Combining Deduction and Algebraic Constraints for Hybrid System Analysis Andr´ e Platzer University of Oldenburg, Department of Computing Science, Germany Verify’07 at CADE’07 Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 1 / 23
Outline Motivation 1 Differential Logic d L 2 Syntax Semantics Verification Calculus Analysis of the European Train Control System 3 Combining Deduction and Algebraic Constraints 4 Nondeterminisms in Branch Selection Nondeterminisms in Formula Selection Nondeterminisms in Mode Selection Iterative Background Closure Strategy Experimental Results 5 Conclusions & Future Work 6 Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 1 / 23
Deductively Verifying Hybrid Systems Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 2 / 23
Deductively Verifying Hybrid Systems Hybrid Systems continuous evolution along differential equations + discrete change z v t Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 2 / 23
Deductively Verifying Hybrid Systems Hybrid Systems continuous evolution along differential equations + discrete change Standard paradigm: model checking z HyTech, CheckMate, PHAVer, . . . find bugs Verification is difficult, because of numerical issues, numerical approximation v t termination of abstraction refinement unbounded regions Parameter SB = 10000? Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 2 / 23
Deductively Verifying Hybrid Systems Hybrid Systems continuous evolution along differential equations + discrete change differential dynamic logic d L = DL + HP Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 2 / 23
Outline Motivation 1 Differential Logic d L 2 Syntax Semantics Verification Calculus Analysis of the European Train Control System 3 Combining Deduction and Algebraic Constraints 4 Nondeterminisms in Branch Selection Nondeterminisms in Formula Selection Nondeterminisms in Mode Selection Iterative Background Closure Strategy Experimental Results 5 Conclusions & Future Work 6 Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 2 / 23
Outline Motivation 1 Differential Logic d L 2 Syntax Semantics Verification Calculus Analysis of the European Train Control System 3 Combining Deduction and Algebraic Constraints 4 Nondeterminisms in Branch Selection Nondeterminisms in Formula Selection Nondeterminisms in Mode Selection Iterative Background Closure Strategy Experimental Results 5 Conclusions & Future Work 6 Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 2 / 23
Differential Logic d L : Syntax Definition (Hybrid program α ) x ′ = f ( x ) (continuous evolution) x := θ (discrete jump) ? χ (conditional execution) α ; β (seq. composition) α ∪ β (nondet. choice) α ∗ (nondet. repetition) Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 3 / 23
Differential Logic d L : Syntax Definition (Hybrid program α ) x ′ = f ( x ) (continuous evolution) x := θ (discrete jump) ? χ (conditional execution) α ; β (seq. composition) α ∪ β (nondet. choice) α ∗ (nondet. repetition) ETCS ≡ ( ctrl ; drive ) ∗ ctrl ≡ (? MA − z ≤ SB ; a := − b ) ∪ (? MA − z ≥ SB ; a := . . . ) drive ≡ z ′′ = a Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 3 / 23
Differential Logic d L : Syntax Definition (Formulas φ ) ¬ , ∧ , ∨ , → , ∀ x , ∃ x , = , ≤ , + , · ( R -first-order part) [ α ] φ, � α � φ (dynamic part) ψ → [( ctrl ; drive ) ∗ ] z ≤ MA All trains respect MA ⇒ system safe Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 3 / 23
Differential Logic d L : Semantics Definition (Formulas φ ) φ v φ [ α ] φ φ α -transitions Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 4 / 23
Differential Logic d L : Semantics Definition (Formulas φ ) v φ � α � φ α -transitions Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 4 / 23
Verification Calculus for Differential Logic d L Dynamic Rules 11 dynamic rules φ ∧ ψ φ ∨ � α ; α ∗ � φ (D1) (D5) � ? φ � ψ � α ∗ � φ φ → ψ φ ∧ [ α ; α ∗ ] φ ∃ t ≥ 0 (¯ χ ∧ � x := y x (D2) (D6) (D9) � x ′ = θ & χ � φ [? φ ] ψ [ α ∗ ] φ � α � φ ∨ � β � φ � α �� β � φ ∀ t ≥ 0 (¯ χ → [ x := y (D3) (D7) (D10) [ x ′ = θ & χ ] φ � α ∪ β � φ � α ; β � φ φ θ [ α ] φ ∧ [ β ] φ x (D4) (D8) [ α ∪ β ] φ � x := θ � φ ⊢ p ⊢ [ α ∗ ]( p → [ α ] p ) (D11) ⊢ [ α ∗ ] p Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 5 / 23
Verification Calculus for d L Propositional/Quantifier Rules 9 propositional rules + 4 quantifier rules ⊢ φ φ, ψ ⊢ φ ⊢ ψ ⊢ (P1) (P4) (P7) ¬ φ ⊢ φ ∧ ψ ⊢ φ ∨ ψ ⊢ φ ⊢ ⊢ φ ⊢ ψ ⊢ φ, ψ (P2) (P5) (P8) ⊢ ¬ φ ⊢ φ ∧ ψ ⊢ φ ∨ ψ φ ⊢ ψ ⊢ φ ψ ⊢ (P3) (P6) (P9) ⊢ φ → ψ φ → ψ ⊢ φ ⊢ φ QE( ∃ x � i (Γ i ⊢ ∆ i )) QE( ∀ x � i (Γ i ⊢ ∆ i )) (F1) (F3) Γ ⊢ ∆ , ∃ x φ Γ ⊢ ∆ , ∀ x φ QE( ∀ x � QE( ∃ x � i (Γ i ⊢ ∆ i )) i (Γ i ⊢ ∆ i )) (F2) (F4) Γ , ∃ x φ ⊢ ∆ Γ , ∀ x φ ⊢ ∆ Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 6 / 23
Concise Theory! But End of the Story? Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 6 / 23
Outline Motivation 1 Differential Logic d L 2 Syntax Semantics Verification Calculus Analysis of the European Train Control System 3 Combining Deduction and Algebraic Constraints 4 Nondeterminisms in Branch Selection Nondeterminisms in Formula Selection Nondeterminisms in Mode Selection Iterative Background Closure Strategy Experimental Results 5 Conclusions & Future Work 6 Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 6 / 23
Analysing European Train Control System (ETCS) ψ → [( ctrl ; drive ) ∗ ] z ≤ MA ctrl ≡ (? MA − z < SB ; a := − b ) ∪ (? MA − z ≥ SB ; a := 0) drive ≡ τ := 0; z ′ = v , v ′ = a , τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 7 / 23
Analysing European Train Control System (ETCS) provable automatically using invariant! ψ → [( ctrl ; drive ) ∗ ] z ≤ MA ctrl ≡ (? MA − z < SB ; a := − b ) ∪ (? MA − z ≥ SB ; a := 0) drive ≡ τ := 0; z ′ = v , v ′ = a , τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε . . . p , MA − z ≥ SB ⊢ v 2 ≤ 2 b ( MA − ε v − z ) p , MA − z ≥ SB ⊢ ∀ t ≥ 0 ( � τ := t � τ ≤ ε → � z := vt + p , MA − z ≥ SB ⊢ � τ := 0 �∀ t ≥ 0 ( � τ := t + τ � τ ≤ ε p , MA − z ≥ SB ⊢ � τ := 0 � [ z ′ = v , v ′ = 0 , τ ′ = 1 & ∗ 2 t 2 + vt + z ; v := − bt + v � p ) p ⊢ ∀ t ≥ 0 ( � v := − bt + v � v ≥ 0 → � z := − b p , MA − z ≥ SB ⊢ � a := 0 �� τ := 0 � [ z ′ = v , v ′ = a , τ p ⊢ [ z ′ = v , v ′ = − b & v ≥ 0] p p , MA − z ≥ SB ⊢ � a := 0 � [ drive ] p p ⊢ � a := − b � [ drive ] p p ⊢ [? MA − z ≥ SB ; a := 0][ drive ] p p ⊢ [ ctrl ][ drive ] p p ⊢ [ ctrl ; drive ] p Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 7 / 23
Full European Train Control System (ETCS) � ∗ � system : poll; (negot ∪ (speedControl; atp; move)) init : drive := 0; brake := 1 � a max �� a max 2 ε 2 + ε v SB := v 2 − d 2 � poll : + + 1 ; ST := ∗ 2 b b negot : (? m − z > ST ) ∪ (? m − z ≤ ST ; rbc) rbc : ( vdes := ∗ ; ? vdes > 0) ∪ ( state := brake ) � ∪ d old := d ; m old := m ; m := ∗ ; d := ∗ ; old − d 2 ≤ 2 b ( m − m old ) ? d ≥ 0 ∧ d 2 � speedCtrl : (? state = brake ; a := − b ) � ∪ ? state = drive ; � (? v ≤ v des ; a := ∗ ; ? − b ≤ a ≤ a max ) �� ∪ (? v ≥ v des ; a := ∗ ; ?0 > a ≥ − b ) atp : (? m − z ≤ SB ; a := − b ) ∪ (? m − z > SB ) v = a , ˙ move : t := 0; { ˙ z = v , ˙ t = 1 , ( v ≥ 0 ∧ t ≤ ε ) } Andr´ e Platzer (University of Oldenburg) Combining Deduction & Algebraic Constraints / Hybrid Systems Verify’07 8 / 23
Recommend
More recommend
