Logical Foundations of Cyber-Physical Systems Andr Platzer Andr - - PowerPoint PPT Presentation

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Logical Foundations of Cyber-Physical Systems Andr Platzer Andr - - PowerPoint PPT Presentation

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09: Reactions & Delays Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 1 / 17 Outline Learning

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SLIDE 1

09: Reactions & Delays

Logical Foundations of Cyber-Physical Systems

Logical Foundations of Cyber-Physical Systems

André Platzer

André Platzer

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 1 / 17

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SLIDE 2

Outline

1

Learning Objectives

2

Delays in Control The Impact of Delays on Event Detection Cartesian Demon Model-Predictive Control Basics Design-by-Invariant Controlling the Control Points Sequencing and Prioritizing Reactions Time-Triggered Verification

3

Summary

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 2 / 17

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SLIDE 3

Outline

1

Learning Objectives

2

Delays in Control The Impact of Delays on Event Detection Cartesian Demon Model-Predictive Control Basics Design-by-Invariant Controlling the Control Points Sequencing and Prioritizing Reactions Time-Triggered Verification

3

Summary

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 2 / 17

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SLIDE 4

Learning Objectives

Reactions & Delays

CT M&C CPS using loop invariants design time-triggered control design-by-invariant modeling CPS designing controls time-triggered control reaction delays discrete sensing semantics of time-triggered control

  • perational effect

finding control constraints model-predictive control

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 3 / 17

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SLIDE 5

Outline

1

Learning Objectives

2

Delays in Control The Impact of Delays on Event Detection Cartesian Demon Model-Predictive Control Basics Design-by-Invariant Controlling the Control Points Sequencing and Prioritizing Reactions Time-Triggered Verification

3

Summary

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 3 / 17

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SLIDE 6

Quantum’s Ping-Pong Proof Invariants

Proposition (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • ({x′ = v,v′ = −g &x ≥ 0∧ x≤5}∪{x′ = v,v′ = −g &x≥5});

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof @invariant(0≤x≤5∧(x = 5 → v≤0))

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 4 / 17

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SLIDE 7

Quantum’s Ping-Pong Proof Invariants

Proposition (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • ({x′ = v,v′ = −g &x ≥ 0∧ x≤5}∪{x′ = v,v′ = −g &x≥5});

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof @invariant(0≤x≤5∧(x = 5 → v≤0)) Just can’t implement . . .

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 4 / 17

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SLIDE 8

Physical vs. Controller Events

Physical vs. Controller Events

1

Justifiable: Physical events (on ground x = 0)

2

Justifiable: Physical evolution domains (above ground x ≥ 0)

3

Questionable: Controller evolution domain (x ≤ 5)

4

Unlike physics, controllers won’t run all the time. Just fairly often.

5

Controllers cannot sense and compute all the time. If you expect the world to change for your controller’s sake, you may be in for a surprise.

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 5 / 17

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SLIDE 9

Back to the Drawing Desk: Quantum the Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • {x′ = v,v′ = −g &x ≥ 0};

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof? Ask René Descartes

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 6 / 17

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SLIDE 10

Back to the Drawing Desk: Quantum the Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • {x′ = v,v′ = −g &x ≥ 0};

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof? Ask René Descartes who says no! Could miss if-then event

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 6 / 17

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SLIDE 11

Back to the Drawing Desk: Quantum the Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • {x′ = v,v′ = −g &x ≥ 0∧ t≤1};

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof?

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 6 / 17

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SLIDE 12

Back to the Drawing Desk: Quantum the Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • {x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1};

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof?

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 6 / 17

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SLIDE 13

Back to the Drawing Desk: Quantum the Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1};

if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv

∗ (0≤x≤5)

Proof? Ask René Descartes Wind up a clock

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 6 / 17

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SLIDE 14

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes Control action before physics

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 7 / 17

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SLIDE 15

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes Could act early or late

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 8 / 17

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SLIDE 16

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says no! Could miss event off control cycle

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 8 / 17

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SLIDE 17

Delays May Miss Events

Delays vs. Events

1

Periodically/frequently monitor for an event with a polling frequency / reaction time.

2

Delays may make the controller miss events.

3

Discrepancy between event-triggered idea vs. real time-triggered implementation.

4

Issues indicate poor event abstraction.

5

Slow controllers monitoring small regions of a fast moving system.

6

Controller needs to be aware of its own delay.

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 9 / 17

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SLIDE 18

Cartesian Doubt: Descartes’s Cartesian Demon 1641

Outwit the Cartesian Demon

Skeptical about the truth of all beliefs until justification has been found.

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 10 / 17

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SLIDE 19

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g > 0∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(4≤x≤5∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says no! Could miss event off control cycle

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 11 / 17

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SLIDE 20

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(x>5 1

2−v∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes predict 1s: x + v − g

2 > 5

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 11 / 17

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SLIDE 21

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(x>5 1

2−v∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says no! Safe after 1 s but not until then All depends on sampling

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 11 / 17

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SLIDE 22

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif(x>5 1

2−v∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says no! Safe after 1 s but not until then All depends on sampling

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 11 / 17

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SLIDE 23

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2gx = 2gH − v2 ∧ x ≥ 0 ∧c = 1∧ g > 0 bouncing ball invariant

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 24

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2gx = 2gH − v2 ∧ x ≥ 0 ∧c = 1∧ g = 1 simplify arithmetic

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 25

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2x = 2H − v2 ∧ x ≥ 0

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 26

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2x = 2· H − v2 ∧ x ≥ 0

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 27

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2x = 2· 5− v2 ∧ x ≥ 0 critical height

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 28

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2x > 2· 5− v2 ∧ x ≥ 0 potential exceeds safe height

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 29

Quantum Discovers Design-by-Invariant

Design-by-Invariant

2x > 2· 5− v2 ∧ x ≥ 0 use invariant for control

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 12 / 17

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SLIDE 30

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 ≥ c ≥ 0∧ f ≥ 0 →

  • if(x=0)v :=−cv elseif((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 13 / 17

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SLIDE 31

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ f = 1 →

  • if(x=0)v :=−cv elseif((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes Just for simplicity

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 13 / 17

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SLIDE 32

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ f = 1 →

  • if(x=0)v :=−cv elseif((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says no! No control near ground

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 13 / 17

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SLIDE 33

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ 1 = f →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes Control despite ground

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 14 / 17

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SLIDE 34

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ 1 = f →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says yes

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 15 / 17

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SLIDE 35

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ 1 = f →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says yes but should have said no!

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 15 / 17

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SLIDE 36

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ 1 = f →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨2x>2·5−v2)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says yes but should have said no! Invariants are invariants! True ever ❀ true always

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 15 / 17

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SLIDE 37

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ 1 = f →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨2x>2·5−v2∧v<1)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes Slow turnaround v(t) = v−gt = v−t < 0

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 15 / 17

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SLIDE 38

Quantum the Time-triggered Ping-Pong Ball

Conjecture (Quantum can play ping-pong safely)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g = 1∧ 1 = c ∧ 1 = f →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨2x>2·5−v2∧v<1)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t≤1}

∗ (0≤x≤5)

Proof? Ask René Descartes who says yes

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 15 / 17

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SLIDE 39

Quantum’s Time-triggered Ping-Pong Proof Invariants

Proposition ( Quantum can play ping-pong safely in real-time)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g=1>0∧ 1=c≥0∧ 1=f≥0 →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨ 2x>2·5−v2∧v<1)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t ≤ 1}

∗ (0≤x≤5)

Proof

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 16 / 17

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SLIDE 40

Quantum’s Time-triggered Ping-Pong Proof Invariants

Proposition ( Quantum can play ping-pong safely in real-time)

0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g=1>0∧ 1=c≥0∧ 1=f≥0 →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨ 2x>2·5−v2∧v<1)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t ≤ 1}

∗ (0≤x≤5)

Proof @invariant(2x = 2H − v2 ∧ x ≥ 0∧ x≤5)

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 16 / 17

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SLIDE 41

Quantum’s Time-triggered Ping-Pong Proof Invariants

Proposition ( Quantum can play ping-pong safely in real-time)

2x = 2H − v2 ∧ 0 ≤ x ∧ x ≤ 5∧ v ≤ 0∧ g=1>0∧ 1=c≥0∧ 1=f≥0 →

  • if(x=0)v :=−cv;if((x>5 1

2−v ∨ 2x>2·5−v2∧v<1)∧v≥0)v :=−fv;

t := 0;{x′ = v,v′ = −g,t′ = 1&x ≥ 0∧ t ≤ 1}

∗ (0≤x≤5)

Proof @invariant(2x = 2H − v2 ∧ x ≥ 0∧ x≤5)

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 16 / 17

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SLIDE 42

Outline

1

Learning Objectives

2

Delays in Control The Impact of Delays on Event Detection Cartesian Demon Model-Predictive Control Basics Design-by-Invariant Controlling the Control Points Sequencing and Prioritizing Reactions Time-Triggered Verification

3

Summary

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 16 / 17

slide-43
SLIDE 43

Summary: Time-triggered Control

1

Common paradigm for designing real controllers

2

Periodical or pseudo-periodical control (jitter)

3

Expects delays, expects inertia

4

Implementation: discrete-time sensing

5

Predict events, not just: if(eventnow(x)) ...

6

Safe controllers know their own reaction delays

7

Burden of event detection brought to attention of CPS programmer

8

Time-triggered controls are implementable and more robust, but make design and verification more challenging!

9

Use knowledge gained from verified event-triggered model as a basis for designing a time-triggered controller

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 17 / 17

slide-44
SLIDE 44

Outline

4

Appendix Zeno’s Quantum Turtles

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 17 / 17

slide-45
SLIDE 45

How Quantum Met Achilles and His Tortoise

Example (Quantum the Bouncing Ball)

  • {x′ = v,v′ = −g &x ≥ 0};

if(x = 0)v :=−cv

∗

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-46
SLIDE 46

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball)

  • {x′ = v,v′ = −g &x ≥ 0};

if(x = 0)v :=−cv

∗

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-47
SLIDE 47

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball)

  • {x′ = v,v′ = −g &x ≥ 0};

if(x = 0)v :=−cv

∗

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-48
SLIDE 48

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +...

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-49
SLIDE 49

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-50
SLIDE 50

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-51
SLIDE 51

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-52
SLIDE 52

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2 < ∞

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-53
SLIDE 53

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2 < ∞

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-54
SLIDE 54

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2 < ∞

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-55
SLIDE 55

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

I don’t exist

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2 < ∞

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 18 / 17

slide-56
SLIDE 56
slide-57
SLIDE 57

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

I don’t exist

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2 < ∞

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 20 / 17

slide-58
SLIDE 58

How Quantum Met Achilles and His Tortoise

t x j 2 4 6 8 10 12 t0 t1 t2 t3 t4 t5 t6

t j 1 2 3 4 5 6 7 8 9 10 11 12 t0 t1 t2 t3 t4 t5 t6

Example (Quantum the Bouncing Ball experiences time)

1+ 1 2 + 1 4 + 1 8 +... =

∞

∑

i=0

1 2i = 1 1− 1

2

= 2 < ∞

Zeno Paradox Quantum’s model causes a time freeze

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 20 / 17

slide-59
SLIDE 59

André Platzer. Logical Foundations of Cyber-Physical Systems. Springer, Switzerland, 2018. URL: http://www.springer.com/978-3-319-63587-3,

doi:10.1007/978-3-319-63588-0.

André Platzer. Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer, Heidelberg, 2010.

doi:10.1007/978-3-642-14509-4.

Sarah M. Loos and André Platzer. Differential refinement logic. In Martin Grohe, Eric Koskinen, and Natarajan Shankar, editors, LICS, pages 505–514, New York, 2016. ACM.

doi:10.1145/2933575.2934555.

André Platzer (CMU) LFCPS/09: Reactions & Delays LFCPS/09 20 / 17