On the Total Variation Distance of SMCs Giorgio Bacci, Giovanni - PowerPoint PPT Presentation

On the Total Variation Distance of SMCs Giorgio Bacci, Giovanni Bacci, Kim G. Larsen, Radu Mardare Aalborg University, Denmark 17 December 2014 - Bologna, Italy FOCUS seminars 1/28

Before to start... Given μ , ν : Σ → ℝ + measures on (X, Σ ) Total Variation Distance || μ - ν || = sup | μ (E) - ν (E)| E ∈ Σ 2/28

Before to start... Given μ , ν : Σ → ℝ + measures on (X, Σ ) Total Variation Distance || μ - ν || = sup | μ (E) - ν (E)| E ∈ Σ The largest possible difference that μ and ν assign to the same event 2/28

Outline • Semi-Markov Chains (SMCs) • Total Variation vs Model Checking of SMCs • An Approximation Algorithm • Concluding Remarks 3/28

semi-Markov Chains 1 1/3 1/3 s 0 s 2 s 1 1/3 2/3 1/3 1 s 3 s 4 1 4/28

semi-Markov Chains 1 1/3 1/3 s 0 s 2 s 1 p,r q p,r 1/3 2/3 1/3 1 s 3 s 4 q,r q,r 1 4/28

semi-Markov Chains 1 Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r q p,r 1/3 2/3 1/3 1 s 3 s 4 q,r q,r 1 4/28

semi-Markov Chains N(2,3) 1 Exp(3) Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r q p,r 1/3 2/3 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 4/28

semi-Markov Chains N(2,3) 1 Exp(3) Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r q p,r 1/3 2/3 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 Given an initial state, SMCs can be interpreted as “machines” that emit timed traces of states with a certain probability 4/28

Timed paths & Events t n-1 t 0 ... π : 𝕯 ( S 0 , R 0 , ... , R n-1 , S n ) ∈ s n-1 s n s 0 s 1 “probability that, starting from s , P[s]( 𝕯 ( S 0 , R 0 , ... , R n-1 , S n )) = the SMC emits a timed path with prefix in S 0 × R 0 × ... × R n-1 × S n ” 5/28

Timed paths & Events residence-time t n-1 t 0 ... π : 𝕯 ( S 0 , R 0 , ... , R n-1 , S n ) ∈ s n-1 s n s 0 s 1 “probability that, starting from s , P[s]( 𝕯 ( S 0 , R 0 , ... , R n-1 , S n )) = the SMC emits a timed path with prefix in S 0 × R 0 × ... × R n-1 × S n ” 5/28

Timed paths & Events Cylinder set residence-time (s i ∈ S i , t i ∈ R i and R i Borel set) t n-1 t 0 ... π : 𝕯 ( S 0 , R 0 , ... , R n-1 , S n ) ∈ s n-1 s n s 0 s 1 “probability that, starting from s , P[s]( 𝕯 ( S 0 , R 0 , ... , R n-1 , S n )) = the SMC emits a timed path with prefix in S 0 × R 0 × ... × R n-1 × S n ” 5/28

Probabilistic Trace Equiv. N(2,3) 1 Exp(3) Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r q p,r 1/3 2/3 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 6/28

Probabilistic Trace Equiv. N(2,3) 1 Exp(3) Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r p,r q 1/3 2/3 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 6/28

Probabilistic Trace Equiv. N(2,3) 1 Exp(3) Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r p,r q 1/3 2/3 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 P[ s 0 ]( 𝕯 ( , R 0 , ... , R n-1 , )) = P[ s 1 ]( 𝕯 ( , R 0 , ... , R n-1 , )) L n L 0 L 0 L n 6/28

Probabilistic Trace Equiv. N(2,3) 1 Exp(3) Exp(3) 1/3 1/3 s 0 s 2 s 1 p,r p,r q 1/3 2/3 1/3 1 U(2) U(2) s 3 s 4 q,r q,r Trace Cylinders (up to label equiv.) 1 P[ s 0 ]( 𝕯 ( , R 0 , ... , R n-1 , )) = P[ s 1 ]( 𝕯 ( , R 0 , ... , R n-1 , )) L n L 0 L 0 L n 6/28

Probabilistic Trace Equiv. N(2,3) 1 Exp(3) Exp(3) 1/3+ ε 1/3 s 0 s 2 s 1 p,r q p,r 1/3 2/3- ε 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 P[ s 0 ]( 𝕯 ( , ℝ , ,)) =1/3+ ε ≠ 1/3 = P[ s 1 ] ( 𝕯 ( , ℝ , ,)) p,r q p,r q 7/28

Probabilistic Trace Equiv. N(2,3) 1 Exp(3) Exp(3) 1/3+ ε 1/3 s 0 s 2 s 1 FRAGILE p,r q p,r 1/3 2/3- ε 1/3 1 U(2) U(2) s 3 s 4 q,r q,r 1 P[ s 0 ]( 𝕯 ( , ℝ , ,)) =1/3+ ε ≠ 1/3 = P[ s 1 ] ( 𝕯 ( , ℝ , ,)) p,r q p,r q 7/28

Trace Pseudometric d(s,s’) = sup |P[s](E) - P[s’](E)| E ∈ σ ( 𝓤 ) σ -algebra generated from Trace Cylinders 8/28

Trace Pseudometric d(s,s’) = sup |P[s](E) - P[s’](E)| E ∈ σ ( 𝓤 ) σ -algebra generated from Trace Cylinders It’s a Behavioral Distance! d(s,s’) = 0 iff s ≈ s’ T 8/28

Distance = Approx. Error 9/28

? Distance = Approx. Error 9/28

? Distance = Approx. Error Example: Probabilistic Model Checking probability of M 0 satisfying φ P[M 0 ]({s ⊨ φ }) 0 1 9/28

? Distance = Approx. Error Example: Probabilistic Model Checking probability of M 0 M 1 satisfying φ P[M 1 ]({s ⊨ φ }) P[M 0 ]({s ⊨ φ }) 0 1 |P[M 0 ]({s ⊨ φ }) - P[M 0 ]({s ⊨ φ })| 9/28

? Distance = Approx. Error Example: Probabilistic Model Checking ε probability of M 0 M 1 satisfying φ P[M 1 ]({s ⊨ φ }) P[M 0 ]({s ⊨ φ }) 0 1 |P[M 0 ]({s ⊨ φ }) - P[M 0 ]({s ⊨ φ })| 9/28

? Distance = Approx. Error Example: Probabilistic Model Checking ε probability of M 0 M 1 satisfying φ P[M 1 ]({s ⊨ φ }) P[M 0 ]({s ⊨ φ }) distance bounds 0 1 ε ε the abs. error |P[M 0 ]({s ⊨ φ }) - P[M 0 ]({s ⊨ φ })| ≤ ε 9/28

Trace Distance vs. Model Checking (i.e., does it provide a good approximation error?) 10/28

Model Checking SMCs i.e., measuring the likelihood that a a linear real-time property is satisfied by the SMC SMC ⊨ Linear Real-time Spec. 11/28

Model Checking SMCs i.e., measuring the likelihood that a a linear real-time property is satisfied by the SMC SMC ⊨ Linear Real-time Spec. represented as Metric Temporal Logic formulas 11/28

Model Checking SMCs i.e., measuring the likelihood that a a linear real-time property is satisfied by the SMC SMC ⊨ Linear Real-time Spec. ... or languages represented as recognized Metric Temporal Logic by Timed Automata formulas 11/28

Model Checking SMCs i.e., measuring the likelihood that a a linear real-time property is satisfied by the SMC a proper measurable set! SMC ⊨ Linear Real-time Spec. ... or languages represented as recognized Metric Temporal Logic by Timed Automata formulas 11/28

(Alur-Henzinger) Metric Temporal Logic Next Until φ ≔ p | ⊥ | φ→φ | X φ | φ U φ I I (*) I ⊆ ℝ closed interval with rational endpoints 12/28

(Alur-Henzinger) Metric Temporal Logic Next Until φ ≔ p | ⊥ | φ→φ | X φ | φ U φ I I (*) I ⊆ ℝ closed interval with rational endpoints ψ within time t ∈ I ... t i-1 ∈ I t 0 + + φ U ψ I π : ⊨ ... φ φ ψ φ 12/28

MTL distance (max error w.r.t. MTL properties) set of timed paths that satisfy φ MTL(s,s’) = sup |P[s]({ π ⊨ φ }) - P[s’]({ π ⊨ φ })| φ ∈ MTL 13/28

MTL distance (max error w.r.t. MTL properties) measurable set of timed paths in σ ( 𝓤 ) that satisfy φ MTL(s,s’) = sup |P[s]({ π ⊨ φ }) - P[s’]({ π ⊨ φ })| φ ∈ MTL Relation with Trace Distance MTL(s,s’) ≤ d(s,s’) = sup |P[s](E) - P[s’](E)| E ∈ σ ( 𝓤 ) 13/28

MTL distance (max error w.r.t. MTL properties) measurable set of timed paths in σ ( 𝓤 ) that satisfy φ MTL(s,s’) = sup |P[s]({ π ⊨ φ }) - P[s’]({ π ⊨ φ })| φ ∈ MTL Relation with Trace Distance = MTL(s,s’) ≤ d(s,s’) = sup |P[s](E) - P[s’](E)| E ∈ σ ( 𝓤 ) 13/28

(Alur-Dill) (Muller)Timed Automata without invariants Clocks = {x,y} q x ≥ 1/4, {x} p,r Clock Guards ℓ 1 x ≥ 5, {x} g ≔ x ⋈ q | g ∧ g q p,r ℓ 0 for ⋈ ∈ {<, ≤ ,>, ≥ }, q ∈ ℚ y ≤ 1/2 x ≤ 1/2 ℓ 2 p,r , x<3, {y} accepted! p,r q q , 1/2 , 2 , 1/2 x=0 x=2 x=2.5 ( ℓ 0 , ) ( ℓ 2 , ) ( ℓ 1 , ) ... y=0 y=0 y=0.5 14/28

TA distance (max error w.r.t. timed regular properties) set of timed paths accepted by 𝓑 TA(s,s’) = sup |P[s]({ π ∈ L( 𝓑 )}) - P[s’]({ π ∈ L( 𝓑 )})| 𝓑 ∈ TA 15/28

TA distance (max error w.r.t. timed regular properties) measurable set of timed paths in σ ( 𝓤 ) accepted by 𝓑 TA(s,s’) = sup |P[s]({ π ∈ L( 𝓑 )}) - P[s’]({ π ∈ L( 𝓑 )})| 𝓑 ∈ TA Relation with Trace Distance TA(s,s’) ≤ d(s,s’) = sup |P[s](E) - P[s’](E)| E ∈ σ ( 𝓤 ) 15/28

TA distance (max error w.r.t. timed regular properties) measurable set of timed paths in σ ( 𝓤 ) accepted by 𝓑 TA(s,s’) = sup |P[s]({ π ∈ L( 𝓑 )}) - P[s’]({ π ∈ L( 𝓑 )})| 𝓑 ∈ TA Relation with Trace Distance = TA(s,s’) ≤ d(s,s’) = sup |P[s](E) - P[s’](E)| E ∈ σ ( 𝓤 ) 15/28

The theorem behind... For μ , ν : Σ → ℝ + finite measures on (X, Σ ) and F ⊆ Σ field such that σ (F)= Σ Representation Theorem || μ - ν || = sup | μ (E) - ν (E)| E ∈ F 16/28

The theorem behind... For μ , ν : Σ → ℝ + finite measures on (X, Σ ) and F ⊆ Σ field such that σ (F)= Σ Representation Theorem || μ - ν || = sup | μ (E) - ν (E)| E ∈ F F is much simpler than Σ , nevertheless it suffices to attain the supremum! 16/28

A series of characterizations MTL(s,s’) = MTL (s,s’) ¬U TA(s,s’) = DTA(s,s’) = 1-DTA(s,s’) = 1-RDTA(s,s’) 17/28

A series of characterizations max error w.r.t. φ ∈ MTL without Until MTL(s,s’) = MTL (s,s’) ¬U TA(s,s’) = DTA(s,s’) = 1-DTA(s,s’) = 1-RDTA(s,s’) 17/28

A series of characterizations max error w.r.t. φ ∈ MTL without Until MTL(s,s’) = MTL (s,s’) ¬U max error w.r.t. Deterministic TAs TA(s,s’) = DTA(s,s’) = 1-DTA(s,s’) = 1-RDTA(s,s’) 17/28