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
Outline Introduction 1 Parametric Chemical Reaction Networks Signal Temporal Logic Verification: a statistical approach Bayesian Threshold Synthesis Problem 2 Definition Algorithm Test Case and Results 3 Conclusions and Future Works 4 L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 2 / 25
Introduction Parametric Chemical Reaction Networks Models: Parametric Chemical Reaction Networks Consider a Parametric Chemical Reaction Network (PCRN) as a tuple M = ( S , X , D , x 0 , R , Θ) α 1 = k i · X s · X i α 1 r 1 : S + I − − → 2 I N α 2 r 2 : I − − → R α 2 = k r · X i θ = ( θ 1 , . . . , θ k ) is the vector of (kinetic) parameters, taking values in a compact hyperrectangle Θ ⊂ R k a trajectory is a function x θ : T → D L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 3 / 25
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 : R n → R is a continuous function. the syntax is φ := ⊥ | ⊤ | µ | ¬ φ | φ ∨ φ | φ U [ T 1 , T 2 ] φ, (1) Eventually and Globally Operators F [ T 1 , T 2 ] φ ≡ ⊤ U [ T 1 , T 2 ] φ and G [ T 1 , T 2 ] φ ≡ ¬ F [ T 1 , T 2 ] ¬ φ Interpretation (Boolean semantics) ( x θ , 0 ) | = F [ 0 , 50 ] | X 1 − X 2 | > 10 L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 4 / 25
Introduction Verification: a statistical approach Verification Problem Which is the probability that a trajectory x θ generated by M θ satisfies φ ∈ STL ? P φ ( θ ) ≡ P ( φ | M θ ) := P ( { x θ ( t ) ∈ Path M θ | ( x θ , 0 ) | = φ } ) L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 5 / 25
Introduction Verification: a statistical approach Verification Problem Which is the probability that a trajectory x θ generated by M θ satisfies φ ∈ STL ? P φ ( θ ) ≡ P ( φ | M θ ) := P ( { x θ ( t ) ∈ Path M θ | ( x θ , 0 ) | = φ } ) Classical Verification compute or estimate the satisfaction probability P φ ( θ ) for a fixed θ L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 5 / 25
Introduction Verification: a statistical approach Verification Problem Which is the probability that a trajectory x θ generated by M θ satisfies φ ∈ STL ? P φ ( θ ) ≡ P ( φ | M θ ) := P ( { x θ ( t ) ∈ Path M θ | ( x θ , 0 ) | = φ } ) Parametric Verification Classical Verification compute or estimate the satisfaction compute or estimate the satisfaction probability P φ ( θ ) as a function of θ ∈ probability P φ ( θ ) for a fixed θ Θ . L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 5 / 25
Introduction Verification: a statistical approach Verification Problem Which is the probability that a trajectory x θ generated by M θ satisfies φ ∈ STL ? P φ ( θ ) ≡ P ( φ | M θ ) := P ( { x θ ( t ) ∈ Path M θ | ( x θ , 0 ) | = φ } ) Parametric Verification Classical Verification compute or estimate the satisfaction compute or estimate the satisfaction probability P φ ( θ ) as a function of θ ∈ probability P φ ( θ ) for a fixed θ Θ . Numerical approaches Numerical Integration ⇒ STATE SPACE EXPLOSION! L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 5 / 25
Introduction Verification: a statistical approach Verification Problem Which is the probability that a trajectory x θ generated by M θ satisfies φ ∈ STL ? P φ ( θ ) ≡ P ( φ | M θ ) := P ( { x θ ( t ) ∈ Path M θ | ( x θ , 0 ) | = φ } ) Parametric Verification Classical Verification compute or estimate the satisfaction compute or estimate the satisfaction probability P φ ( θ ) as a function of θ ∈ probability P φ ( θ ) for a fixed θ Θ . Statistical Approaches Statistical Model Checking (SMC) Smoothed Model Checking (SmMC) L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 6 / 25
Introduction Verification: a statistical approach Gaussian Processes Definition A random variable f ( θ ) , θ ∈ Θ is a GP f ∼ GP ( m , k ) ⇐ ⇒ ( f ( θ 1 ) , f ( θ 2 ) , . . . , f ( θ n )) ∼ N ( m , K ) where m = ( m ( θ 1 ; h 1 ) , m ( θ 2 ; h 1 ) , . . . , m ( θ n ; h 1 )) and K ij = k ( f ( θ i ) , f ( θ j ); h 2 ) Prediction , f ( θ ′ ) } ∼ N ( m ′ , K ′ ) { f ( θ 1 ) , . . . , f ( θ n ) � �� � f · K − 1 · f E ( f ( θ ′ )) = ( k ( θ 1 , θ ′ ) , . . . , k ( θ N , θ ′ , )) � �� � k var ( f ( θ ′ )) = k ( θ ′ , θ ′ ) − k · K − 1 · k T L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 7 / 25
Introduction Verification: a statistical approach Smoothed Model Checking Hypothesis: The reaction rate α j ( x , θ ) depends smoothly on θ and polynomially on x . Goal: Approximate θ → P φ ( θ ) with a surrogate model θ → ˜ P φ ( θ ) . Problem: We cannot apply GP directly. L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 8 / 25
Introduction Verification: a statistical approach Smoothed Model Checking Hypothesis: The reaction rate α j ( x , θ ) depends smoothly on θ and polynomially on x . Goal: Approximate θ → P φ ( θ ) with a surrogate model θ → ˜ P φ ( θ ) . Problem: We cannot apply GP directly. Idea Reconstructing a real-valued latent function f ( θ ) , which is related to P φ ( θ ) and which can be approximated through GP regression. � f ( θ ) ( P φ ( θ )) ⇐ ⇒ P φ ( θ ) = N ( 0 , 1 ) f ( θ ) = Ψ ���� −∞ probit L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 8 / 25
Introduction Verification: a statistical approach Smoothed Model Checking Hypothesis: The reaction rate α j ( x , θ ) depends smoothly on θ and polynomially on x . Goal: Approximate θ → P φ ( θ ) with a surrogate model θ → ˜ P φ ( θ ) . Problem: We cannot apply GP directly. Idea Reconstructing a real-valued latent function f ( θ ) , which is related to P φ ( θ ) and which can be approximated through GP regression. � f ( θ ) ( P φ ( θ )) ⇐ ⇒ P φ ( θ ) = N ( 0 , 1 ) f ( θ ) = Ψ ���� −∞ probit Statistical Surrogates Model From { P φ ( θ 1 ) , . . . , P φ ( θ n ) } we obtain ˜ P φ ( θ ) as a statistical surrogate models of P φ ( θ ) . � � ˜ P φ ( θ ) ∈ [ λ − , λ + ] we can calculate: p , mean, variance, etc. large training set ⇒ a more accurate model. L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 8 / 25
Bayesian Threshold Synthesis Problem Problem Definition Partitioning the parameter space Θ in three classes P α (positive), N α (negative) and U α (undefined) Threshold Synthesis Problem - ˇ Ceška et al. 1 P α = { θ ∈ Θ | P φ ( θ ) > α } N α = { θ ∈ Θ | P φ ( θ ) < α } U α = Θ \ ( P α ∪ N α ) , vol ( U α ) vol (Θ) < ǫ . 1 M. 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
Bayesian Threshold Synthesis Problem Problem Definition Partitioning the parameter space Θ in three classes P α (positive), N α (negative) and U α (undefined) Bayesian Threshold Synthesis Problem P α = { θ ∈ Θ | ˜ P φ ( θ ) > α } N α = { θ ∈ Θ | ˜ P φ ( θ ) < α } U α = Θ \ ( P α ∪ N α ) , vol ( U α ) vol (Θ) < ǫ . L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 10 / 25
Bayesian Threshold Synthesis Problem Problem Definition Partitioning the parameter space Θ in three classes P α (positive), N α (negative) and U α (undefined) Bayesian Threshold Synthesis Problem P α = { θ ∈ Θ | p (˜ P φ ( θ ) > α ) > δ } N α = { θ ∈ Θ | p (˜ P φ ( θ ) < α ) > δ } U α = Θ \ ( P α ∪ N α ) , vol ( U α ) vol (Θ) < ǫ . L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 11 / 25
Bayesian Threshold Synthesis Problem Definition Problem Definition Bayesian Threshold Synthesis Problem Lower and Upper Bound functions P α = { θ ∈ Θ | p (˜ P φ ( θ ) > α ) > δ } � � ˜ P φ ( θ ) < λ + ( θ, δ ) p > δ N α = { θ ∈ Θ | p (˜ P φ ( θ ) < α ) > δ } � � ˜ P φ ( θ ) > λ − ( θ, δ ) p > δ U α = Θ \ ( P α ∪ N α ) , vol ( U α ) vol (Θ) < ǫ . δ ∈ ( 0 , 1 ) is the confidence probability. L. Bortolussi, S. Silvetti TACAS 2018 April 19, 2018 12 / 25
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