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Differential Dynamic Logic for Verifying Parametric Hybrid Systems e Platzer 1 , 2 Andr 1 University of Oldenburg, Department of Computing Science, Germany 2 Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA

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Differential Dynamic Logic for Verifying Parametric Hybrid Systems

Andr´ e Platzer1,2

1University of Oldenburg, Department of Computing Science, Germany 2Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA

Tableaux’07

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 1 / 18

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SLIDE 2

Outline

1

Motivation

2

Differential Logic dL Design Motives Syntax Transition Semantics Speed Supervision in Train Control

3

Verification Calculus for dL Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 1 / 18

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SLIDE 3

Verifying Parametric Hybrid Systems

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 4

Verifying Parametric Hybrid Systems

Hybrid Systems

continuous evolution along differential equations + discrete change

t z v

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 5

Verifying Parametric Hybrid Systems

Parametric Hybrid Systems

continuous evolution along differential equations + discrete change Fix parameter SB = 10000 and hope? Handle SB as free symbolic parameter? Which constraints for SB? ∀MA ∃SB all(train-runs)safe

t z v

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 6

Verifying Parametric Hybrid Systems

Parametric Hybrid Systems

continuous evolution along differential equations + discrete change

differential dynamic logic

dL = DL + HP

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 7
  • J. M. Davoren and A. Nerode.

Logics for hybrid systems. Proceedings of the IEEE, 88(7):985–1010, July 2000.

  • M. R¨
  • nkk¨
  • , A. P. Ravn, and K. Sere.

Hybrid action systems.

  • Theor. Comput. Sci., 290(1):937–973, 2003.
  • W. C. Rounds.

A spatial logic for the hybrid π-calculus. In R. Alur and G. J. Pappas, editors, HSCC, volume 2993 of LNCS, pages 508–522. Springer, 2004.

  • C. Zhou, A. P. Ravn, and M. R. Hansen.

An extended duration calculus for hybrid real-time systems. In R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, editors, Hybrid Systems, volume 736 of LNCS, pages 36–59. Springer, 1992.

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 8

Outline

1

Motivation

2

Differential Logic dL Design Motives Syntax Transition Semantics Speed Supervision in Train Control

3

Verification Calculus for dL Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 9

Outline

1

Motivation

2

Differential Logic dL Design Motives Syntax Transition Semantics Speed Supervision in Train Control

3

Verification Calculus for dL Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

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SLIDE 10

dL Motives:

differential dynamic logic

dL = DL + HP

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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dL Motives: Regions in First-order Logic

differential dynamic logic

dL = FOLR MA − z v MA v2 ≤ 2b(MA − z)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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dL Motives: Regions in First-order Logic

differential dynamic logic

dL = FOLR MA − z v MA v2 ≤ 2b(MA − z) ∀t after(train-runs(t))(v2 ≤ 2b(MA − z))

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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dL Motives: State Transitions in Dynamic Logic

differential dynamic logic

dL = FOLR + DL MA − z v MA v2 ≤ 2b(MA − z) ∀t after(train-runs(t))(v2 ≤ 2b(MA − z)) [train-runs]v2 ≤ 2b(MA − z)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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dL Motives: Hybrid Programs as Uniform Model

differential dynamic logic

dL = FOLR + DL + HP [train-runs]v2 ≤ 2b(MA − z)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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SLIDE 15

dL Motives: Hybrid Programs as Uniform Model

differential dynamic logic

dL = FOLR + DL + HP [ ]v2 ≤ 2b(MA − z)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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SLIDE 16

dL Motives: Hybrid Programs as Uniform Model

differential dynamic logic

dL = FOLR + DL + HP far neg cor rec fsa

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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SLIDE 17

dL Motives: Hybrid Programs as Uniform Model

differential dynamic logic

dL = FOLR + DL + HP neg far neg cor rec fsa

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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SLIDE 18

dL Motives: Hybrid Programs as Uniform Model

differential dynamic logic

dL = FOLR + DL + HP neg far neg cor rec fsa not compositional

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

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Differential Logic dL: 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) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

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Differential Logic dL: 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 ≡ (cor ; drive)∗ cor ≡ (?MA − z ≤ SB; a := −b) ∪ (?MA − z ≥ SB; a := 0) drive ≡ τ := 0; z′′ = a & v ≥ 0 ∧ τ ≤ ε

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

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Differential Logic dL: 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 ≡ (cor ; drive)∗ cor ≡ (?MA − z ≤ SB; a := −b) ∪ (?MA − z ≥ SB; a ≤ amax) drive ≡ τ := 0; z′′ = a & v ≥ 0 ∧ τ ≤ ε

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

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SLIDE 22

Differential Logic dL: 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 ≡ (cor ; drive)∗ cor ≡ (?MA − z ≤ SB; a := −b) ∪ (?MA − z ≥ SB; a ≤ amax) drive ≡ τ := 0; z′ = v, v′ = a, τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

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Differential Logic dL: Syntax

Definition (Hybrid program α)

x′ = f (x) & χ (continuous evolution within invariant region) x := θ (discrete jump) ?χ (conditional execution) α; β (seq. composition) α ∪ β (nondet. choice) α∗ (nondet. repetition) ETCS ≡ (cor ; drive)∗ cor ≡ (?MA − z ≤ SB; a := −b) ∪ (?MA − z ≥ SB; a ≤ amax) drive ≡ τ := 0; z′ = v, v′ = a, τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

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Differential Logic dL: Syntax

Definition (Formulas φ)

¬, ∧, ∨, →, ∀x , ∃x , =, ≤, +, · (R-first-order part) [α]φ, αφ (dynamic part) ψ → [(cor ; drive)∗] z ≤ MA All trains respect MA ⇒ system safe

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v w x := θ x . = val(v, θ)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v w x′ = f (x) t x w v ϕ(t) x′ = f (x)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v w x′ = f (x) & χ t x χ w v ϕ(t) x′ = f (x)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v w x′ = f (x) & χ t x χ w v ϕ(t) x′ = f (x)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v s w α; β α β

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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SLIDE 30

Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v s w α; β α β t x s v w

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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SLIDE 31

Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v s w α; β α β t x s v w

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v s1 s2 sn w α∗ α α α

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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SLIDE 33

Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v s1 s2 sn w α∗ α α α t x v w

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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SLIDE 34

Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v w1 w2 α β α ∪ β

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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SLIDE 35

Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v w1 w2 α β α ∪ β t x v w1 w2

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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Differential Logic dL: Transition Semantics

Definition (Hybrid programs α: transition semantics)

v ?χ if v | = χ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

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SLIDE 37

Differential Logic dL: Transition Semantics

Definition (Formulas φ)

v [α]φ φ φ φ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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SLIDE 38

Differential Logic dL: Transition Semantics

Definition (Formulas φ)

v αφ φ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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SLIDE 39

Differential Logic dL: Transition Semantics

Definition (Formulas φ)

v α-span [α]φ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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SLIDE 40

Differential Logic dL: Transition Semantics

Definition (Formulas φ)

v α-span [α]φ βφ β-span

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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SLIDE 41

Differential Logic dL: Transition Semantics

Definition (Formulas φ)

v α-span [α]φ βφ β-span β[α]-span

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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Differential Logic dL: Transition Semantics

Definition (Formulas φ)

v α-span [α]φ βφ β-span β[α]-span compositional semantics!

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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SLIDE 43

Outline

1

Motivation

2

Differential Logic dL Design Motives Syntax Transition Semantics Speed Supervision in Train Control

3

Verification Calculus for dL Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 6 / 18

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SLIDE 44

Verification Calculus for dL

φθ

x

[x := θ]φ v w φθ

x

x := θ φ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 7 / 18

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SLIDE 45

Verification Calculus for dL

φθ

x

[x := θ]φ v w φθ

x

x := θ φ ∃t≥0 x := yx(t)φ x′ = f (x)φ v w x′ = f (x) φ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 7 / 18

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SLIDE 46

Verification Calculus for dL

φθ

x

[x := θ]φ v w φθ

x

x := θ φ ∃t≥0 x := yx(t)φ x′ = f (x)φ v w x′ = f (x) φ x := yx(t)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 7 / 18

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SLIDE 47

Verification Calculus for dL

Differential Analyser for Solving Differential Equations invented 1876 built 1927

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 8 / 18

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SLIDE 48

Verification Calculus for dL

φθ

x

x := θφ v w φθ

x

x := θ φ ∃t≥0 x := yx(t)φ x′ = f (x)φ v w x′ = f (x) φ x := yx(t)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 9 / 18

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SLIDE 49

Verification Calculus for dL

φθ

x

x := θφ v w φθ

x

x := θ φ ∃t≥0 x := yx(t)φ x′ = f (x)φ v w x′ = f (x) φ x := yx(t) ∃t≥0 (¯ χ ∧ x := yx(t)φ) x′ = f (x) & χφ v w x′ = f (x) & χ φ x := yx(t) x : = yx ( s ) χ ¯ χ ≡ ∀0≤s≤t x := yx(s)χ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 9 / 18

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SLIDE 50

Verification Calculus for dL

compositional semantics ⇒ other rules as usual!

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 10 / 18

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SLIDE 51

Verification Calculus for dL

[α]φ ∧ [β]φ [α ∪ β]φ v w1 w2 α φ β φ α ∪ β

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 10 / 18

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SLIDE 52

Verification Calculus for dL

[α]φ ∧ [β]φ [α ∪ β]φ v w1 w2 α φ β φ α ∪ β [α][β]φ [α; β]φ v s w α; β [α][β]φ α [β]φ β φ

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 10 / 18

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SLIDE 53

Verification Calculus for dL

[α]φ ∧ [β]φ [α ∪ β]φ v w1 w2 α φ β φ α ∪ β [α][β]φ [α; β]φ v s w α; β [α][β]φ α [β]φ β φ ⊢ p ⊢ (p → [α]p) ⊢ [α∗]p v w α∗ p α p → [α]p α α p

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 10 / 18

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SLIDE 54

Modular Combination by Side Deduction

⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 55

Modular Combination by Side Deduction

v > 0, z < MA ⊢ ∃t≥0 z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ z′ = v, v′ = −bz ≥ MA ⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 56

Modular Combination by Side Deduction

v > 0, z < MA ⊢ ∃t≥0 z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ z′ = v, v′ = −bz ≥ MA ⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA QE not applicable!

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 57

Modular Combination by Side Deduction

v > 0, z < MA ⊢ ∃t≥0 z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ z′ = v, v′ = −bz ≥ MA ⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA v > 0, z < MA ⊢ t ≥ 0 ∧ z := −b

2t2 + vt + zz ≥ MA

start side

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 58

Modular Combination by Side Deduction

v > 0, z < MA ⊢ ∃t≥0 z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ z′ = v, v′ = −bz ≥ MA ⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA v > 0, z < MA ⊢ t≥0 v > 0, z < MA ⊢ −b

2t2 + vt + z ≥ MA

v > 0, z < MA ⊢ z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ t ≥ 0 ∧ z := −b

2t2 + vt + zz ≥ MA

start side

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 59

Modular Combination by Side Deduction

v > 0, z < MA ⊢ v2 ≥ 2b(MA − z) v > 0, z < MA ⊢ ∃t≥0 z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ z′ = v, v′ = −bz ≥ MA ⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA v > 0, z < MA ⊢ t≥0 v > 0, z < MA ⊢ −b

2t2 + vt + z ≥ MA

v > 0, z < MA ⊢ z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ t ≥ 0 ∧ z := −b

2t2 + vt + zz ≥ MA

start side QE

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 60

Modular Combination by Side Deduction

v > 0, z < MA ⊢ v2 ≥ 2b(MA − z) v > 0, z < MA ⊢ ∃t≥0 z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ z′ = v, v′ = −bz ≥ MA ⊢ v > 0 ∧ z < MA → z′ = v, v′ = −b z ≥ MA v > 0, z < MA ⊢ t≥0 v > 0, z < MA ⊢ −b

2t2 + vt + z ≥ MA

v > 0, z < MA ⊢ z := −b

2t2 + vt + zz ≥ MA

v > 0, z < MA ⊢ t ≥ 0 ∧ z := −b

2t2 + vt + zz ≥ MA

start side QE

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 11 / 18

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SLIDE 61

Verification Calculus for dL

Dynamic Rules

11 dynamic rules (D1) φ ∧ ψ ?φψ (D2) φ → ψ [?φ]ψ (D3) αφ ∨ βφ α ∪ βφ (D4) [α]φ ∧ [β]φ [α ∪ β]φ (D5) φ ∨ α; α∗φ α∗φ (D6) φ ∧ [α; α∗]φ [α∗]φ (D7) αβφ α; βφ (D8) φθ

x

x := θφ (D9) ∃t≥0 (¯ χ ∧ x := yx x′ = θ & χφ (D10) ∀t≥0 (¯ χ → [x := y [x′ = θ & χ]φ (D11) ⊢ p ⊢ [α∗](p → [α]p) ⊢ [α∗]p

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 12 / 18

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SLIDE 62

Verification Calculus for dL

Propositional/Quantifier Rules

9 propositional rules + 4 quantifier rules (P1) ⊢ φ ¬φ ⊢ (P2) φ ⊢ ⊢ ¬φ (P3) φ ⊢ ψ ⊢ φ → ψ (P4) φ, ψ ⊢ φ ∧ ψ ⊢ (P5) ⊢ φ ⊢ ψ ⊢ φ ∧ ψ (P6) ⊢ φ ψ ⊢ φ → ψ ⊢ (P7) φ ⊢ ψ ⊢ φ ∨ ψ ⊢ (P8) ⊢ φ, ψ ⊢ φ ∨ ψ (P9) φ ⊢ φ (F1) QE(∃x

i(Γi ⊢ ∆i))

Γ ⊢ ∆, ∃x φ (F2) QE(∀x

i(Γi ⊢ ∆i))

Γ, ∃x φ ⊢ ∆ (F3) QE(∀x

i(Γi ⊢ ∆i))

Γ ⊢ ∆, ∀x φ (F4) QE(∃x

i(Γi ⊢ ∆i))

Γ, ∀x φ ⊢ ∆

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 13 / 18

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SLIDE 63

Verify Safety in Train Control

ψ → [(cor ; drive)∗] z ≤ MA cor ≡ (?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) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

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SLIDE 64

Verify Safety in Train Control

ψ → [(cor ; drive)∗] z ≤ MA cor ≡ (?MA − z < SB; a := −b) ∪ (?MA − z ≥ SB; a := 0) drive ≡ τ := 0; z′ = v, v ′ = a, τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε

∗ p⊢ ∀t≥0 (v := −bt + vv ≥ 0 → z := − b

2 t2 + vt + z; v := −bt + vp)

p⊢ [z′ = v, v′ = −b & v ≥ 0]p p⊢ a := −b[drive]p . . . p, MA−z≥SB⊢ v2 ≤ 2b(MA − εv − z) p, MA−z≥SB⊢ ∀t≥0 (τ := tτ ≤ ε → z := vt + p, MA−z≥SB⊢ τ := 0∀t≥0 (τ := t + ττ ≤ ε p, MA−z≥SB⊢ τ := 0[z′ = v, v′ = 0, τ′ = 1 & p, MA−z≥SB⊢ a := 0τ := 0[z′ = v, v′ = a, τ p, MA−z≥SB⊢ a := 0[drive]p p⊢ [?MA−z≥SB; a := 0][drive]p p⊢ [cor][drive]p p⊢ [cor ; drive]p Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

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SLIDE 65

Verify Safety in Train Control

v2 ≤ 2b(MA − εv − z)

∗ p⊢ ∀t≥0 (v := −bt + vv ≥ 0 → z := − b

2 t2 + vt + z; v := −bt + vp)

p⊢ [z′ = v, v′ = −b & v ≥ 0]p p⊢ a := −b[drive]p . . . p, MA−z≥SB⊢ v2 ≤ 2b(MA − εv − z) p, MA−z≥SB⊢ ∀t≥0 (τ := tτ ≤ ε → z := vt + p, MA−z≥SB⊢ τ := 0∀t≥0 (τ := t + ττ ≤ ε p, MA−z≥SB⊢ τ := 0[z′ = v, v′ = 0, τ′ = 1 & p, MA−z≥SB⊢ a := 0τ := 0[z′ = v, v′ = a, τ p, MA−z≥SB⊢ a := 0[drive]p p⊢ [?MA−z≥SB; a := 0][drive]p p⊢ [cor][drive]p p⊢ [cor ; drive]p Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

slide-66
SLIDE 66

Verify Safety in Train Control

v2 ≤ 2b(MA − εv − z) SB ≥ εv + v2

2b

QE

∗ p⊢ ∀t≥0 (v := −bt + vv ≥ 0 → z := − b

2 t2 + vt + z; v := −bt + vp)

p⊢ [z′ = v, v′ = −b & v ≥ 0]p p⊢ a := −b[drive]p . . . p, MA−z≥SB⊢ v2 ≤ 2b(MA − εv − z) p, MA−z≥SB⊢ ∀t≥0 (τ := tτ ≤ ε → z := vt + p, MA−z≥SB⊢ τ := 0∀t≥0 (τ := t + ττ ≤ ε p, MA−z≥SB⊢ τ := 0[z′ = v, v′ = 0, τ′ = 1 & p, MA−z≥SB⊢ a := 0τ := 0[z′ = v, v′ = a, τ p, MA−z≥SB⊢ a := 0[drive]p p⊢ [?MA−z≥SB; a := 0][drive]p p⊢ [cor][drive]p p⊢ [cor ; drive]p Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

slide-67
SLIDE 67

Verify Safety in Train Control

v2 ≤ 2b(MA − εv − z) SB ≥

v2 2b +

a

b + 1

a

2ε2 + εv

  • QE

∗ p⊢ ∀t≥0 (v := −bt + vv ≥ 0 → z := − b

2 t2 + vt + z; v := −bt + vp)

p⊢ [z′ = v, v′ = −b & v ≥ 0]p p⊢ a := −b[drive]p . . . p, MA−z≥SB⊢ v2 ≤ 2b(MA − εv − z) p, MA−z≥SB⊢ ∀t≥0 (τ := tτ ≤ ε → z := vt + p, MA−z≥SB⊢ τ := 0∀t≥0 (τ := t + ττ ≤ ε p, MA−z≥SB⊢ τ := 0[z′ = v, v′ = 0, τ′ = 1 & p, MA−z≥SB⊢ a := 0τ := 0[z′ = v, v′ = a, τ p, MA−z≥SB⊢ a := 0[drive]p p⊢ [?MA−z≥SB; a := 0][drive]p p⊢ [cor][drive]p p⊢ [cor ; drive]p Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

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SLIDE 68

Verify Safety in Train Control

inv ≡ v2 ≤ 2b(MA − z)

∗ p⊢ ∀t≥0 (v := −bt + vv ≥ 0 → z := − b

2 t2 + vt + z; v := −bt + vp)

p⊢ [z′ = v, v′ = −b & v ≥ 0]p p⊢ a := −b[drive]p . . . p, MA−z≥SB⊢ v2 ≤ 2b(MA − εv − z) p, MA−z≥SB⊢ ∀t≥0 (τ := tτ ≤ ε → z := vt + p, MA−z≥SB⊢ τ := 0∀t≥0 (τ := t + ττ ≤ ε p, MA−z≥SB⊢ τ := 0[z′ = v, v′ = 0, τ′ = 1 & p, MA−z≥SB⊢ a := 0τ := 0[z′ = v, v′ = a, τ p, MA−z≥SB⊢ a := 0[drive]p p⊢ [?MA−z≥SB; a := 0][drive]p p⊢ [cor][drive]p p⊢ [cor ; drive]p Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

slide-69
SLIDE 69

Verify Safety in Train Control

inv ≡ v2 ≤ 2b(MA − z) MA − z v

∗ p⊢ ∀t≥0 (v := −bt + vv ≥ 0 → z := − b

2 t2 + vt + z; v := −bt + vp)

p⊢ [z′ = v, v′ = −b & v ≥ 0]p p⊢ a := −b[drive]p . . . p, MA−z≥SB⊢ v2 ≤ 2b(MA − εv − z) p, MA−z≥SB⊢ ∀t≥0 (τ := tτ ≤ ε → z := vt + p, MA−z≥SB⊢ τ := 0∀t≥0 (τ := t + ττ ≤ ε p, MA−z≥SB⊢ τ := 0[z′ = v, v′ = 0, τ′ = 1 & p, MA−z≥SB⊢ a := 0τ := 0[z′ = v, v′ = a, τ p, MA−z≥SB⊢ a := 0[drive]p p⊢ [?MA−z≥SB; a := 0][drive]p p⊢ [cor][drive]p p⊢ [cor ; drive]p Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 14 / 18

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SLIDE 70

Soundness

Theorem (Soundness)

dL calculus is sound. x′ = f (x) Side deductions

Proposition (Incompleteness)

The discrete or continuous fragments of dL are inherently incomplete. (Yet, reachability in hybrid systems is undecidable)

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 15 / 18

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SLIDE 71

Soundness

Proof

(x := x + 1)∗ x = n

+1 +1 +1 +1 +1

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 15 / 18

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SLIDE 72

Soundness

Proof

(x := x + 1)∗ x = n

+1 +1 +1 +1 +1

s′′ = −s, τ ′ = 1(s = 0 ∧ τ = n) s = sin

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 15 / 18

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SLIDE 73

Verified ETCS Model

system :

  • poll; (negot ∪ (speedControl; atp; move))

∗ init : drive := 0; brake := 1 poll : SB := v2−d2

2b

+ amax

b

+ 1 amax

2 ε2 + εv

  • ; ST := ∗

negot : (?m − z > ST) ∪ (?m − z ≤ ST; rbc) rbc : (vdes := ∗; ?vdes > 0) ∪ (state := brake) ∪

  • dold := d; mold := m; m := ∗; d := ∗;

?d ≥ 0 ∧ d2

  • ld − d2 ≤ 2b(m − mold)
  • speedCtrl

: (?state = brake; a := −b) ∪

  • ?state = drive;
  • (?v ≤ vdes; a := ∗; ? − b ≤ a ≤ amax)

∪(?v ≥ vdes; a := ∗; ?0 > a ≥ − b)

  • atp

: (?m − z ≤ SB; a := −b) ∪ (?m − z > SB) move : t := 0; {˙ z = v, ˙ v = a, ˙ t = 1, (v ≥ 0 ∧ t ≤ ε)}

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 16 / 18

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SLIDE 74

Outline

1

Motivation

2

Differential Logic dL Design Motives Syntax Transition Semantics Speed Supervision in Train Control

3

Verification Calculus for dL Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 16 / 18

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SLIDE 75

Future Work

Prove relative completeness of dL/ODE Dynamic reconfiguration of system structures

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 17 / 18

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SLIDE 76

Conclusions

differential dynamic logic

dL = DL + HP Deductively verify hybrid systems Train control (ETCS) verification Constructive deduction modulo by side deduction Verification tool HyKeY Parameter discovery

Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 18 / 18