quantified differential dynamic logic for distributed hybrid systems › 4a5a ›...
TRANSCRIPT
![Page 1: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/1.jpg)
Quantified Differential Dynamic Logicfor
Distributed Hybrid Systems
Andre Platzer
Carnegie Mellon University, Pittsburgh, PA
0.20.4
0.60.8
1.00.1
0.2
0.3
0.4
0.5
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 1 / 16
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Outline
1 Motivation
2 Quantified Differential Dynamic Logic QdLDesignSyntaxSemantics
3 Proof Calculus for Distributed Hybrid SystemsCompositional Verification CalculusDeduction Modulo with Free Variables & SkolemizationActual Existence and CreationSoundness and Completeness
4 Conclusions
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 1 / 16
![Page 3: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/3.jpg)
Complex Physical Systems:
Hybrid Systems
Q: I want to verify my car
A: Hybrid systems Q: But there’s a lot of cars!
Challenge
(Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 2 / 16
![Page 4: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/4.jpg)
Complex Physical Systems: Hybrid Systems
Q: I want to verify my car A: Hybrid systems
Q: But there’s a lot of cars!
Challenge (Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
1 2 3 4t
-2
-1
1
2a
1 2 3 4t
0.5
1.0
1.5
2.0
2.5
3.0v
1 2 3 4t
1
2
3
4
5
6z
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 2 / 16
![Page 5: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/5.jpg)
Complex Physical Systems: Hybrid Systems
Q: I want to verify my car A: Hybrid systems Q: But there’s a lot of cars!
Challenge (Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
1 2 3 4t
-2
-1
1
2a
1 2 3 4t
0.5
1.0
1.5
2.0
2.5
3.0v
1 2 3 4t
1
2
3
4
5
6z
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 2 / 16
![Page 6: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/6.jpg)
Complex Physical Systems:
Distributed Systems
Q: I want to verify a lot of cars
A: Distributed systems Q: But they move!
Challenge
(Distributed Systems)
Local computation(finite state automaton)
Remote communication(network graph)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 3 / 16
![Page 7: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/7.jpg)
Complex Physical Systems: Distributed Systems
Q: I want to verify a lot of cars A: Distributed systems
Q: But they move!
Challenge (Distributed Systems)
Local computation(finite state automaton)
Remote communication(network graph)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 3 / 16
![Page 8: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/8.jpg)
Complex Physical Systems: Distributed Systems
Q: I want to verify a lot of cars A: Distributed systems Q: But they move!
Challenge (Distributed Systems)
Local computation(finite state automaton)
Remote communication(network graph)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 3 / 16
![Page 9: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/9.jpg)
Complex Physical Systems:
Distributed Hybrid Systems
Q: I want to verify lots of moving cars
A: Distributed hybrid systems Q: How?
Challenge
(Distributed Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
Structural dynamics(remote communication)
Dimensional dynamics(appearance)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16
![Page 10: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/10.jpg)
Complex Physical Systems: Distributed Hybrid Systems
Q: I want to verify lots of moving cars A: Distributed hybrid systems
Q: How?
Challenge (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
Structural dynamics(remote communication)
Dimensional dynamics(appearance)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16
![Page 11: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/11.jpg)
Complex Physical Systems: Distributed Hybrid Systems
Q: I want to verify lots of moving cars A: Distributed hybrid systems
Q: How?
Challenge (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
Structural dynamics(remote communication)
Dimensional dynamics(appearance)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16
![Page 12: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/12.jpg)
Complex Physical Systems: Distributed Hybrid Systems
Q: I want to verify lots of moving cars A: Distributed hybrid systems Q: How?
Challenge (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
Structural dynamics(remote communication)
Dimensional dynamics(appearance)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16
![Page 13: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/13.jpg)
State of the Art:
Modeling and Simulation
No formal verification of distributed hybrid systems
Shift [DGV96] The Hybrid SystemSimulation ProgrammingLanguage
R-Charon [KSPL06] ModelingLanguage for ReconfigurableHybrid Systems
Hybrid CSP [CJR95] Semantics inExtended Duration Calculus
HyPA [CR05] Translate fragmentinto normal form.
χ process algebra [vBMR+06]Simulation, translation offragments to PHAVER, UPPAAL
Φ-calculus [Rou04] Semantics in richset theory
ACPsrths [BM05] Modeling languageproposal
OBSHS [MS06] Partial randomsimulation of objects
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 5 / 16
![Page 14: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/14.jpg)
State of the Art: Modeling and Simulation
No formal verification of distributed hybrid systems
Shift [DGV96] The Hybrid SystemSimulation ProgrammingLanguage
R-Charon [KSPL06] ModelingLanguage for ReconfigurableHybrid Systems
Hybrid CSP [CJR95] Semantics inExtended Duration Calculus
HyPA [CR05] Translate fragmentinto normal form.
χ process algebra [vBMR+06]Simulation, translation offragments to PHAVER, UPPAAL
Φ-calculus [Rou04] Semantics in richset theory
ACPsrths [BM05] Modeling languageproposal
OBSHS [MS06] Partial randomsimulation of objects
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 5 / 16
![Page 15: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/15.jpg)
State of the Art: Modeling and Simulation
No formal verification of distributed hybrid systems
Shift [DGV96] The Hybrid SystemSimulation ProgrammingLanguage
R-Charon [KSPL06] ModelingLanguage for ReconfigurableHybrid Systems
Hybrid CSP [CJR95] Semantics inExtended Duration Calculus
HyPA [CR05] Translate fragmentinto normal form.
χ process algebra [vBMR+06]Simulation, translation offragments to PHAVER, UPPAAL
Φ-calculus [Rou04] Semantics in richset theory
ACPsrths [BM05] Modeling languageproposal
OBSHS [MS06] Partial randomsimulation of objects
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 5 / 16
![Page 16: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/16.jpg)
Contributions
1 System model and semantics for distributed hybrid systems: QHP
2 Specification and verification logic: QdL3 Proof calculus for QdL4 First verification approach for distributed hybrid systems
5 Sound and complete axiomatization relative to differential equations
6 Prove collision freedom in a (simple) distributed car control system,where new cars may appear dynamically on the road
7 Logical foundation for analysis of distributed hybrid systems
8 Fundamental extension: first-order x(i) versus primitive x
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 6 / 16
![Page 17: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/17.jpg)
Outline
1 Motivation
2 Quantified Differential Dynamic Logic QdLDesignSyntaxSemantics
3 Proof Calculus for Distributed Hybrid SystemsCompositional Verification CalculusDeduction Modulo with Free Variables & SkolemizationActual Existence and CreationSoundness and Completeness
4 Conclusions
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 6 / 16
![Page 18: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/18.jpg)
Outline (Conceptual Approach)
1 Motivation
2 Quantified Differential Dynamic Logic QdLDesignSyntaxSemantics
3 Proof Calculus for Distributed Hybrid SystemsCompositional Verification CalculusDeduction Modulo with Free Variables & SkolemizationActual Existence and CreationSoundness and Completeness
4 Conclusions
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 6 / 16
![Page 19: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/19.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
Discrete dynamics(control decisions)
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 20: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/20.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
x ′′ = a
Discrete dynamics(control decisions)
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 21: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/21.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
x ′′ = a
Discrete dynamics(control decisions)
a := if .. thenA else−b
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 22: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/22.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
x ′′ = a
Discrete dynamics(control decisions)
a := if .. thenA else−b
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 23: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/23.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
x ′′ = a
Discrete dynamics(control decisions)
a := if .. thenA else−b
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
(4) (4) (3) (3) (2) (2) (1) (1)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 24: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/24.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)
∀i
x(i)′′ = a(i)
Discrete dynamics(control decisions)
∀i
a(i) := if .. thenA else−b
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
(4) (4) (3) (3) (2) (2) (1) (1)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 25: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/25.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)∀i x(i)′′ = a(i)
Discrete dynamics(control decisions)
∀i a(i) := if .. thenA else−b
Structural dynamics(communication/coupling)
Dimensional dynamics(appearance)
n := newCar
(4) (4) (3) (3) (2) (2) (1) (1)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 26: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/26.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems
A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)∀i x(i)′′ = a(i)
Discrete dynamics(control decisions)
∀i a(i) := if .. thenA else−b
Structural dynamics(communication/coupling)
`(i) := carInFrontOf(i)
Dimensional dynamics(appearance)
n := newCar
(4) (4) (3) (3) (2) (2) (1) (1)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 27: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/27.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)∀i x(i)′′ = a(i)
Discrete dynamics(control decisions)
∀i a(i) := if .. thenA else−b
Structural dynamics(communication/coupling)
`(i) := carInFrontOf(i)
Dimensional dynamics(appearance)
n := newCar
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 28: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/28.jpg)
Model for Distributed Hybrid Systems
Q: How to model distributed hybrid systems A: Quantified Hybrid Programs
Model (Distributed Hybrid Systems)
Continuous dynamics(differential equations)∀i x(i)′′ = a(i)
Discrete dynamics(control decisions)
∀i a(i) := if .. thenA else−b
Structural dynamics(communication/coupling)
`(i) := carInFrontOf(i)
Dimensional dynamics(appearance)
n := newCarAndre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16
![Page 29: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/29.jpg)
Quantified Differential Dynamic Logic QdL: Syntax
Definition (Quantified hybrid program α)
∀i : C x(s)′ = θ (quantified ODE)∀i : C x(s) := θ (quantified assignment)
}jump & test?χ (conditional execution)
α;β (seq. composition) }Kleene algebraα ∪ β (nondet. choice)
α∗ (nondet. repetition)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 8 / 16
![Page 30: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/30.jpg)
Quantified Differential Dynamic Logic QdL: Syntax
Definition (Quantified hybrid program α)
∀i : C x(s)′ = θ (quantified ODE)∀i : C x(s) := θ (quantified assignment)
}jump & test?χ (conditional execution)
α;β (seq. composition) }Kleene algebraα ∪ β (nondet. choice)
α∗ (nondet. repetition)
DCCS ≡ (ctrl ; drive)∗
appear ≡ n := newC ; ?(∀j : C far(j , n))
ctrl ≡ ∀i : C a(i) := if∀j : C far(i , j) thenA else−b
drive ≡ ∀i : C x(i)′′ = a(i)
newC is definable!
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 8 / 16
![Page 31: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/31.jpg)
Quantified Differential Dynamic Logic QdL: Syntax
Definition (Quantified hybrid program α)
∀i : C x(s)′ = θ (quantified ODE)∀i : C x(s) := θ (quantified assignment)
}jump & test?χ (conditional execution)
α;β (seq. composition) }Kleene algebraα ∪ β (nondet. choice)
α∗ (nondet. repetition)
DCCS ≡ (appear ; ctrl ; drive)∗
appear ≡ n := newC ; ?(∀j : C far(j , n))
ctrl ≡ ∀i : C a(i) := if∀j : C far(i , j) thenA else−b
drive ≡ ∀i : C x(i)′′ = a(i)
newC is definable!
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 8 / 16
![Page 32: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/32.jpg)
Quantified Differential Dynamic Logic QdL: Syntax
Definition (Quantified hybrid program α)
∀i : C x(s)′ = θ (quantified ODE)∀i : C x(s) := θ (quantified assignment)
}jump & test?χ (conditional execution)
α;β (seq. composition) }Kleene algebraα ∪ β (nondet. choice)
α∗ (nondet. repetition)
DCCS ≡ (appear ; ctrl ; drive)∗
appear ≡ n := newC ; ?(∀j : C far(j , n))
ctrl ≡ ∀i : C a(i) := if∀j : C far(i , j) thenA else−b
drive ≡ ∀i : C x(i)′′ = a(i)
newC is definable!
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 8 / 16
![Page 33: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/33.jpg)
Quantified Differential Dynamic Logic QdL: Syntax
Definition (QdL Formula φ)
¬,∧,∨,→, ∀x ,∃x , =,≤, +, · (R-first-order part)[α]φ, 〈α〉φ (dynamic part)
∀i , j : C far(i , j)→ [(appear ; ctrl ; drive)∗] ∀i 6=j : C x(i) 6= x(j)
far(i , j) ≡ i 6= j → x(i) < x(j) ∧ v(i) ≤ v(j) ∧ a(i) ≤ a(j)
∨ x(i) > x(j) ∧ v(i) ≥ v(j) ∧ a(i) ≥ a(j) . . .
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 8 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v w∀i : C x(s) := θ
Details
t
x
0
v
wif w(x)(v e
i [[s]]) = v ei [[θ]] (for all e)
and otherwise unchanged
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v w∀i : C x(s)′ = θ
∧ χ
Details
t
x
χ
w
v
ϕ(t)
∀i x(s)′ = θ
dϕ(t)ei [[x(s)]]
dt(ζ) = ϕ(ζ)ei [[θ]] (for all e)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v s w
α;β
α β
Details
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v s w
α;β
α β
Details
t
x
s
v w
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v s w
α;β
α β
Details
t
x
s
v w
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v s1 s2 sn w
α∗
α α α
Details
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v s1 s2 sn w
α∗
α α α
Details
t
xv
w
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v
w1
w2
α
β
α ∪ β
Details
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v
w1
w2
α
β
α ∪ β
Details
t
xv w1
w2
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v
?χ
if v |= χ
if v 6|= χ
Details
t
x
0
v no change if v |= χotherwise no transition
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (Quantified hybrid program α: transition semantics)
v
?χ
if v |= χ
if v 6|= χ
Details
t
x
0
v no change if v |= χotherwise no transition
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 9 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (QdL Formula φ)
v[α]φ
φ
φ
φ
α-span
[α]φ
〈β〉φ
β-span
〈β〉[α
]-sp
an
Details
compositional semantics ⇒ compositional calculus!
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (QdL Formula φ)
v〈α〉φ
φ
α-span
[α]φ
〈β〉φ
β-span
〈β〉[α
]-sp
an
Details
compositional semantics ⇒ compositional calculus!
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (QdL Formula φ)
v α-span
[α]φ
〈β〉φ
β-span
〈β〉[α
]-sp
an
Details
compositional semantics ⇒ compositional calculus!
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (QdL Formula φ)
v α-span
[α]φ
〈β〉φ
β-span
〈β〉[α
]-sp
an
Details
compositional semantics ⇒ compositional calculus!
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (QdL Formula φ)
v α-span
[α]φ
〈β〉φ
β-span
〈β〉[α
]-sp
an
Details
compositional semantics ⇒ compositional calculus!
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Quantified Differential Dynamic Logic QdL: Semantics
Definition (QdL Formula φ)
v α-span
[α]φ
〈β〉φ
β-span
〈β〉[α
]-sp
an
Details
compositional semantics ⇒ compositional calculus!
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Outline (Verification Approach)
1 Motivation
2 Quantified Differential Dynamic Logic QdLDesignSyntaxSemantics
3 Proof Calculus for Distributed Hybrid SystemsCompositional Verification CalculusDeduction Modulo with Free Variables & SkolemizationActual Existence and CreationSoundness and Completeness
4 Conclusions
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 10 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = u then∀i (s = u → φ(θ)) elseφ(x(u))
φ([∀i x(s) := θ︸ ︷︷ ︸]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ
∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = u then∀i (s = u → φ(θ)) elseφ(x(u))
φ([∀i x(s) := θ︸ ︷︷ ︸]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ
∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = u then∀i (s = u → φ(θ)) elseφ(x(u))
φ([∀i x(s) := θ︸ ︷︷ ︸]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ
∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = u then∀i (s = u → φ(θ)) elseφ(x(u))
φ([∀i x(s) := θ︸ ︷︷ ︸]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ
∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = [A]u then∀i (s = [A]u → φ(θ)) elseφ(x([A]u))
φ([∀i x(s) := θ︸ ︷︷ ︸A
]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ
∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = [A]u then∀i (s = [A]u → φ(θ)) elseφ(x([A]u))
φ([∀i x(s) := θ︸ ︷︷ ︸A
]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ
∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
if ∃i s = [A]u then∀i (s = [A]u → φ(θ)) elseφ(x([A]u))
φ([∀i x(s) := θ︸ ︷︷ ︸A
]x(u))
v w∀i x(s) := θ
φ
∃t≥0 〈∀i S(t)〉φ〈∀i x(s)′ = θ〉φ
v w∀i x(s)′ = θ
φ∀i S(t)
∃t≥0 (χ ∧ 〈x := yx(t)〉φ)
〈x ′ = f (x)〉φ
v wx ′ = f (x)
φ
x := yx(t)x := yx (s)
χ
χ ≡ ∀0≤s≤t 〈x := yx(s)〉χ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 11 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
compositional semantics ⇒ compositional rules!
[α]φ ∧ [β]φ
[α ∪ β]φv
w1
w2
αφ
βφ
α ∪ β
[α][β]φ
[α;β]φv s w
α;β
[α][β]φα
[β]φβ
φ
φ (φ→ [α]φ)
[α∗]φ v w
α∗
φ
α
φ→ [α]φ
α α
φ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 12 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
[α]φ ∧ [β]φ
[α ∪ β]φv
w1
w2
αφ
βφ
α ∪ β
[α][β]φ
[α;β]φv s w
α;β
[α][β]φα
[β]φβ
φ
φ (φ→ [α]φ)
[α∗]φ v w
α∗
φ
α
φ→ [α]φ
α α
φ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 12 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
[α]φ ∧ [β]φ
[α ∪ β]φv
w1
w2
αφ
βφ
α ∪ β
[α][β]φ
[α;β]φv s w
α;β
[α][β]φα
[β]φβ
φ
φ (φ→ [α]φ)
[α∗]φ v w
α∗
φ
α
φ→ [α]φ
α α
φ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 12 / 16
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Proof Calculus for Quantified Differential Dynamic Logic
[α]φ ∧ [β]φ
[α ∪ β]φv
w1
w2
αφ
βφ
α ∪ β
[α][β]φ
[α;β]φv s w
α;β
[α][β]φα
[β]φβ
φ
φ (φ→ [α]φ)
[α∗]φ v w
α∗
φ
α
φ→ [α]φ
α α
φ
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 12 / 16
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Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 64: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/64.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 65: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/65.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 66: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/66.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 67: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/67.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 68: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/68.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 69: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/69.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k
QE
∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 70: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/70.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k QE∀s≥0(−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 71: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/71.jpg)
Deduction Modulo with Free Variables & Skolemization
∀i 6=j x(i)6=x(j) →∀j 6=k (x(j)≤x(k)∧v(j)≤v(k) ∨ x(j)≥x(k)∧v(j)≥v(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 72: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/72.jpg)
Deduction Modulo with Free Variables & Skolemization
∀X ,Y ,V ,W (X 6=Y → X≤Y∧V≤W ∨ X≥Y∧V≥W )
∀i 6=j x(i)6=x(j) →∀j 6=k (x(j)≤x(k)∧v(j)≤v(k) ∨ x(j)≥x(k)∧v(j)≥v(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 73: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/73.jpg)
Deduction Modulo with Free Variables & Skolemization
∀X ,Y ,V ,W (X 6=Y → X≤Y∧V≤W ∨ X≥Y∧V≥W )
∀i 6=j x(i)6=x(j) →∀j 6=k (x(j)≤x(k)∧v(j)≤v(k) ∨ x(j)≥x(k)∧v(j)≥v(k))
∀i 6=j x(i)6=x(j),s≥0 →∀j 6=k (−b2 s2 + v(j)s + x(j) 6= −b
2 s2 + v(k)s + x(k))
∀i 6=j x(i)6=x(j),s≥0 →[∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j) 6=x(k)
∀i 6=j x(i)6=x(j) →s≥0→ [∀i x(i) :=−b2 s2 + v(i)s + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →∀t≥0 [∀i x(i) :=−b2 t2 + v(i)t + x(i)]∀j 6=k x(j)6=x(k)
∀i 6=j x(i)6=x(j) →[∀i x(i)′ = v(i), v(i)′ = −b] ∀j 6=k x(j) 6=x(k)
∀i 6=j x(i) 6=x(j)→ [∀i x(i)′′ = −b]∀j 6=k x(j)6=x(k)
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 13 / 16
![Page 74: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/74.jpg)
Actual Existence and Creation
Actual Existence Function
∃
(·)
∃
(i) =
{0 if i denotes a possible object
1 if i denotes an actively existing objects
[(∀j : C n := j);
?(
∃
(n) = 0);
∃
(n) := 1
]φ
[n := newC ]φ
∀i : C ! φ ≡ ∀i : C (
∃
(i) = 1→ φ)
∀i : C ! f (s) := θ ≡ ∀i : C f (s) := (if
∃
(i) = 1 then θ else f (s))
∀i : C ! f (s)′ = θ ≡ ∀i : C f (s)′ =
∃
(i)θ
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 14 / 16
![Page 75: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/75.jpg)
Actual Existence and Creation
Actual Existence Function
∃
(·)
∃
(i) =
{0 if i denotes a possible object
1 if i denotes an actively existing objects
[(∀j : C n := j);
?(
∃
(n) = 0);
∃
(n) := 1
]φ
[n := newC ]φ
∀i : C ! φ ≡ ∀i : C (
∃
(i) = 1→ φ)
∀i : C ! f (s) := θ ≡ ∀i : C f (s) := (if
∃
(i) = 1 then θ else f (s))
∀i : C ! f (s)′ = θ ≡ ∀i : C f (s)′ =
∃
(i)θ
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 14 / 16
![Page 76: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/76.jpg)
Actual Existence and Creation
Actual Existence Function
∃
(·)
∃
(i) =
{0 if i denotes a possible object
1 if i denotes an actively existing objects
[(∀j : C n := j);
?(
∃
(n) = 0);
∃
(n) := 1
]φ
[n := newC ]φ
∀i : C ! φ ≡ ∀i : C (
∃
(i) = 1→ φ)
∀i : C ! f (s) := θ ≡ ∀i : C f (s) := (if
∃
(i) = 1 then θ else f (s))
∀i : C ! f (s)′ = θ ≡ ∀i : C f (s)′ =
∃
(i)θ
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 14 / 16
![Page 77: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/77.jpg)
Actual Existence and Creation
Actual Existence Function
∃
(·)
∃
(i) =
{0 if i denotes a possible object
1 if i denotes an actively existing objects
[(∀j : C n := j); ?(
∃
(n) = 0);
∃
(n) := 1
]φ
[n := newC ]φ
∀i : C ! φ ≡ ∀i : C (
∃
(i) = 1→ φ)
∀i : C ! f (s) := θ ≡ ∀i : C f (s) := (if
∃
(i) = 1 then θ else f (s))
∀i : C ! f (s)′ = θ ≡ ∀i : C f (s)′ =
∃
(i)θ
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 14 / 16
![Page 78: Quantified Differential Dynamic Logic for Distributed Hybrid Systems › 4a5a › 86d42bab32531b... · 2017-05-04 · Outline 1 Motivation 2 Quanti ed Di erential Dynamic Logic QdL](https://reader033.vdocument.in/reader033/viewer/2022060407/5f0fa74b7e708231d4453a41/html5/thumbnails/78.jpg)
Actual Existence and Creation
Actual Existence Function
∃
(·)
∃
(i) =
{0 if i denotes a possible object
1 if i denotes an actively existing objects
[(∀j : C n := j); ?(
∃
(n) = 0);
∃
(n) := 1]φ
[n := newC ]φ
∀i : C ! φ ≡ ∀i : C (
∃
(i) = 1→ φ)
∀i : C ! f (s) := θ ≡ ∀i : C f (s) := (if
∃
(i) = 1 then θ else f (s))
∀i : C ! f (s)′ = θ ≡ ∀i : C f (s)′ =
∃
(i)θ
( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
Andre Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 14 / 16
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Actual Existence and Creation
Actual Existence Function
∃
(·)
∃
(i) =
{0 if i denotes a possible object
1 if i denotes an actively existing objects
[(∀j : C n := j); ?(
∃
(n) = 0);
∃
(n) := 1]φ
[n := newC ]φ
∀i : C ! φ ≡ ∀i : C (
∃
(i) = 1→ φ)
∀i : C ! f (s) := θ ≡ ∀i : C f (s) := (if
∃
(i) = 1 then θ else f (s))
∀i : C ! f (s)′ = θ ≡ ∀i : C f (s)′ =
∃
(i)θ ( ) ( )
(2) (2) (1) (1) (3) (3) (4) (4)
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Soundness and Completeness
Theorem (Relative Completeness)
QdL calculus is a sound & complete axiomatisation of distributed hybridsystems relative to quantified differential equations. Proof 16p.
Corollary (Proof-theoretical Alignment)
proving distributed hybrid systems = proving dynamical systems!
Corollary (Yes, we can!)
distributed hybrid systems can be verified by recursive decomposition
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Soundness and Completeness
Theorem (Relative Completeness)
QdL calculus is a sound & complete axiomatisation of distributed hybridsystems relative to quantified differential equations. Proof 16p.
Corollary (Proof-theoretical Alignment)
proving distributed hybrid systems = proving dynamical systems!
Corollary (Yes, we can!)
distributed hybrid systems can be verified by recursive decomposition
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Soundness and Completeness
Theorem (Relative Completeness)
QdL calculus is a sound & complete axiomatisation of distributed hybridsystems relative to quantified differential equations. Proof 16p.
Corollary (Proof-theoretical Alignment)
proving distributed hybrid systems = proving dynamical systems!
Corollary (Yes, we can!)
distributed hybrid systems can be verified by recursive decomposition
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Outline
1 Motivation
2 Quantified Differential Dynamic Logic QdLDesignSyntaxSemantics
3 Proof Calculus for Distributed Hybrid SystemsCompositional Verification CalculusDeduction Modulo with Free Variables & SkolemizationActual Existence and CreationSoundness and Completeness
4 Conclusions
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Conclusions
quantified differential dynamic logic
QdL = FOL + DL + QHP[α]φ φ
α
Distributed hybrid systems everywhere
System model and semantics
Logic for distributed hybrid systems
Compositional proof calculus
First verification approach
Sound & complete / diff. eqn.
Simple distributed car control verified
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Conclusions
quantified differential dynamic logic
QdL = FOL + DL + QHP[α]φ φ
α
Distributed hybrid systems everywhere
System model and semantics
Logic for distributed hybrid systems
Compositional proof calculus
First verification approach
Sound & complete / diff. eqn.
Simple distributed car control verified
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Andre Platzer.
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