[PPT] - 03: Choice & Control 15-424: Foundations of Cyber-Physical PowerPoint Presentation

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03: Choice & Control

15-424: Foundations of Cyber-Physical Systems Andr´ e Platzer

aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA

0.2 0.4 0.6 0.8 1.0

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Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 1 / 10

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Outline

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Learning Objectives

2

Gradual Introduction to Hybrid Programs

3

Notational Convention

4

Semantics of Hybrid Programs

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 2 / 10

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Outline

1

Learning Objectives

2

Gradual Introduction to Hybrid Programs

3

Notational Convention

4

Semantics of Hybrid Programs

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 2 / 10

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Learning Objectives

Choice & Control

CT M&C CPS nondeterminism abstraction programming languages for CPS semantics compositionality models core principles discrete+ continuous

perational effect
perational precision

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 3 / 10

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Outline

1

Learning Objectives

2

Gradual Introduction to Hybrid Programs

3

Notational Convention

4

Semantics of Hybrid Programs

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 3 / 10

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Playing with Acceleration and Braking

Example (Speedy the point)

a := a + 1; {x′ = v, v′ = a}

1 2 3 4 5 6 7 t 2.5 2.0 1.5 1.0 0.5 0.0 0.5a 1 2 3 4 5 6 7 t 2 2 4 6v

m

1 2 3 4 5 6 7 t 2 2 4 6 8 10

x

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 4 / 10

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Playing with Acceleration and Braking

Example (Speedy the point)

a := −2; {x′ = v, v′ = a}; a := 0.25; {x′ = v, v′ = a}; a := −2; {x′ = v, v′ = a}; a := 0.25; {x′ = v, v′ = a}; a := −2; {x′ = v, v′ = a}; a := 0.25; {x′ = v, v′ = a}

1 2 3 4 5 6 7 t 2.5 2.0 1.5 1.0 0.5 0.0 0.5a 1 2 3 4 5 6 7 t 2 2 4 6v

m

1 2 3 4 5 6 7 t 2 2 4 6 8 10

x

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 4 / 10

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Outline

1

Learning Objectives

2

Gradual Introduction to Hybrid Programs

3

Notational Convention

4

Semantics of Hybrid Programs

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 4 / 10

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Notational Conventions: Names

Example (Naming Conventions)

Letters Convention x, y, z variables e, ˜ e terms P, Q formulas α, β programs c constant symbols f , g, h function symbols p, q, r predicate symbols In CPS applications, all bets are off because names follow application: x position v velocity and a acceleration variables

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 5 / 10

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Notational Conventions: Precedence

Convention (Operator Precedence)

1 Unary operators (including ∗, ¬ and ∀x, ∃x) bind stronger than binary. 2 ∧ bind stronger than ∨, which binds stronger than →, ↔ 3 ; bind stronger than ∪ 4 Arithmetic operators +, −, · associate to the left 5 Logical and program operators associate to the right

Example (Operator Precedence)

∀x P ∧ Q ≡ (∀x P) ∧ Q ∀x P → Q ≡ (∀x P) → Q. α; β ∪ γ ≡ (α; β) ∪ γ α ∪ β; γ ≡ α ∪ (β; γ) α; β∗ ≡ α; (β∗) P → Q → R ≡ P → (Q → R). But →, ↔ expect explicit parentheses. Illegal: P → Q ↔ R P ↔ Q → R

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 6 / 10

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Outline

1

Learning Objectives

2

Gradual Introduction to Hybrid Programs

3

Notational Convention

4

Semantics of Hybrid Programs

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 6 / 10

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Hybrid Program: Semantics

Definition (Hybrid program semantics) ([ [·] ] : HP → ℘(S × S))

[ [x := e] ] = {(ω, ν) : ν = ω except ν[ [x] ] = ω[ [e] ]} [ [?Q] ] = {(ω, ω) : ω ∈ [ [Q] ]} [ [x′ = f (x)] ] = {(ϕ(0), ϕ(r)) : ϕ | = x′ = f (x) for some duration r} [ [α ∪ β] ] = [ [α] ] ∪ [ [β] ] [ [α; β] ] = [ [α] ] ◦ [ [β] ] [ [α∗] ] =

n∈N

[ [αn] ]

Andr´ e Platzer (CMU) FCPS / 03: Choice & Control 7 / 10

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