The Image Computation Problem in Hybrid Systems Model Checking e - PowerPoint PPT Presentation

The Image Computation Problem in Hybrid Systems Model Checking e Platzer 1 , 2 Edmund M. Clarke 2 Andr´ 1 University of Oldenburg, Department of Computing Science 2 Carnegie Mellon University, Computer Science Department Hybrid Systems: Computation and Control (HSCC’2007) Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 1 / 13

Image Computation in Hybrid Systems I Analyse image computation problem in hybrid systems Approximation refinement techniques and their limits Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 2 / 13

Image Computation in Hybrid Systems I H Analyse image computation problem in hybrid systems Approximation refinement techniques and their limits Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 2 / 13

Image Computation in Hybrid Systems I H H Analyse image computation problem in hybrid systems Approximation refinement techniques and their limits Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 2 / 13

Image Computation in Hybrid Systems I H H B Analyse image computation problem in hybrid systems Approximation refinement techniques and their limits Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 2 / 13

Image Computation in Hybrid Systems Computation I Model Checking Image H H B Analyse image computation problem in hybrid systems Approximation refinement techniques and their limits Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 2 / 13

Air Traffic Management Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 3 / 13

Air Traffic Management Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 3 / 13

ATM: Roundabout Maneuver Automaton  x ˙ = − v 1 + v 2 cos φ + ω 1 y  y ˙ = v 2 sin φ − ω 1 x   ˙ − ω 1 φ = ω 2 close close rot[ - θ, - θ ] Cruise rot[ θ, θ ] ω i := 0 back back LCircle RCircle ω i := ω ω i := − ω rot[ θ, θ ] rot[ - θ, - θ ] Details Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 4 / 13

Outline Motivation 1 Image Computation in Hybrid Systems Air Traffic Management Approximation in Model Checking 2 Approximation Refinement Model Checking Exact Image Computation: Polynomials and Beyond Image Approximation Flow Approximation 3 Bounded Flow Approximation Continuous Image Computation Probabilistic Model Checking Differential Flow Approximation Experimental Results 4 Conclusions and Future Work 5 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 4 / 13

Outline Motivation 1 Image Computation in Hybrid Systems Air Traffic Management Approximation in Model Checking 2 Approximation Refinement Model Checking Exact Image Computation: Polynomials and Beyond Image Approximation Flow Approximation 3 Bounded Flow Approximation Continuous Image Computation Probabilistic Model Checking Differential Flow Approximation Experimental Results 4 Conclusions and Future Work 5 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 4 / 13

AMC: Approximation Refinement Model Checking AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe I B Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Approximation Refinement Model Checking AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe I H B Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Approximation Refinement Model Checking AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe I H A B Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Approximation Refinement Model Checking AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe I H A + ǫ B Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Approximation Refinement Model Checking AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe I H A + ǫ B Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Approximation Refinement Model Checking AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe I H A + ǫ B Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Exact Image Computation AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe Proposition check and blur can be implemented for I and B semialgebraic A with polynomial flows over R +Piecewise definitions +Rational extensions (e.g. multivariate rational splines) Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

AMC: Image Approximation AMC( B reachable from I in H ): 1 A := approx( H ) uniformly 2 blur by uniform approximation error + ǫ 3 check( B reachable from I in A + ǫ ) 4 B not reachable ⇒ H safe Proposition approx exists for all uniform errors ǫ > 0 when using polynomials to build A Flows ϕ ∈ C ( D , R n ) of H D ⊂ R × R n compact closure of an open set Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

Outline Motivation 1 Image Computation in Hybrid Systems Air Traffic Management Approximation in Model Checking 2 Approximation Refinement Model Checking Exact Image Computation: Polynomials and Beyond Image Approximation Flow Approximation 3 Bounded Flow Approximation Continuous Image Computation Probabilistic Model Checking Differential Flow Approximation Experimental Results 4 Conclusions and Future Work 5 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 5 / 13

Bounded Flow Approximation Proposition (Effective Weierstraß approximation) Flows ϕ ∈ C 1 ( D , R n ) Bounds b := max x ∈ D � ˙ ϕ ( x ) � ⇒ approx computable, hence image computation decidable Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 6 / 13

Continuous Image Computation = ∅ B Numerical B � R -Turing Machine � = ∅ x 1 x 2 x 3 ϕ ( x ) x ∈ R ϕ ( x ) ˙ ϕ Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 7 / 13

Continuous Image Computation = ∅ B Numerical B � R -Turing Machine � = ∅ x 1 x 2 x 3 ϕ ( x ) g x ∈ R ϕ ( x ) ˙ ϕ Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 7 / 13

Continuous Image Computation = ∅ B Numerical B � R -Turing Machine � = ∅ x 1 x 2 x 3 ϕ ( x ) g x ∈ R ϕ ( x ) ˙ ϕ Proposition (Image computation undecidable for. . . ) arbitrarily effective flow ϕ ∈ C k ( D ⊆ R n , R m ) ; D, B effective tolerate error ǫ > 0 in decisions Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 7 / 13

Continuous Image Computation = ∅ B Numerical B � R -Turing Machine � = ∅ x 1 x 2 x 3 ϕ ( x ) g x ∈ R ϕ ( x ) ˙ ϕ Proposition (Image computation undecidable for. . . ) arbitrarily effective flow ϕ ∈ C k ( D ⊆ R n , R m ) ; D, B effective tolerate error ǫ > 0 in decisions ϕ smooth polynomial function with Q -coefficients Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 7 / 13

Probabilistic Model Checking B � x 1 x 2 x 3 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 8 / 13

Probabilistic Model Checking B � x 1 x 2 x 3 x 4 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 8 / 13

Probabilistic Model Checking B � x 1 x 2 x 3 x 4 x 5 x 6 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 8 / 13

Probabilistic Model Checking B � x 1 x 2 x 3 x 4 x 5 x 6 Proposition P ( � ˙ ϕ � ∞ > b ) → 0 as b → ∞ ϕ evaluated on finite subset X = { x i } of open or compact D ⇒ P ( decision correct ) → 1 as � d ( · , X ) � ∞ → 0 Andr´ e Platzer, Edmund M. Clarke (CMU) Image Computation in Hybrid Systems HSCC’07 8 / 13