(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 16/28
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 17/28
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 ) 17/28
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 ) 17/28
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 18/28
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! 18/28
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’) 19/28
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’) 19/28
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’) 19/28
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’) max error w.r.t. single-clock DTAs 19/28
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. max error w.r.t. Deterministic TAs Resetting 1-DTAs TA(s,s’) = DTA(s,s’) = 1-DTA(s,s’) = 1-RDTA(s,s’) max error w.r.t. single-clock DTAs 19/28
Approximation Algorithm for the Trace Distance 20/28
generalizes Chan-Kiefer LICS’14 with timed-event Approximation Algorithm for the Trace Distance 20/28
generalizes Chan-Kiefer LICS’14 with timed-event Approximation Algorithm for the Trace Distance NP-hardness [Lyngsø-Pedersen JCSS’02] Approximating the trace distance easy to adapt up to any ε >0 whose size is polynomial to MCs... in the size of the Interval MC is NP-hard. 20/28
generalizes Chan-Kiefer LICS’14 with timed-event Approximation Decidability still an open Algorithm for problem! the Trace Distance NP-hardness [Lyngsø-Pedersen JCSS’02] Approximating the trace distance easy to adapt up to any ε >0 whose size is polynomial to MCs... in the size of the Interval MC is NP-hard. 20/28
Approximation Algorithm d(s,s’) trace distance 21/28
Approximation Algorithm d(s,s’) trace distance ε 21/28
Approximation Algorithm d(s,s’) trace distance ε l 0 l 1 ... l k lower approximants 21/28
Approximation Algorithm d(s,s’) trace distance ε l 0 l 1 ... ... u 1 u 0 l k u k upper approximants lower approximants 21/28
Approximation Algorithm d(s,s’) trace distance ε l 0 l 1 ... ... u 1 u 0 l k u k upper approximants lower approximants • l i and u i must converge to d(s,s’) • For all i ∈ ℕ , l i and u i must be computable . 21/28
Approximation Algorithm (general version) total variation || μ - ν || distance ε l 0 l 1 ... ... u 1 u 0 l k u k upper approximants lower approximants • l i and u i must converge to || μ - ν || • For all i ∈ ℕ , l i and u i must be computable . 21/28
... from below 22/28
... from below recall that... Representation Theorem || μ - ν || = sup | μ (E) - ν (E)| E ∈ F 22/28
... from below recall that... Representation Theorem || μ - ν || = sup | μ (E) - ν (E)| E ∈ F F field that generates Σ 22/28
... from below recall that... Representation Theorem || μ - ν || = sup | μ (E) - ν (E)| E ∈ F F field that generates Σ We need F 0 ⊆ F 1 ⊆ F 2 ⊆ ... such that U i F i = F l i = sup | μ (E) - ν (E)| E ∈ F i 22/28
... from below recall that... Representation Theorem || μ - ν || = sup | μ (E) - ν (E)| E ∈ F F field that generates Σ We need F 0 ⊆ F 1 ⊆ F 2 ⊆ ... such that U i F i = F l i = sup | μ (E) - ν (E)| E ∈ F i so that ∀ i ≥ 0, l i ≤ l i+1 & sup i l i = || μ - ν || increasing limiting 22/28
Trace dist. (from below) ...seen before Provide F 0 ⊆ F 1 ⊆ F 2 ⊆ ... such that U i F i is a field for σ ( 𝓤 ) 23/28
Trace dist. (from below) ...seen before Provide F 0 ⊆ F 1 ⊆ F 2 ⊆ ... such that U i F i is a field for σ ( 𝓤 ) Take F i to be the collection of finite unions of cylinders 𝕯 ( ,R 0 , ... ,R i-1 , ) ∈ 𝓤 L 0 L i n n+1 where R j ∈ {[ , ) | 0 ≤ n ≤ i2 i } ⋃ {[i, ∞ )} 2 i 2 i 23/28
Trace dist. (from below) ...seen before Provide F 0 ⊆ F 1 ⊆ F 2 ⊆ ... such that U i F i is a field for σ ( 𝓤 ) Take F i to be the collection of finite unions of cylinders 𝕯 ( ,R 0 , ... ,R i-1 , ) ∈ 𝓤 L 0 L i n n+1 where R j ∈ {[ , ) | 0 ≤ n ≤ i2 i } ⋃ {[i, ∞ )} 2 i 2 i each repartitioned in 2 i [closed-open) intervals [ )[ )[ )[ ) [ )[ )[ ... ℝ + 0 1 2 3 4 i-2 i-1 i 23/28
... from above 24/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} 24/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} Coupling as a transportation schedule... μ ν s 0 t 0 s 1 t 1 s 2 t 2 s 3 t 3 s 4 t 4 24/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} Coupling as a transportation schedule... μ ν w(s 0 ,t 2 ) s 0 t 0 s 1 t 1 s 2 t 2 s 3 t 3 s 4 t 4 24/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} Coupling as a transportation schedule... μ ν w(s 0 ,t 2 ) s 0 t 0 s 1 t 1 s 2 t 2 s 3 t 3 s 4 t 4 24/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} 25/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} We need Ω 0 ⊆ Ω 1 ⊆ Ω 2 ⊆ ... such that U i Ω i dense in Ω ( μ , ν ) w.r.t. total variation u i = inf {w( ≄ ) | w ∈ Ω i } 25/28
... from above it is know Coupling Characterization that... || μ - ν || = min {w( ≄ ) | w ∈ Ω ( μ , ν )} We need Ω 0 ⊆ Ω 1 ⊆ Ω 2 ⊆ ... such that U i Ω i dense in Ω ( μ , ν ) w.r.t. total variation u i = inf {w( ≄ ) | w ∈ Ω i } so that ∀ i ≥ 0, u i ≥ u i+1 & inf i u i = || μ - ν || decreasing limiting 25/28
Trace dist. (from above) ...seen before Provide Ω 0 ⊆ Ω 1 ⊆ Ω 2 ⊆ ... such that U i Ω i is dense in Ω (P[s],P[s’]) 26/28
Trace dist. (from above) ...seen before Provide Ω 0 ⊆ Ω 1 ⊆ Ω 2 ⊆ ... such that U i Ω i is dense in Ω (P[s],P[s’]) coupling structure Stochastic process of rank k generating pairs of timed 𝓓 : S × S →Δ (S k × S k ) paths divided in multisteps of length k such that 𝓓 (s,s’) ∈ Ω (P[s] k ,P[s’] k ) 26/28
Trace dist. (from above) ...seen before Provide Ω 0 ⊆ Ω 1 ⊆ Ω 2 ⊆ ... such that U i Ω i is dense in Ω (P[s],P[s’]) coupling structure Stochastic process of rank k generating pairs of timed 𝓓 : S × S →Δ (S k × S k ) paths divided in multisteps of length k such that 𝓓 (s,s’) ∈ Ω (P[s] k ,P[s’] k ) Take Ω i = {P 𝓓 [s,s’] ∈ Ω (P[s],P[s’]) | 𝓓 of rank 2 i } where P 𝓓 [s,s’] is the probability generated by 𝓓 26/28
Decidability • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable 27/28
Decidability Not that strong! • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable 27/28
Decidability Not that strong! • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable Exp( λ ) 27/28
Decidability Not that strong! • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable N(a,b) Exp( λ ) 27/28
Decidability Not that strong! • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable N(a,b) U(a,b) Exp( λ ) 27/28
Decidability Not that strong! • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable N(a,b) U(a,b) Exp( λ ) ... 27/28
Decidability Not that strong! • A1: rational transition probabilities & residence-time distributions are computable on [q,q’) with q,q’ ∈ ℚ + • A2: total variation between residence-time distributions is computable N(a,b) U(a,b) Exp( λ ) ... For any ε >0, the approximation procedure for the trace distance is decidable. 27/28
Concluding Remarks 28/28
Concluding Remarks • Trace Distance vs Model Checking 28/28
Concluding Remarks • Trace Distance vs Model Checking • MTL Formulas & Timed Automata 28/28
Concluding Remarks • Trace Distance vs Model Checking • MTL Formulas & Timed Automata • several new characterizations 28/28
Concluding Remarks • Trace Distance vs Model Checking • MTL Formulas & Timed Automata • several new characterizations • Approx. algorithm for Trace Distance 28/28
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