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

SLIDE 1

15: Winning Strategies & Regions

Logical Foundations of Cyber-Physical Systems

Logical Foundations of Cyber-Physical Systems

André Platzer

André Platzer

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 1 / 23

SLIDE 2

Outline

1

Learning Objectives

2

Denotational Semantics Differential Game Logic Semantics Hybrid Game Semantics

3

Semantics of Repetition Repetition with Advance Notice Infinite Iterations and Inflationary Semantics Ordinals Inflationary Semantics of Repetitions Implicit Definitions vs. Explicit Constructions +1 Argument Fixpoints and Pre-fixpoints Comparing Fixpoints Characterizing Winning Repetitions Implicitly

4

Summary

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 2 / 23

SLIDE 3

Outline

1

Learning Objectives

2

Denotational Semantics Differential Game Logic Semantics Hybrid Game Semantics

3

Semantics of Repetition Repetition with Advance Notice Infinite Iterations and Inflationary Semantics Ordinals Inflationary Semantics of Repetitions Implicit Definitions vs. Explicit Constructions +1 Argument Fixpoints and Pre-fixpoints Comparing Fixpoints Characterizing Winning Repetitions Implicitly

4

Summary

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 2 / 23

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Learning Objectives

Winning Strategies & Regions

CT M&C CPS fundamental principles of computational thinking logical extensions PL modularity principles compositional extensions differential game logic denotational vs. operational semantics adversarial dynamics adversarial semantics adversarial repetitions fixpoints CPS semantics multi-agent operational-effects mutual reactions complementary hybrid systems

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 3 / 23

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Differential Game Logic: Syntax

Definition (Hybrid game α) α,β ::= x := e | ?Q | x′ = f(x)&Q | α ∪β | α;β | α∗ | αd Definition (dGL Formula P)

P,Q ::= e ≥ ˜ e | ¬P | P ∧ Q | ∀x P | ∃x P | αP | [α]P

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

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Differential Game Logic: Syntax

Definition (Hybrid game α) α,β ::= x := e | ?Q | x′ = f(x)&Q | α ∪β | α;β | α∗ | αd Definition (dGL Formula P)

P,Q ::= e ≥ ˜ e | ¬P | P ∧ Q | ∀x P | ∃x P | αP | [α]P Discrete Assign Test Game Differential Equation Choice Game Seq. Game Repeat Game All Reals Some Reals

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

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Differential Game Logic: Syntax

Definition (Hybrid game α) α,β ::= x := e | ?Q | x′ = f(x)&Q | α ∪β | α;β | α∗ | αd Definition (dGL Formula P)

P,Q ::= e ≥ ˜ e | ¬P | P ∧ Q | ∀x P | ∃x P | αP | [α]P Discrete Assign Test Game Differential Equation Choice Game Seq. Game Repeat Game All Reals Some Reals Dual Game

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

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Differential Game Logic: Syntax

Definition (Hybrid game α) α,β ::= x := e | ?Q | x′ = f(x)&Q | α ∪β | α;β | α∗ | αd Definition (dGL Formula P)

P,Q ::= e ≥ ˜ e | ¬P | P ∧ Q | ∀x P | ∃x P | αP | [α]P Discrete Assign Test Game Differential Equation Choice Game Seq. Game Repeat Game All Reals Some Reals Dual Game Angel Wins

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

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Differential Game Logic: Syntax

Definition (Hybrid game α) α,β ::= x := e | ?Q | x′ = f(x)&Q | α ∪β | α;β | α∗ | αd Definition (dGL Formula P)

P,Q ::= e ≥ ˜ e | ¬P | P ∧ Q | ∀x P | ∃x P | αP | [α]P Discrete Assign Test Game Differential Equation Choice Game Seq. Game Repeat Game All Reals Some Reals Dual Game Angel Wins Demon Wins

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

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Differential Game Logic: Syntax

Definition (Hybrid game α) α,β ::= x := e | ?Q | x′ = f(x)&Q | α ∪β | α;β | α∗ | αd Definition (dGL Formula P)

P,Q ::= e ≥ ˜ e | ¬P | P ∧ Q | ∀x P | ∃x P | αP | [α]P Discrete Assign Test Game Differential Equation Choice Game Seq. Game Repeat Game All Reals Some Reals Dual Game Angel Wins Demon Wins “Angel has Wings α”

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

SLIDE 11

Outline

1

Learning Objectives

2

Denotational Semantics Differential Game Logic Semantics Hybrid Game Semantics

3

Semantics of Repetition Repetition with Advance Notice Infinite Iterations and Inflationary Semantics Ordinals Inflationary Semantics of Repetitions Implicit Definitions vs. Explicit Constructions +1 Argument Fixpoints and Pre-fixpoints Comparing Fixpoints Characterizing Winning Repetitions Implicitly

4

Summary

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 4 / 23

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Differential Game Logic: Denotational Semantics

Definition (dGL Formula P)

[ [·] ] : Fml →℘(S)

[ [e1 ≥ e2] ] = {ω ∈ S : ω[ [e1] ] ≥ ω[ [e2] ]} [ [¬P] ] = ([ [P] ])∁ [ [P ∧ Q] ] = [ [P] ]∩[ [Q] ] [ [αP] ] = ςα([ [P] ]) {ω:ν ∈ [ [P] ] for some ν with (ω,ν) ∈ [ [α] ]} ??? [ [[α]P] ] = δα([ [P] ])

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 5 / 23

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Differential Game Logic: Denotational Semantics

Definition (dGL Formula P)

[ [·] ] : Fml →℘(S)

[ [e1 ≥ e2] ] = {ω ∈ S : ω[ [e1] ] ≥ ω[ [e2] ]} [ [¬P] ] = ([ [P] ])∁ [ [P ∧ Q] ] = [ [P] ]∩[ [Q] ] [ [αP] ] = ςα([ [P] ]) {ω:ν ∈ [ [P] ] for some ν with (ω,ν) ∈ [ [α] ]} ??? [ [[α]P] ] = δα([ [P] ])

Only for HPs. No interactive play!

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 5 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςx:=e(X) =

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X}

X

ςx:=e(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςx′=f(x)&Q(X) =

X x′ = f(x)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςx′=f(x)&Q(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for an r and ϕ | = x′ = f(x)∧ Q}

X x′ = f(x)

ςx′=f(x)(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ς?Q(X) =

X

[ [Q] ]

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ς?Q(X) = [ [Q] ]∩ X

X

[ [Q] ] ς?Q(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςα∪β(X) = ςα (X ) ςβ (X )

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςα∪β(X) = ςα(X)∪ςβ(X) ςα (X ) ςβ (X )

X

ςα∪β(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςα;β(X) =

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςα;β(X) = ςα(ςβ(X)) ςα(ςβ(X)) ςβ(X)

X

ςα;β(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςαd(X) =

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςαd(X) =

X ∁ X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςαd(X) =

X ∁ X

ςα(X ∁) ςα(X ∁)∁

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) ςαd(X) = (ςα(X ∁))∁

X ∁ X

ςα(X ∁) ςα(X ∁)∁ ςαd(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 6 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δx:=e(X) =

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

SLIDE 29

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X}

X

δx:=e(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δx′=f(x)&Q(X) =

X x′ = f(x)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δx′=f(x)&Q(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for all r with ϕ | = x′ = f(x)∧ Q}

X x′ = f(x)

δx′=f(x)(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δ?Q(X) =

X

[ [Q] ]

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δ?Q(X) = [ [Q] ]∁ ∪ X

X

[ [Q] ] δ?Q(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

SLIDE 34

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δα∪β(X) = δα (X ) δβ (X )

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δα∪β(X) = δα(X)∩δβ(X) δα (X ) δβ (X )

X

δα∪β(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

SLIDE 36

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δα;β(X) =

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

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Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δα;β(X) = δα(δβ(X)) δα(δβ(X)) δβ(X) X δα;β(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

SLIDE 38

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δαd(X) =

X ∁ X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

SLIDE 39

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α: denotational semantics) δαd(X) = (δα(X ∁))∁

X ∁ X

δα(X ∁) δα(X ∁)∁ δαd(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 7 / 23

SLIDE 40

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α)

[ [·] ] : HG → (℘(S) →℘(S))

ςx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X} ςx′=f(x)(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for some r ≥ 0 and ϕ | = x′ = f(x)} ς?Q(X) = [ [Q] ]∩ X ςα∪β(X) = ςα(X)∪ςβ(X) ςα;β(X) = ςα(ςβ(X)) ςα∗(X) = ςαd(X) = (ςα(X ∁))∁ Definition (dGL Formula P)

[ [·] ] : Fml →℘(S)

[ [e1 ≥ e2] ] = {ω ∈ S : ω[ [e1] ] ≥ ω[ [e2] ]} [ [¬P] ] = ([ [P] ])∁ [ [P ∧ Q] ] = [ [P] ]∩[ [Q] ] [ [αP] ] = ςα([ [P] ]) [ [[α]P] ] = δα([ [P] ])

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 8 / 23

SLIDE 41

Monotonicity

Lemma (Monotonicity) ςα(X) ⊆ ςα(Y) and δα(X) ⊆ δα(Y) for all X ⊆ Y

X Y

ςα(X) ςα(Y)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 9 / 23

SLIDE 42

Monotonicity

Lemma (Monotonicity) ςα(X) ⊆ ςα(Y) and δα(X) ⊆ δα(Y) for all X ⊆ Y Definition (Hybrid game α)

[ [·] ] : HG → (℘(S) →℘(S))

ςx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X} ςx′=f(x)(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for some r ≥ 0 and ϕ | = x′ = f(x)} ς?Q(X) = [ [Q] ]∩ X ςα∪β(X) = ςα(X)∪ςβ(X) ςα;β(X) = ςα(ςβ(X)) ςα∗(X) = ςαd(X) = (ςα(X ∁))∁

X Y

ςα(X) ςα(Y)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 9 / 23

SLIDE 43

Outline

1

Learning Objectives

2

Denotational Semantics Differential Game Logic Semantics Hybrid Game Semantics

3

Semantics of Repetition Repetition with Advance Notice Infinite Iterations and Inflationary Semantics Ordinals Inflationary Semantics of Repetitions Implicit Definitions vs. Explicit Constructions +1 Argument Fixpoints and Pre-fixpoints Comparing Fixpoints Characterizing Winning Repetitions Implicitly

4

Summary

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 9 / 23

SLIDE 44

Filibusters & The Significance of Finitude

(x := 0∩ x := 1)∗x = 0

wfd

false unless x = 0

X X 1 1 1 1

⋄

r e p e a t

⋄

s t

• p

repeat 1

⋄

stop 1

⋄

repeat

⋄

stop repeat X stop

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 10 / 23

SLIDE 45

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) =

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 11 / 23

SLIDE 46

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) =

n∈N ςαn(X)

[ [α∗] ] =

n∈N [

[αn] ]

where αn+1 ≡ αn;α

α0 ≡?true

for HP α

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 11 / 23

SLIDE 47

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) =

n∈N ςαn(X)

11 11 01 01 01

⋄

10 10 00

⋄

00

⋄

repeat 10

⋄

stop r e p e a t 01

⋄

s t

• p

10 10 00

⋄

00

⋄

r e p e a t 10

⋄

s t

• p

repeat 11

⋄

stop 11 11 01 01 01 01

⋄

10

⋄

10 00

⋄

00

⋄

10 00 00

⋄

00

⋄

00 00

⋄

00

⋄

10 00 00 00

⋄

00

⋄

00 00

⋄

00

⋄

00 00 00

⋄

00

⋄

00 00

⋄

00

⋄

4 11 01 01 01

⋄

10

⋄

10 00

⋄

00

⋄

10 00 00

⋄

00

⋄

00 00

⋄

00

⋄

3 11 01 01

⋄

10

⋄

10 00

⋄

00

⋄

2 11 01

⋄

10

⋄

1 11

⋄

. . .

x = 1∧ a = 1 → ((x := a;a:= 0)∩ x := 0)∗x = 1

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 11 / 23

SLIDE 48

Semantics of Repetition Advance Notice Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ςαn(X)

advance notice semantics?

11 11 01 01 01

⋄

10 10 00

⋄

00

⋄

repeat 10

⋄

stop r e p e a t 01

⋄

s t

• p

10 10 00

⋄

00

⋄

r e p e a t 10

⋄

s t

• p

repeat 11

⋄

stop 11 11 01 01 01 01

⋄

10

⋄

10 00

⋄

00

⋄

10 00 00

⋄

00

⋄

00 00

⋄

00

⋄

10 00 00 00

⋄

00

⋄

00 00

⋄

00

⋄

00 00 00

⋄

00

⋄

00 00

⋄

00

⋄

4 11 01 01 01

⋄

10

⋄

10 00

⋄

00

⋄

10 00 00

⋄

00

⋄

00 00

⋄

00

⋄

3 11 01 01

⋄

10

⋄

10 00

⋄

00

⋄

2 11 01

⋄

10

⋄

1 11

⋄

. . .

x = 1∧ a = 1 → ((x := a;a:= 0)∩ x := 0)∗x = 1

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 11 / 23

SLIDE 49

Semantics of Repetition Advance Notice Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ςαn(X)

too hard to predict all iterations!

11 11 01 01 01

⋄

10 10 00

⋄

00

⋄

repeat 10

⋄

stop r e p e a t 01

⋄

s t

• p

10 10 00

⋄

00

⋄

r e p e a t 10

⋄

s t

• p

repeat 11

⋄

stop 11 11 01 01 01 01

⋄

10

⋄

10 00

⋄

00

⋄

10 00 00

⋄

00

⋄

00 00

⋄

00

⋄

10 00 00 00

⋄

00

⋄

00 00

⋄

00

⋄

00 00 00

⋄

00

⋄

00 00

⋄

00

⋄

4 11 01 01 01

⋄

10

⋄

10 00

⋄

00

⋄

10 00 00

⋄

00

⋄

00 00

⋄

00

⋄

3 11 01 01

⋄

10

⋄

10 00

⋄

00

⋄

2 11 01

⋄

10

⋄

1 11

⋄

. . .

x = 1∧ a = 1 → ((x := a;a:= 0)∩ x := 0)∗x = 1

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 11 / 23

SLIDE 50

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Since ςα(Y) is just one more round away from Y.

ςα(Y)\ςα∗(X) / ςα∗(X) ςα(Y)

Y

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 12 / 23

SLIDE 51

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 52

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 53

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ς 2

α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 54

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ς 3

α(X) ς 2 α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 55

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

n outside the game so Demon won’t know

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ς 3

α(X) ς 2 α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 56

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

Example (x := 1;x′ = 1d ∪ x := x − 1)∗(0 ≤ x < 1)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 57

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

Example (x := 1;x′ = 1d ∪ x := x − 1)∗(0 ≤ x < 1) ς n

α([0,1)) = [0,n+1) = R

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 58

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ω-semantics ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ςλ

α (X)

def

=

• κ<λ

ςκ

α (X)

λ = 0 a limit ordinal Example (x := 1;x′ = 1d ∪ x := x − 1)∗(0 ≤ x < 1) ς n

α([0,1)) = [0,n+1) = R

ςω

α ([0,1)) =

n∈N ς n

α([0,1)) = [0,∞) = R

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 59

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ω-semantics ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ςλ

α (X)

def

=

• κ<λ

ςκ

α (X)

λ = 0 a limit ordinal Example (x := 1;x′ = 1d ∪ x := x − 1)∗(0 ≤ x < 1) ς n

α([0,1)) = [0,n+1) = R

ςω+1

α

([0,1)) = ςα([0,∞)) = R ςω

α ([0,1)) =

n∈N ς n

α([0,1)) = [0,∞) = R

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 60

Semantics of Repetition

ω-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

ω-semantics ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ςλ

α (X)

def

=

• κ<λ

ςκ

α (X)

λ = 0 a limit ordinal ςω

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 61

Semantics of Repetition

(ω + 1)-Semantics

Definition (Hybrid game α) ςα∗(X) =

n∈N ς n

α(X)

missing winning strategies

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ςλ

α (X)

def

=

• κ<λ

ςκ

α (X)

λ = 0 a limit ordinal ςω+1

α

(X) ςω

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 13 / 23

SLIDE 62

Strategic Closure Ordinal

≥ ωCK

1

Theorem

Hybrid game closure ordinal >ωω

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 14 / 23

SLIDE 63

Expedition: Ordinal Arithmetic

ι + 0 = ι ι +(κ+1) = (ι +κ)+ 1

successor κ+1

ι +λ =

• κ<λ

ι +κ

limit λ

ι · 0 = 0 ι ·(κ+1) = (ι ·κ)+ι

successor κ+1

ι ·λ =

• κ<λ

ι ·κ

limit λ

ι0 = 1 ικ+1 = ικ ·ι

successor κ+1

ιλ =

• κ<λ

ικ

limit λ 2·ω = 4·ω = ω · 2 < ω · 4

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 15 / 23

SLIDE 64

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

ς 0

α(X)

def

= X ςκ+1

α

(X)

def

= X ∪ςα(ςκ

α (X))

ςλ

α (X)

def

=

• κ<λ

ςκ

α (X)

λ = 0 a limit ordinal

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 65

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 66

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 67

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

ς 2

α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 68

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

ς 3

α(X) ς 2 α(X) ςα(X) X

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 69

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 70

Semantics of Repetition Inflationary Semantics

Definition (Hybrid game α) ςα∗(X) =

κ<∞ ςκ α (X)

requires transfinite patience

ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 16 / 23

SLIDE 71

The Power of Implicit Definitions

Implicit Definitions

The advantages of implicit definition

• ver construction are roughly those of

theft over honest toil. — Bertrand Russell

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 17 / 23

SLIDE 72

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Since ςα(Y) is just one more round away from Y.

ςα(Y)\ςα∗(X) / ςα∗(X) ςα(Y)

Y

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 73

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Z

def

= ςα∗(X) then ςα(Z) ⊆ ςα∗(X) = Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 74

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Z

def

= ςα∗(X) then ςα(Z) ⊆ ςα∗(X) = Z

Which Z with ςα(Z) ⊆ Z is the right one? Are there multiple such Z? Does such a Z exist?

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 75

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Z

def

= ςα∗(X) then ςα(Z) ⊆ ςα∗(X) = Z

Which Z with ςα(Z) ⊆ Z is the right one? Are there multiple such Z? Does such a Z exist? Existence: Z = /

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 76

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Z

def

= ςα∗(X) then ςα(Z) ⊆ ςα∗(X) = Z

Which Z with ςα(Z) ⊆ Z is the right one? Are there multiple such Z? Does such a Z exist? Existence: Z = / No wait, dual tests: ς?Qd(/

0) = ς?Q(/ 0∁)∁ = ([ [Q] ]∩S)∁ = [ [Q] ]∁ ⊆ /

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 77

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Z

def

= ςα∗(X) then ςα(Z) ⊆ ςα∗(X) = Z

Which Z with ςα(Z) ⊆ Z is the right one? Are there multiple such Z? Does such a Z exist? Existence: Z = / No wait, dual tests: ς?Qd(/

0) = ς?Q(/ 0∁)∁ = ([ [Q] ]∩S)∁ = [ [Q] ]∁ ⊆ /

Then: ς?Qd([

[¬Q] ]) = ς?Q([ [¬Q] ]∁)∁ = ([ [Q] ]∩[ [Q] ])∁ = [ [¬Q] ] ⊆ [ [¬Q] ]

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 78

+1 Argument

Note (+1 argument)

Y ⊆ ςα∗(X) then ςα(Y) ⊆ ςα∗(X) Z

def

= ςα∗(X) then ςα(Z) ⊆ ςα∗(X) = Z

Which Z with ςα(Z) ⊆ Z is the right one? Are there multiple such Z? Does such a Z exist? Existence: Z = / No wait, dual tests: ς?Qd(/

0) = ς?Q(/ 0∁)∁ = ([ [Q] ]∩S)∁ = [ [Q] ]∁ ⊆ /

Then: ς?Qd([

[¬Q] ]) = ς?Q([ [¬Q] ]∁)∁ = ([ [Q] ]∩[ [Q] ])∁ = [ [¬Q] ] ⊆ [ [¬Q] ]

Still too small: X ⊆ Z since Angel may decide not to repeat

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 18 / 23

SLIDE 79

Fixpoints and Pre-Fixpoints

Definition (Pre-fixpoint)

X ∪ςα(Z) ⊆ Z for the winning region Z

def

= ςα∗(X) ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 19 / 23

SLIDE 80

Fixpoints and Pre-Fixpoints

Definition (Pre-fixpoint)

X ∪ςα(Z) ⊆ Z for the winning region Z

def

= ςα∗(X) ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Which Z is the right one? Are there multiple such Z? Does such a Z exist?

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 19 / 23

SLIDE 81

Fixpoints and Pre-Fixpoints

Definition (Pre-fixpoint)

X ∪ςα(Z) ⊆ Z for the winning region Z

def

= ςα∗(X) ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Which Z is the right one? Are there multiple such Z? Does such a Z exist? Existence: Z = S

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 19 / 23

SLIDE 82

Fixpoints and Pre-Fixpoints

Definition (Pre-fixpoint)

X ∪ςα(Z) ⊆ Z for the winning region Z

def

= ςα∗(X) ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Which Z is the right one? Are there multiple such Z? Does such a Z exist? Existence: Z = S but that’s too big and independent of α

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 19 / 23

SLIDE 83

Comparing (Pre-)Fixpoints

Lemma ( )

X ∪ςα(Y) ⊆ Y X ∪ςα(Z) ⊆ Z are pre-fixpoints, then

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 20 / 23

SLIDE 84

Comparing (Pre-)Fixpoints

Lemma (Intersection closure)

X ∪ςα(Y) ⊆ Y X ∪ςα(Z) ⊆ Z are pre-fixpoints, then Y ∩ Z is a smaller pre-fixpoint.

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 20 / 23

SLIDE 85

Comparing (Pre-)Fixpoints

Lemma (Intersection closure)

X ∪ςα(Y) ⊆ Y X ∪ςα(Z) ⊆ Z are pre-fixpoints, then Y ∩ Z is a smaller pre-fixpoint.

Proof.

X ∪ςα(Y ∩ Z)

mon

⊆ X ∪(ςα(Y)∩ςα(Z))

above

⊆ Y ∩ Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 20 / 23

SLIDE 86

Comparing (Pre-)Fixpoints

Lemma (Intersection closure)

X ∪ςα(Y) ⊆ Y X ∪ςα(Z) ⊆ Z are pre-fixpoints, then Y ∩ Z is a smaller pre-fixpoint.

Proof.

X ∪ςα(Y ∩ Z)

mon

⊆ X ∪(ςα(Y)∩ςα(Z))

above

⊆ Y ∩ Z

Even: The intersection of any family of pre-fixpoints is a pre-fixpoint!

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 20 / 23

SLIDE 87

Comparing (Pre-)Fixpoints

Lemma (Intersection closure)

X ∪ςα(Y) ⊆ Y X ∪ςα(Z) ⊆ Z are pre-fixpoints, then Y ∩ Z is a smaller pre-fixpoint.

Proof.

X ∪ςα(Y ∩ Z)

mon

⊆ X ∪(ςα(Y)∩ςα(Z))

above

⊆ Y ∩ Z

Even: The intersection of any family of pre-fixpoints is a pre-fixpoint! So: repetition semantics is the smallest pre-fixpoint (well-founded)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 20 / 23

SLIDE 88

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

X ∪ςα(ςα∗(X)) ⊆ ςα∗(X)

ςα∗(X) intersection of solutions

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 89

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions ςα(Z) ⊆ ςα(ςα∗(X))

by mon since Z ⊆ ςα∗(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 90

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions

X ∪ςα(Z) ⊆ X ∪ςα(ςα∗(X)) = Z by mon since Z ⊆ ςα∗(X)

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 91

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions

X ∪ςα(Z) ⊆ X ∪ςα(ςα∗(X)) = Z by mon since Z ⊆ ςα∗(X)

ςα∗(X) ⊆ X ∪ςα(ςα∗(X)) = Z

since ςα∗(X) smallest such Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 92

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions

X ∪ςα(Z) ⊆ X ∪ςα(ςα∗(X)) = Z by mon since Z ⊆ ςα∗(X)

ςα∗(X) ⊆ X ∪ςα(ςα∗(X)) = Z

since ςα∗(X) smallest such Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 93

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions

X ∪ςα(Z) ⊆ X ∪ςα(ςα∗(X)) = Z by mon since Z ⊆ ςα∗(X)

ςα∗(X) = X ∪ςα(ςα∗(X)) = Z

since ςα∗(X) smallest such Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 94

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) = Z} ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions

X ∪ςα(Z) ⊆ X ∪ςα(ςα∗(X)) = Z by mon since Z ⊆ ςα∗(X)

ςα∗(X) = X ∪ςα(ςα∗(X)) = Z

since ςα∗(X) smallest such Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 95

Semantics of Repetition

Definition (Hybrid game α) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) = Z} =

κ<∞ ςκ α (X)

by Knaster-Tarski

ςα(ςα∗(X))\ςα∗(X) / ς∞

α (X) ···

ς 3

α(X) ς 2 α(X) ςα(X) X

ςα∗(X)

Z

def

= X ∪ςα(ςα∗(X)) ⊆ ςα∗(X) ςα∗(X) intersection of solutions

X ∪ςα(Z) ⊆ X ∪ςα(ςα∗(X)) = Z by mon since Z ⊆ ςα∗(X)

ςα∗(X) = X ∪ςα(ςα∗(X)) = Z

since ςα∗(X) smallest such Z

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 96

Outline

1

Learning Objectives

2

Denotational Semantics Differential Game Logic Semantics Hybrid Game Semantics

3

Semantics of Repetition Repetition with Advance Notice Infinite Iterations and Inflationary Semantics Ordinals Inflationary Semantics of Repetitions Implicit Definitions vs. Explicit Constructions +1 Argument Fixpoints and Pre-fixpoints Comparing Fixpoints Characterizing Winning Repetitions Implicitly

4

Summary

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 21 / 23

SLIDE 97

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α)

[ [·] ] : HG → (℘(S) →℘(S))

ςx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X} ςx′=f(x)(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for some r ≥ 0 and ϕ | = x′ = f(x)} ς?Q(X) = [ [Q] ]∩ X ςα∪β(X) = ςα(X)∪ςβ(X) ςα;β(X) = ςα(ςβ(X)) ςα∗(X) =

κ<∞ ςκ α (X)

ςαd(X) = (ςα(X ∁))∁ Definition (dGL Formula P)

[ [·] ] : Fml →℘(S)

[ [e1 ≥ e2] ] = {ω ∈ S : ω[ [e1] ] ≥ ω[ [e2] ]} [ [¬P] ] = ([ [P] ])∁ [ [P ∧ Q] ] = [ [P] ]∩[ [Q] ] [ [αP] ] = ςα([ [P] ]) [ [[α]P] ] = δα([ [P] ])

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 22 / 23

SLIDE 98

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α)

[ [·] ] : HG → (℘(S) →℘(S))

ςx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X} ςx′=f(x)(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for some r ≥ 0 and ϕ | = x′ = f(x)} ς?Q(X) = [ [Q] ]∩ X ςα∪β(X) = ςα(X)∪ςβ(X) ςα;β(X) = ςα(ςβ(X)) ςα∗(X) =

κ<∞ ςκ α (X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z}

ςαd(X) = (ςα(X ∁))∁ Definition (dGL Formula P)

[ [·] ] : Fml →℘(S)

[ [e1 ≥ e2] ] = {ω ∈ S : ω[ [e1] ] ≥ ω[ [e2] ]} [ [¬P] ] = ([ [P] ])∁ [ [P ∧ Q] ] = [ [P] ]∩[ [Q] ] [ [αP] ] = ςα([ [P] ]) [ [[α]P] ] = δα([ [P] ])

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 22 / 23

SLIDE 99

Differential Game Logic: Denotational Semantics

Definition (Hybrid game α)

[ [·] ] : HG → (℘(S) →℘(S))

ςx:=e(X) = {ω ∈ S : ωω[

[e] ]

x

∈ X} ςx′=f(x)(X) = {ϕ(0) ∈ S : ϕ(r) ∈ X for some r ≥ 0 and ϕ | = x′ = f(x)} ς?Q(X) = [ [Q] ]∩ X ςα∪β(X) = ςα(X)∪ςβ(X) ςα;β(X) = ςα(ςβ(X)) ςα∗(X) = {Z ⊆ S : X ∪ςα(Z) ⊆ Z} ςαd(X) = (ςα(X ∁))∁ Definition (dGL Formula P)

[ [·] ] : Fml →℘(S)

[ [e1 ≥ e2] ] = {ω ∈ S : ω[ [e1] ] ≥ ω[ [e2] ]} [ [¬P] ] = ([ [P] ])∁ [ [P ∧ Q] ] = [ [P] ]∩[ [Q] ] [ [αP] ] = ςα([ [P] ]) [ [[α]P] ] = δα([ [P] ])

André Platzer (CMU) LFCPS/15: Winning Strategies & Regions LFCPS/15 22 / 23

SLIDE 100

Summary

differential game logic

dGL = GL+ HG = dL+ d αϕ ϕ Semantics for differential game logic Simple compositional denotational semantics Meaning is a simple function of its pieces Outlier: repetition is subtle higher-ordinal iteration Better: repetition means least fixpoint Next chapter

1

Axiomatics

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How to win and prove hybrid games

d i s c r e t e c

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t i n u

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s nondet stochastic a d v e r s a r i a l

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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. Differential game logic. ACM Trans. Comput. Log., 17(1):1:1–1:51, 2015.

doi:10.1145/2817824.

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