Problem Definition Algorithm

3

Test Case and Results

4

Conclusions and Future Works

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 2 / 25

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Introduction Parametric Chemical Reaction Networks

Models: Parametric Chemical Reaction Networks

Consider a Parametric Chemical Reaction Network (PCRN) as a tuple M = (S, X, D, x0, R, Θ) r1 : S + I

α1

− − → 2I α1 = ki · Xs · Xi N r2 : I

α2

− − → R α2 = kr · Xi θ = (θ1, . . . , θk) is the vector of (kinetic) parameters, taking values in a compact hyperrectangle Θ ⊂ Rk a trajectory is a function xθ : T → D

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 3 / 25

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Introduction Signal Temporal Logic

The requirements: Signal Temporal Logic (STL)

Signal temporal logic is: a linear continuous time temporal logic. the atomic predicates are of the form µ(X):=[g(X) ≥ 0] where g : Rn → R is a continuous function. the syntax is φ := ⊥ | ⊤ | µ | ¬φ | φ ∨ φ | φU[T1,T2]φ, (1) Eventually and Globally Operators F[T1,T2]φ ≡ ⊤U[T1,T2]φ and G[T1,T2]φ ≡ ¬F[T1,T2]¬φ Interpretation (Boolean semantics) (xθ, 0) | = F[0,50]|X1 − X2| > 10

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 4 / 25

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Introduction Verification: a statistical approach

Verification

Problem Which is the probability that a trajectory xθ generated by Mθ satisfies φ ∈ STL? Pφ(θ) ≡ P(φ | Mθ) := P({xθ(t) ∈ PathMθ | (xθ, 0) | = φ})

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 5 / 25

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Introduction Verification: a statistical approach

Verification

Problem Which is the probability that a trajectory xθ generated by Mθ satisfies φ ∈ STL? Pφ(θ) ≡ P(φ | Mθ) := P({xθ(t) ∈ PathMθ | (xθ, 0) | = φ}) Classical Verification compute or estimate the satisfaction probability Pφ(θ) for a fixed θ

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 5 / 25

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Introduction Verification: a statistical approach

Verification

Problem Which is the probability that a trajectory xθ generated by Mθ satisfies φ ∈ STL? Pφ(θ) ≡ P(φ | Mθ) := P({xθ(t) ∈ PathMθ | (xθ, 0) | = φ}) Classical Verification compute or estimate the satisfaction probability Pφ(θ) for a fixed θ Parametric Verification compute or estimate the satisfaction probability Pφ(θ) as a function of θ ∈ Θ.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 5 / 25

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Introduction Verification: a statistical approach

Verification

Problem Which is the probability that a trajectory xθ generated by Mθ satisfies φ ∈ STL? Pφ(θ) ≡ P(φ | Mθ) := P({xθ(t) ∈ PathMθ | (xθ, 0) | = φ}) Classical Verification compute or estimate the satisfaction probability Pφ(θ) for a fixed θ Parametric Verification compute or estimate the satisfaction probability Pφ(θ) as a function of θ ∈ Θ. Numerical approaches Numerical Integration ⇒ STATE SPACE EXPLOSION!

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 5 / 25

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Introduction Verification: a statistical approach

Verification

Problem Which is the probability that a trajectory xθ generated by Mθ satisfies φ ∈ STL? Pφ(θ) ≡ P(φ | Mθ) := P({xθ(t) ∈ PathMθ | (xθ, 0) | = φ}) Classical Verification compute or estimate the satisfaction probability Pφ(θ) for a fixed θ Parametric Verification compute or estimate the satisfaction probability Pφ(θ) as a function of θ ∈ Θ. Statistical Approaches Statistical Model Checking (SMC) Smoothed Model Checking (SmMC)

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 6 / 25

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Introduction Verification: a statistical approach

Gaussian Processes

Definition A random variable f(θ), θ ∈ Θ is a GP f ∼ GP(m, k) ⇐ ⇒ (f(θ1), f(θ2), . . . , f(θn)) ∼ N(m, K) where m = (m(θ1; h1), m(θ2; h1), . . . , m(θn; h1)) and Kij = k(f(θi), f(θj); h2) Prediction {f(θ1), . . . , f(θn)

  • f

, f(θ′)} ∼ N(m′, K ′) E(f(θ′)) = (k(θ1, θ′), . . . , k(θN, θ′, ))

  • k

·K −1 · f var(f(θ′)) = k(θ′, θ′) − k · K −1 · kT

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 7 / 25

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Introduction Verification: a statistical approach

Smoothed Model Checking

Hypothesis: The reaction rate αj(x, θ) depends smoothly on θ and polynomially on x. Goal: Approximate θ → Pφ(θ) with a surrogate model θ → ˜ Pφ(θ). Problem: We cannot apply GP directly.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 8 / 25

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Introduction Verification: a statistical approach

Smoothed Model Checking

Hypothesis: The reaction rate αj(x, θ) depends smoothly on θ and polynomially on x. Goal: Approximate θ → Pφ(θ) with a surrogate model θ → ˜ Pφ(θ). Problem: We cannot apply GP directly. Idea Reconstructing a real-valued latent function f(θ), which is related to Pφ(θ) and which can be approximated through GP regression. f(θ) = Ψ

  • probit

(Pφ(θ)) ⇐ ⇒ Pφ(θ) = f(θ)

−∞

N(0, 1)

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 8 / 25

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Introduction Verification: a statistical approach

Smoothed Model Checking

Hypothesis: The reaction rate αj(x, θ) depends smoothly on θ and polynomially on x. Goal: Approximate θ → Pφ(θ) with a surrogate model θ → ˜ Pφ(θ). Problem: We cannot apply GP directly. Idea Reconstructing a real-valued latent function f(θ), which is related to Pφ(θ) and which can be approximated through GP regression. f(θ) = Ψ

  • probit

(Pφ(θ)) ⇐ ⇒ Pφ(θ) = f(θ)

−∞

N(0, 1) Statistical Surrogates Model From {Pφ(θ1), . . . , Pφ(θn)} we obtain ˜ Pφ(θ) as a statistical surrogate models of Pφ(θ). we can calculate: p

  • ˜

Pφ(θ) ∈ [λ−, λ+]

  • , mean, variance, etc.

large training set ⇒ a more accurate model.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 8 / 25

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Bayesian Threshold Synthesis Problem

Problem Definition

Partitioning the parameter space Θ in three classes Pα (positive), Nα (negative) and Uα (undefined) Threshold Synthesis Problem - ˇ Ceška et al.1 Pα = {θ ∈ Θ | Pφ(θ) > α} Nα = {θ ∈ Θ | Pφ(θ) < α} Uα = Θ \ (Pα ∪ Nα), vol(Uα)

vol(Θ) < ǫ.

  • 1M. ˇ

Ceška, F. Dannenberg, N. Paoletti, M. Kwiatkowska and L. Brim, Precise parameter synthesis for stochastic biochemical systems, Acta Informatica 56(6), 2017, 589 - 623.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 9 / 25

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Bayesian Threshold Synthesis Problem

Problem Definition

Partitioning the parameter space Θ in three classes Pα (positive), Nα (negative) and Uα (undefined) Bayesian Threshold Synthesis Problem Pα = {θ ∈ Θ | ˜ Pφ(θ) > α} Nα = {θ ∈ Θ | ˜ Pφ(θ) < α} Uα = Θ \ (Pα ∪ Nα), vol(Uα)

vol(Θ) < ǫ.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 10 / 25

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Bayesian Threshold Synthesis Problem

Problem Definition

Partitioning the parameter space Θ in three classes Pα (positive), Nα (negative) and Uα (undefined) Bayesian Threshold Synthesis Problem Pα = {θ ∈ Θ | p(˜ Pφ(θ) > α) > δ} Nα = {θ ∈ Θ | p(˜ Pφ(θ) < α) > δ} Uα = Θ \ (Pα ∪ Nα), vol(Uα)

vol(Θ) < ǫ.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 11 / 25

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Bayesian Threshold Synthesis Problem Definition

Problem Definition

Bayesian Threshold Synthesis Problem Pα = {θ ∈ Θ | p(˜ Pφ(θ) > α) > δ} Nα = {θ ∈ Θ | p(˜ Pφ(θ) < α) > δ} Uα = Θ \ (Pα ∪ Nα), vol(Uα)

vol(Θ) < ǫ.

δ ∈ (0, 1) is the confidence probability. Lower and Upper Bound functions p

  • ˜

Pφ(θ) < λ+(θ, δ)

  • > δ

p

  • ˜

Pφ(θ) > λ−(θ, δ)

  • > δ
  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 12 / 25

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Bayesian Threshold Synthesis Problem Definition

Problem Definition

Bayesian Threshold Synthesis Problem Pα = {θ ∈ Θ | p(˜ Pφ(θ) > α) > δ} Nα = {θ ∈ Θ | p(˜ Pφ(θ) < α) > δ} Uα = Θ \ (Pα ∪ Nα), vol(Uα)

vol(Θ) < ǫ.

δ ∈ (0, 1) is the confidence probability. Lower and Upper Bound functions p

  • ˜

Pφ(θ) < λ+(θ, δ)

  • > δ

p

  • ˜

Pφ(θ) > λ−(θ, δ)

  • > δ

Bayesian Threshold Synthesis Problem Pα = {θ ∈ Θ | λ−(θ, δ) > α} Nα = {θ ∈ Θ | λ+(θ, δ) < α} Uα = Θ \ (Pα ∪ Nα), vol(Uα)

vol(Θ) < ǫ.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 12 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm I - Grid

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm II - Training Set

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm III - Prediction

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm IV - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm V - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm VI - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm VII - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm VIII - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm IX - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm X - Tessellation

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XI - Active Learning Step

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XII - Update

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XIII - Update

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XIV - Update

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XV - Active Learning Step

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XVI- Update

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XVII - Update

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Algorithm XVIII - Update

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 13 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Problem: the guarantees

A guarantee Problem

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 14 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Problem: the guarantees

A guarantee Problem

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 15 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Problem: the guarantees

A guarantee Problem Solution Continuity of Gaussian Processes.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 16 / 25

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Bayesian Threshold Synthesis Problem Algorithm

Problem: the guarantees

A guarantee Problem global/local Lipschitz approach:

  • ˆ

p − L ∗ d

2

  • > α

Heuristic Approach: s ≪ GP length scale and δ = 0.99 Solution Continuity of Gaussian Processes.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 17 / 25

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Test Case and Results

Epidemic Model: SIR

SIR Model r1 : S + I

α1

− − → 2I α1 = ki · Xs · Xi N r2 : I

α2

− − → R α2 = kr · Xi Disease Extinction φ = (I > 0) U[100,120] (I = 0)

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 18 / 25

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Test Case and Results

Results

Bayesian Threshold Synthesis Problem volume tolerance: ǫ = 0.1 threshold: α = 0.1

Case ki × kr h-grid Time (sec) 1 [0.005, 0.3] × 0.05 0.0007 17.92 ± 2.61 2 0.12 × [0.005, 0.2] 0.0005 4.87 ± 0.01 3 [0.005, 0.3] × [0.005, 0.2] (0.003,0.002) 116.4 ± 4.06

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 19 / 25

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Test Case and Results

Results

Case ki × kr h-grid Time (sec) 1 [0.005, 0.3] × 0.05 0.0007 17.92 ± 2.61 2 0.12 × [0.005, 0.2] 0.0005 4.87 ± 0.01 3 [0.005, 0.3] × [0.005, 0.2] (0.003,0.002) 116.4 ± 4.06

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 20 / 25

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Test Case and Results

Results

Case ki × kr h-grid Time (sec) 1 [0.005, 0.3] × 0.05 0.0007 17.92 ± 2.61 2 0.12 × [0.005, 0.2] 0.0005 4.87 ± 0.01 3 [0.005, 0.3] × [0.005, 0.2] (0.003,0.002) 116.4 ± 4.06

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 21 / 25

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Test Case and Results

Results

Case ki × kr h-grid Time (sec) 1 [0.005, 0.3] × 0.05 0.0007 17.92 ± 2.61 2 0.12 × [0.005, 0.2] 0.0005 4.87 ± 0.01 3 [0.005, 0.3] × [0.005, 0.2] (0.003,0.002) 116.4 ± 4.06

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 22 / 25

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Conclusions and Future Works

Conclusions

Bayesian version of the Threshold Synthesis Problem Smoothed Model Checking + Active Learning Approach good performance in terms of execution time w.r.t [ ˇ Ceška et al.]2, retaining good accuracy at the price of having only statistical guarantees.

  • 2M. ˇ

Ceška, F. Dannenberg, N. Paoletti, M. Kwiatkowska and L. Brim, Precise parameter synthesis for stochastic biochemical systems, Acta Informatica 56(6), 2017, 589 - 623.

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 23 / 25

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Conclusions and Future Works

Future Works

leveraging GPU Computing adaptive grid approach to tessellate the parameter space use GP reconstruction tailored for grid dataset combined approach with numerical methods of [ ˇ Ceška et al.]

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 24 / 25

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Conclusions and Future Works

  • L. Bortolussi, S. Silvetti

TACAS 2018 April 19, 2018 25 / 25