[PPT] - A Temporal Dynamic Logic for Verifying Hybrid System Invariants e PowerPoint Presentation

SLIDE 1

A Temporal Dynamic Logic for Verifying Hybrid System Invariants

Andr´ e Platzer1,2

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

LFCS’07

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 1 / 16

SLIDE 2

Outline

1

Motivation

2

Temporal Dynamic Logic dTL Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control

3

Verification Calculus for dTL Sequent Calculus Verifying Safety Invariants in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 1 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

Hybrid Systems

continuous evolution along differential equations + discrete change

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

Hybrid Systems

continuous evolution along differential equations + discrete change

t z v

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

× no free parameters like ST, SB × no finite-state bisimulation for HS

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

× declaratively axiomatise operational model

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

| = [ETCS] z < MA DL-calculus

×
Andr´

e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

| = [ETCS] z < MA DL-calculus

×
[RBC]partitioned → Train[RBC]safe

× no intermediate states

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

| = [ETCS] z < MA DL-calculus

×
|

= [ETCS] z < MA DTL-calculus

Andr´

e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Verifying Hybrid Systems

RBC MA ST SB negot corr far

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

| = [ETCS] z < MA DL-calculus

×
|

= [ETCS] z < MA DTL-calculus

differential temporal dynamic logic

dTL = TL + DL + HP

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Outline

1

Motivation

2

Temporal Dynamic Logic dTL Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control

3

Verification Calculus for dTL Sequent Calculus Verifying Safety Invariants in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Outline

1

Motivation

2

Temporal Dynamic Logic dTL Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control

3

Verification Calculus for dTL Sequent Calculus Verifying Safety Invariants in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

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Temporal Dynamic Logic dTL: 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) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

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Temporal Dynamic Logic dTL: 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 ≡ negot; corr; z′′ = a negot ≡ z′ = v, ℓ′ = 1 corr ≡ (?MA − z < SB; a := −b) ∪ (?MA − z ≥ SB; a := . . . )

RBC MA ST SB negot corr far Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

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Temporal Dynamic Logic dTL: Syntax

Definition (Formulas / state formulas φ)

¬, ∧, ∨, →, ∀x , ∃x , =, ≤, +, · (first-order part) [α]π, απ (dynamic part)

Definition (Trace formulas π)

φ (non-temporal part) φ, ♦φ (temporal part)

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

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Temporal Dynamic Logic dTL: Syntax

Definition (Formulas / state formulas φ)

¬, ∧, ∨, →, ∀x , ∃x , =, ≤, +, · (first-order part) [α]π, απ (dynamic part)

Definition (Trace formulas π)

φ (non-temporal part) φ, ♦φ (temporal part) [ETCS](ℓ ≤ L → z < MA) ETCS ≡ negot; corr; z′′ = a negot ≡ z′ = v, ℓ′ = 1

RBC MA ST SB negot corr far

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid trace)

Hybrid trace is sequence of continuous functions σi : [0, ri] → Sta V t x Semantics of hybrid program: set of all its hybrid traces σ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 4 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid trace)

Hybrid trace is sequence of continuous functions σi : [0, ri] → Sta V t x Semantics of hybrid program: set of all its hybrid traces σ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 4 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid programs α: trace semantics)

v w x := θ x . = val(v, θ)

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid programs α: trace semantics)

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

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid programs α: trace semantics)

v ?χ if v | = χ limbo ?χ if v | = χ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid programs α: trace semantics)

v s w α; β α β α; β ≡ α

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid programs α: trace semantics)

v s1 s2 sn w α∗ α α α

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Hybrid programs α: trace semantics)

v w1 w2 α β α ∪ β

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (State formulas φ)

v [α]π π π π

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (State formulas φ)

v απ π

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v αφ φ φ φ φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v [α]φ φ φ φ φ φ φ φ φ φ φ φ φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v α♦φ ♦φ φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v [α]♦φ ♦φ φ ♦φ φ ♦φ φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v w αφ φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v α-span [α]π

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v α-span [α]π βπ β-span

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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Temporal Dynamic Logic dTL: Trace Semantics

Definition (Trace formulas φ)

v α-span [α]π βπ β-span β♦[α]-span

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

SLIDE 38

Conservative Extension dTL/dL

Proposition

dTL is conservative extension of non-temporal dL, i.e., trace semantics ≡ transition semantics (without , ♦) v w

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 7 / 16

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Conservative Extension dTL/dL

Proposition

dTL is conservative extension of non-temporal dL, i.e., trace semantics ≡ transition semantics (without , ♦) v w

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 7 / 16

SLIDE 40

Outline

1

Motivation

2

Temporal Dynamic Logic dTL Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control

3

Verification Calculus for dTL Sequent Calculus Verifying Safety Invariants in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 7 / 16

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Verification Calculus for dTL

φ ∧ [x := θ]φ [x := θ]φ v w φ x := θ φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 8 / 16

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Verification Calculus for dTL

φ ∧ [x := θ]φ [x := θ]φ v w φ x := θ φ [α]φ ∧ [α][β]φ [α; β]φ v s w α; β α φ β φ α; β ≡ α φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 8 / 16

SLIDE 43

Verification Calculus for dTL

φ ∧ [x := θ]φ [x := θ]φ v w φ x := θ φ [α]φ ∧ [α][β]φ [α; β]φ v s w α; β α φ β φ α; β ≡ α φ [x′ = θ]φ [x′ = θ]φ v w x′ = f (x) φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 8 / 16

SLIDE 44

Verification Calculus for dTL

φ ∧ [x := θ]φ [x := θ]φ v w φ x := θ φ [α]φ ∧ [α][β]φ [α; β]φ v s w α; β α φ β φ α; β ≡ α φ [x′ = θ]φ [x′ = θ]φ v w s x′ = f (x) φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 8 / 16

SLIDE 45

Verification Calculus for dTL

Temporal Reduction Rules

10 temporal rules (T1) [α]φ ∧ [α][β]φ [α; β]φ (T2) φ [?χ]φ (T3) φ ∧ [x := θ]φ [x := θ]φ (T4) [x′ = θ]φ [x′ = θ]φ (T5) [α; α∗]φ [α∗]φ (T6) α♦φ ∨ αβ♦φ α; β♦φ (T7) φ ?χ♦φ (T8) φ ∨ x := θφ x := θ♦φ (T9) x′ = θφ x′ = θ♦φ (T10) α; α∗♦φ α∗♦φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 9 / 16

SLIDE 46

Verification Calculus for dTL

Non-temporal Rules

10 non-temporal rules (D1) απ ∨ βπ α ∪ βπ (D2) [α]π ∧ [β]π [α ∪ β]π (D3) αβφ α; βφ (D4) χ ∧ φ ?χφ (D5) χ → φ [?χ]φ (D6) φ ∨ α; α∗φ α∗φ (D7) φ ∧ [α; α∗]φ [α∗]φ (D8) F θ

x

x := θF (D9) ∃t≥0 x := yx(t)φ x′ = θφ (D10) ∀t≥0 [x := yx(t)]φ [x′ = θ]φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 10 / 16

SLIDE 47

Verification Calculus for dTL

Propositional/Quantifier Rules

10 propositional 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 (P10) F0 ⊢ G0 F ⊢ G

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 11 / 16

SLIDE 48

Verify Safety in Train Control

ETCS ≡ negot; corr; z′′ = a negot ≡ z′ = v, ℓ′ = 1 corr ≡ (?MA − z < ST; a := −b) ∪ (?MA − z ≥ ST; a := . . . )

RBC MA ST SB negot corr far Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 12 / 16

SLIDE 49

Verify Safety in Train Control

ETCS ≡ negot; corr; z′′ = a negot ≡ z′ = v, ℓ′ = 1 corr ≡ (?MA − z < ST; a := −b) ∪ (?MA − z ≥ ST; a := . . . )

RBC MA ST SB negot corr far

Proof

ψ ⊢ Lv + z < MA ψ ⊢ ∀l≥0 (l ≤ L → lv + z < MA) ψ ⊢ ∀l≥0 z := lv + z, ℓ := lφ ψ ⊢ [negot]φ ψ ⊢ [negot]φ ψ, ℓ≥0 ⊢ v2 < 2b(MA − Lv − z) ψ, ℓ≥0 ⊢ z := ℓv+z, a := -b∀t≥0 (ℓ≤L → a

2 t2+vt+z<MA)

ψ, ℓ≥0 ⊢ z := ℓv+z, a := -b∀t≥0 z := a

2 t2+vt+zφ)

ψ, ℓ≥0 ⊢ z := ℓv+z, a := -b[z′′ = a]φ ⊲ ψ, ℓ≥0 ⊢ z := ℓv+z[corr][z′′ = a]φ ⊲ ψ, ℓ≥0 ⊢ z := ℓv+z[corr; z′′ = a]φ ψ ⊢ ℓ≥0 → z := ℓv+z[corr; z′′ = a]φ ψ ⊢ ∀ℓ≥0 z := ℓv+z[corr; z′′ = a]φ ψ ⊢ [negot][corr; z′′ = a]φ ψ ⊢ [negot; corr; z′′ = a]φ ⊢ ψ → [negot; corr; z′′ = a]φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 12 / 16

SLIDE 50

Verify Safety in Train Control

v2 < 2b(MA−Lv −z) Lv + z < MA

RBC MA ST SB negot corr far

Proof

ψ ⊢ Lv + z < MA ψ ⊢ ∀l≥0 (l ≤ L → lv + z < MA) ψ ⊢ ∀l≥0 z := lv + z, ℓ := lφ ψ ⊢ [negot]φ ψ ⊢ [negot]φ ψ, ℓ≥0 ⊢ v2 < 2b(MA − Lv − z) ψ, ℓ≥0 ⊢ z := ℓv+z, a := -b∀t≥0 (ℓ≤L → a

2 t2+vt+z<MA)

ψ, ℓ≥0 ⊢ z := ℓv+z, a := -b∀t≥0 z := a

2 t2+vt+zφ)

ψ, ℓ≥0 ⊢ z := ℓv+z, a := -b[z′′ = a]φ ⊲ ψ, ℓ≥0 ⊢ z := ℓv+z[corr][z′′ = a]φ ⊲ ψ, ℓ≥0 ⊢ z := ℓv+z[corr; z′′ = a]φ ψ ⊢ ℓ≥0 → z := ℓv+z[corr; z′′ = a]φ ψ ⊢ ∀ℓ≥0 z := ℓv+z[corr; z′′ = a]φ ψ ⊢ [negot][corr; z′′ = a]φ ψ ⊢ [negot; corr; z′′ = a]φ ⊢ ψ → [negot; corr; z′′ = a]φ

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 12 / 16

SLIDE 51

Verify Safety in Train Control

inv ≡ v2 ≤ 2b(MA − z)

RBC MA ST SB negot corr far

ST ≥ Lv + v2

2b

SB ≥

v2 2b +

a

b + 1

a

2ε2 + εv

Andr´

e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 12 / 16

SLIDE 52

Soundness

Theorem (Soundness)

dTL calculus is sound.

Proposition (Incompleteness)

“All” discrete or continuous fragments of dTL are inherently incomplete.

fragment discrete continuous

FOL

[α]φ

× × [α]♦φ × × [α]φ × × (Yet, reachability in hybrid systems is not semidecidable)

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 13 / 16

SLIDE 53

Outline

1

Motivation

2

Temporal Dynamic Logic dTL Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control

3

Verification Calculus for dTL Sequent Calculus Verifying Safety Invariants in Train Control Soundness

4

Conclusions & Future Work

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 13 / 16

SLIDE 54

Future Work

Prove dTL/dL relatively complete “Temporal” induction Improve alternating “liveness” quantifiers [α]♦φ dTL∗ [ETCS](♦sensor → ♦stable)

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 14 / 16

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Conclusions (I)

Deductively verify temporal properties of operational hybrid systems

differential temporal dynamic logic

dTL = TL + DL + HP [α]♦φ ♦φ φ

problem technique OP PAR T closed

ETCS | = z < MA TL-MC

×
×

| = (Ax(ETCS) → z < MA) TL-calculus × . . .

. . .

| = [ETCS] z < MA DL-calculus

×
|

= [ETCS] z < MA dTL-calculus

Andr´

e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 15 / 16

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Conclusions (II)

Train control (ETCS) verification Modular temporal/non-temporal calculus Constructive deduction modulo Verification tool HyKeY Parameter discovery

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 16 / 16

SLIDE 57

B. Beckert and S. Schlager.

A sequent calculus for first-order dynamic logic with trace modalities. In R. Gor´ e, A. Leitsch, and T. Nipkow, editors, IJCAR, volume 2083 of LNCS, pages 626–641. Springer, 2001.

J. M. Davoren, V. Coulthard, N. Markey, and T. Moor.

Non-deterministic temporal logics for general flow systems. In R. Alur and G. J. Pappas, editors, HSCC, volume 2993 of LNCS, pages 280–295. Springer, 2004.

V. Mysore, C. Piazza, and B. Mishra.

Algorithmic algebraic model checking II: Decidability of semi-algebraic model checking and its applications to systems biology. In D. Peled and Y.-K. Tsay, editors, ATVA, volume 3707 of LNCS, pages 217–233. Springer, 2005.

M. R¨
nkk¨
, A. P. Ravn, and K. Sere.

Hybrid action systems.

Theor. Comput. Sci., 290(1):937–973, 2003.

Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 16 / 16