Control & Dual Control Operators ⋄ ⋄ Angel Ops Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve ? Q d ? Q challenge challenge Let Angel be one player Let Demon be another player André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 11 / 24
Control & Dual Control Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge Duality operator d passes control between players André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 11 / 24
Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge Duality operator d passes control between players 8 0Z0Z0s0Z 7 o0Z0Z0j0 6 Po0o0ZpZ 5 Z0oPZ0Z0 4 0Z0Z0Znl 3 Z0Z0Z0Z0 2 0OPZ0OQZ 1 Z0Z0Z0ZB a b c d e f g h André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 11 / 24
Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge Duality operator d passes control between players h g f e d c b a 1 Z0Z0Z0ZB 2 0OPZ0OQZ 3 Z0Z0Z0Z0 4 0Z0Z0Znl 5 Z0oPZ0Z0 6 Po0o0ZpZ 7 o0Z0Z0j0 8 0Z0Z0s0Z André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 11 / 24
Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge Duality operator d passes control between players 8 0Z0Z0s0Z 7 o0Z0Z0j0 6 Po0o0ZpZ 5 Z0oPZ0Z0 4 0Z0Z0Znl 3 Z0Z0Z0Z0 2 0OPZ0OQZ 1 Z0Z0Z0ZB a b c d e f g h André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 11 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ while ( Q ) α ≡ α ∩ β ≡ α × ≡ ( x ′ = f ( x )& Q ) d x ′ = f ( x )& Q ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) while ( Q ) α ≡ α ∩ β ≡ α × ≡ ( x ′ = f ( x )& Q ) d x ′ = f ( x )& Q ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q α ∩ β ≡ α × ≡ ( x ′ = f ( x )& Q ) d x ′ = f ( x )& Q ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) 8 0Z0Z0s0Z while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q 7 o0Z0Z0j0 6 Po0o0ZpZ 5 Z0oPZ0Z0 α ∩ β ≡ 4 0Z0Z0Znl α × ≡ 3 Z0Z0Z0Z0 2 0OPZ0OQZ 1 Z0Z0Z0ZB ( x ′ = f ( x )& Q ) d x ′ = f ( x )& Q a b c d e f g h ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) h g f e d c b a while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q 1 Z0Z0Z0ZB 2 0OPZ0OQZ α ∩ β ≡ ( α d ∪ β d ) d 3 Z0Z0Z0Z0 4 0Z0Z0Znl α × ≡ 5 Z0oPZ0Z0 6 Po0o0ZpZ 7 o0Z0Z0j0 ( x ′ = f ( x )& Q ) d x ′ = f ( x )& Q 8 0Z0Z0s0Z ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q α ∩ β ≡ ( α d ∪ β d ) d α × ≡ (( α d ) ∗ ) d ( x ′ = f ( x )& Q ) d x ′ = f ( x )& Q ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q α ∩ β ≡ ( α d ∪ β d ) d α × ≡ (( α d ) ∗ ) d ( x ′ = f ( x )& Q ) d �≡ x ′ = f ( x )& Q ( x := e ) d x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q α ∩ β ≡ ( α d ∪ β d ) d α × ≡ (( α d ) ∗ ) d ( x ′ = f ( x )& Q ) d �≡ x ′ = f ( x )& Q ( x := e ) d ≡ x := e ? Q d ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Definable Game Operators ⋄ ⋄ Angel Ops d Demon Ops ∪ ∩ choice choice ∗ × repeat repeat x ′ = f ( x ) evolve x ′ = f ( x ) d evolve d ? Q d ? Q challenge challenge if ( Q ) α else β ≡ (? Q ; α ) ∪ (? ¬ Q ; β ) while ( Q ) α ≡ (? Q ; α ) ∗ ;? ¬ Q α ∩ β ≡ ( α d ∪ β d ) d α × ≡ (( α d ) ∗ ) d ( x ′ = f ( x )& Q ) d �≡ x ′ = f ( x )& Q ( x := e ) d ≡ x := e ? Q d �≡ ? Q André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Outline Learning Objectives 1 Motivation 2 A Gradual Introduction to Hybrid Games 3 Choices & Nondeterminism Control & Dual Control Demon’s Derived Controls Differential Game Logic 4 Syntax of Hybrid Games Syntax of Differential Game Logic Formulas Examples Push-around Cart Robot Dance Example: Robot Soccer 5 An Informal Operational Game Tree Semantics Summary 6 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 12 / 24
Hybrid Games: Syntax Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 13 / 24
Hybrid Games: Syntax Discrete Differential Test Choice Seq. Repeat Assign Equation Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 13 / 24
Hybrid Games: Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 13 / 24
Example: Push-around Cart a d x v André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 14 / 24
Example: Push-around Cart a d x v ( a := 1 ∪ a := − 1 );( d := 1 ∪ d := − 1 ) d ; { x ′ = v , v ′ = a + d } � ∗ � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 14 / 24
Example: Push-around Cart a d x v ( a := 1 ∪ a := − 1 );( d := 1 ∪ d := − 1 ) d ; { x ′ = v , v ′ = a + d } � ∗ � ( d := 1 ∪ d := − 1 ) d ;( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 14 / 24
Example: Push-around Cart a d x v ( a := 1 ∪ a := − 1 );( d := 1 ∩ d := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 14 / 24
Example: Push-around Cart a d x v ( a := 1 ∪ a := − 1 );( d := 1 ∩ d := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � ( d := 1 ∪ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � HP André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 14 / 24
Example: Push-around Cart a d x v ( a := 1 ∪ a := − 1 );( d := 1 ∩ d := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � ( d := 1 ∪ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � HP Hybrid systems can’t say that a is Angel’s choice and d is Demon’s André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 14 / 24
Differential Game Logic: Syntax Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 15 / 24
Differential Game Logic: Syntax Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d Definition (dGL Formula P ) P , Q ::= e ≥ ˜ e | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � α � P | [ α ] P André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 15 / 24
Differential Game Logic: Syntax Discrete Differential Test Choice Seq. Repeat Assign Equation Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d Definition (dGL Formula P ) P , Q ::= e ≥ ˜ e | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � α � P | [ α ] P All Some Reals Reals André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 15 / 24
Differential Game Logic: Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d Definition (dGL Formula P ) P , Q ::= e ≥ ˜ e | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � α � P | [ α ] P All Some Reals Reals André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 15 / 24
Differential Game Logic: Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d Definition (dGL Formula P ) P , Q ::= e ≥ ˜ e | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � α � P | [ α ] P All Some Angel Reals Wins Reals André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 15 / 24
Differential Game Logic: Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d Definition (dGL Formula P ) P , Q ::= e ≥ ˜ e | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � α � P | [ α ] P All Some Angel Demon Reals Wins Wins Reals André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 15 / 24
Simple Examples � ( x := x + 1 ;( x ′ = 1 ) d ∪ x := x − 1 ) ∗ � ( 0 ≤ x < 1 ) � ( x := x + 1 ;( x ′ = 1 ) d ∪ ( x := x − 1 ∩ x := x − 2 )) ∗ � ( 0 ≤ x < 1 ) André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 16 / 24
Simple Examples � � ( x := x + 1 ;( x ′ = 1 ) d ∪ x := x − 1 ) ∗ � ( 0 ≤ x < 1 ) � ( x := x + 1 ;( x ′ = 1 ) d ∪ ( x := x − 1 ∩ x := x − 2 )) ∗ � ( 0 ≤ x < 1 ) André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 16 / 24
Simple Examples � � ( x := x + 1 ;( x ′ = 1 ) d ∪ x := x − 1 ) ∗ � ( 0 ≤ x < 1 ) � � ( x := x + 1 ;( x ′ = 1 ) d ∪ ( x := x − 1 ∩ x := x − 2 )) ∗ � ( 0 ≤ x < 1 ) André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 16 / 24
Example: Push-around Cart a d x v v ≥ 1 → ( d := 1 ∪ d := − 1 ) d ;( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 x ≥ 0 ∧ v ≥ 0 → ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � x ≥ 0 ∧ v ≥ 0 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 x ≥ 0 → ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � x ≥ 0 → boring by skip ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � counterstrategy d := − 1 ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � counterstrategy d := − 1 ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 ( d := 1 ∩ d := − 1 ); ( a := 2 ∪ a := − 2 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � counterstrategy d := − 1 ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 ( d := 1 ∩ d := − 1 ); ( a := 2 ∪ a := − 2 ); { x ′ = v , v ′ = a + d } � ∗ � �� � x ≥ 0 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � counterstrategy d := − 1 ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 ( d := 1 ∩ d := − 1 ); ( a := 2 ∪ a := − 2 ); { x ′ = v , v ′ = a + d } � ∗ � �� � x ≥ 0 �� ( d := 2 ∩ d := − 2 ); ( a := 2 ∪ a := − 2 ); t := 0 ; { x ′ = v , v ′ = a + d , t ′ = 1 & t ≤ 1 } x 2 ≥ 100 � ∗ � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: Push-around Cart a d x v � v ≥ 1 → d before a can compensate ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� v ≥ 0 � counterstrategy d := − 1 ( d := 1 ∩ d := − 1 );( a := 1 ∪ a := − 1 ); { x ′ = v , v ′ = a + d } � ∗ � �� x ≥ 0 ( d := 1 ∩ d := − 1 ); ( a := 2 ∪ a := − 2 ); { x ′ = v , v ′ = a + d } � ∗ � �� � x ≥ 0 �� ( d := 2 ∩ d := − 2 ); ( a := 2 ∪ a := − 2 ); a := d then a := 2 sign v � t := 0 ; { x ′ = v , v ′ = a + d , t ′ = 1 & t ≤ 1 } x 2 ≥ 100 � ∗ � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 17 / 24
Example: WALL · E and EVE Robot Dance 1D planet e w v g u f ( w − e ) 2 ≤ 1 ∧ v = f → �� ( u := 1 ∩ u := − 1 ); ( g := 1 ∪ g := − 1 ); t := 0 ; { w ′ = v , v ′ = u , e ′ = f , f ′ = g , t ′ = 1 & t ≤ 1 } d ( w − e ) 2 ≤ 1 � × � EVE at e plays Angel’s part controlling g WALL · E at w plays Demon’s part controlling u André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 18 / 24
Example: WALL · E and EVE Robot Dance and the World 1D planet e w v g u f ( w − e ) 2 ≤ 1 ∧ v = f → �� ( u := 1 ∩ u := − 1 ); ( g := 1 ∪ g := − 1 ); t := 0 ; { w ′ = v , v ′ = u , e ′ = f , f ′ = g , t ′ = 1 & t ≤ 1 } d ( w − e ) 2 ≤ 1 � × � EVE at e plays Angel’s part controlling g WALL · E at w plays Demon’s part controlling u and world time André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 18 / 24
Example: WALL · E and EVE 1D planet e w v g u f ( w − e ) 2 ≤ 1 ∧ v = f → �� ( u := 1 ∩ u := − 1 ); ( g := 1 ∪ g := − 1 ); t := 0 ; { w ′ = v , v ′ = u , e ′ = f , f ′ = g , t ′ = 1 & t ≤ 1 } � × � ( w − e ) 2 > 1 WALL · E at w plays Demon’s part controlling u and world time EVE at e plays Angel’s part controlling g André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 19 / 24
Example: Goalie in Robot Soccer y , g g ( x , y ) x x < 0 ∧ v > 0 ∧ y = g → � ( w :=+ w ∩ w := − w ); ( u :=+ u ∪ u := − u ); { x ′ = v , y ′ = w , g ′ = u } � ∗ � x 2 +( y − g ) 2 ≤ 1 � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Example: Goalie in Robot Soccer y , g ( v , + w ) g ( x , y ) x x < 0 ∧ v > 0 ∧ y = g → � ( w :=+ w ∩ w := − w ); ( u :=+ u ∪ u := − u ); { x ′ = v , y ′ = w , g ′ = u } � ∗ � x 2 +( y − g ) 2 ≤ 1 � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Example: Goalie in Robot Soccer y , g ( v , + w ) g ( x , y ) x ( v , − w ) x < 0 ∧ v > 0 ∧ y = g → � ( w :=+ w ∩ w := − w ); ( u :=+ u ∪ u := − u ); { x ′ = v , y ′ = w , g ′ = u } � ∗ � x 2 +( y − g ) 2 ≤ 1 � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Example: Goalie in Robot Soccer y , g ( v , + w ) + u g ( x , y ) x ( v , − w ) x < 0 ∧ v > 0 ∧ y = g → � ( w :=+ w ∩ w := − w ); ( u :=+ u ∪ u := − u ); { x ′ = v , y ′ = w , g ′ = u } � ∗ � x 2 +( y − g ) 2 ≤ 1 � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Example: Goalie in Robot Soccer y , g ( v , + w ) + u g ( x , y ) x − u ( v , − w ) x < 0 ∧ v > 0 ∧ y = g → � ( w :=+ w ∩ w := − w ); ( u :=+ u ∪ u := − u ); { x ′ = v , y ′ = w , g ′ = u } � ∗ � x 2 +( y − g ) 2 ≤ 1 � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Example: Goalie in Robot Soccer y , g ( v , + w ) + u g ( x , y ) Goalie’s x − u Secret ( v , − w ) � x � 2 ( u − w ) 2 ≤ 1 ∧ v x < 0 ∧ v > 0 ∧ y = g → � ( w :=+ w ∩ w := − w ); ( u :=+ u ∪ u := − u ); { x ′ = v , y ′ = w , g ′ = u } � ∗ � x 2 +( y − g ) 2 ≤ 1 � André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Outline Learning Objectives 1 Motivation 2 A Gradual Introduction to Hybrid Games 3 Choices & Nondeterminism Control & Dual Control Demon’s Derived Controls Differential Game Logic 4 Syntax of Hybrid Games Syntax of Differential Game Logic Formulas Examples Push-around Cart Robot Dance Example: Robot Soccer 5 An Informal Operational Game Tree Semantics Summary 6 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 20 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) x := e ω x := e ω ω [ [ e ] ] x André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) x ′ = f ( x )& Q ω 0 r t ϕ ( 0 ) ϕ ( t ) ϕ ( r ) André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) ? Q ω ] [ Q ] ? Q ω ∈ [ ω André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) α ∪ β ω r t i g f e h l t ω ω α α α β β β s λ s 1 s i t 1 t j t κ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) α ; β ω α α α t 1 t i t λ β β β β β β β β r j r λ 1 r λ i r j r λ 1 r 1 r 1 r 1 1 i λ 1 1 i λ λ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) α ∗ ω repeat p o t s ω ω α α repeat repeat stop stop α α α α r r r r e e e e stop stop stop stop p p p p e e e e a a a a t t t t α α α α α α α α André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) α α d d ω ω s 0 s 0 t 0 t 0 s λ s λ s 1 s i t 1 t j t κ s 1 s i t 1 t j t κ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 21 / 24
Filibusters X repeat stop X X 0 1 repeat repeat stop stop 0 0 1 1 � ( x := 0 ∩ x := 1 ) ∗ � x = 0 ⋄ ⋄ 0 1 0 1 r e p ⋄ p o e t s a t 0 0 ⋄ 0 1 ⋄ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 22 / 24
Filibusters & The Significance of Finitude X repeat stop X X 0 1 repeat repeat stop stop 0 0 1 1 � ( x := 0 ∩ x := 1 ) ∗ � x = 0 ⋄ ⋄ wfd 0 1 0 1 � false unless x = 0 r e p ⋄ p o e t s a t 0 0 ⋄ 0 1 ⋄ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 22 / 24
Filibusters & The Significance of Finitude X repeat stop X X � ( x ′ = 1 d ; x := 0 ) ∗ � x = 0 0 1 repeat repeat � ( x := 0 ; x ′ = 1 d ) ∗ � x = 0 stop stop 0 0 1 1 � ( x := 0 ∩ x := 1 ) ∗ � x = 0 ⋄ ⋄ wfd 0 1 0 1 � false unless x = 0 r e p ⋄ p o e t s a t 0 0 ⋄ 0 1 ⋄ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 22 / 24
Filibusters & The Significance of Finitude X repeat stop < ∞ � true X X � ( x ′ = 1 d ; x := 0 ) ∗ � x = 0 0 1 repeat repeat � ( x := 0 ; x ′ = 1 d ) ∗ � x = 0 stop stop 0 0 1 1 � ( x := 0 ∩ x := 1 ) ∗ � x = 0 ⋄ ⋄ wfd 0 1 0 1 � false unless x = 0 r e p ⋄ p o e t s a t 0 0 ⋄ 0 1 ⋄ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 22 / 24
Filibusters & The Significance of Finitude X repeat stop < ∞ � true X X � ( x ′ = 1 d ; x := 0 ) ∗ � x = 0 0 1 repeat repeat � ( x := 0 ; x ′ = 1 d ) ∗ � x = 0 stop stop 0 0 1 1 � ( x := 0 ∩ x := 1 ) ∗ � x = 0 ⋄ ⋄ wfd 0 1 0 1 � false unless x = 0 r e p ⋄ p o e t s a Well-defined games t 0 0 can’t be postponed forever ⋄ 0 1 ⋄ André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 22 / 24
Outline Learning Objectives 1 Motivation 2 A Gradual Introduction to Hybrid Games 3 Choices & Nondeterminism Control & Dual Control Demon’s Derived Controls Differential Game Logic 4 Syntax of Hybrid Games Syntax of Differential Game Logic Formulas Examples Push-around Cart Robot Dance Example: Robot Soccer 5 An Informal Operational Game Tree Semantics Summary 6 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 22 / 24
Differential Game Logic: Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game α ) α , β ::= x := e | ? Q | x ′ = f ( x )& Q | α ∪ β | α ; β | α ∗ | α d Definition (dGL Formula P ) P , Q ::= e ≥ ˜ e | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � α � P | [ α ] P All Some Angel Demon Reals Wins Wins Reals André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 23 / 24
Summary differential game logic � α � ϕ dGL = GL + HG = dL + d ϕ e o n t c t e i n r Differential game logic u c s o i u Logic for hybrid games d s Compositional PL + logic Discrete + continuous + adversarial l a stochastic i Operational semantics (informally) r a s nondet r e Next chapter v d a Formal semantics 1 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 24 / 24
Outline Example: Robot Factory 7 André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 24 / 24
Example: Robot Factory Decentralized Automation ( r x , r y ) f y e y ( l x , l y ) x b e x f x Model Primary objectives of the robot ( x , y ) robot coordinates within time ε Leave ( v x , v y ) velocities Never leave outer conveyor belts may Challenges instantaneously increase robot’s velocity by ( c x , c y ) Distributed, physical environment Possibly conflicting secondary objectives André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 25 / 24
Robot Factory Automation ( RF ) Example (Robot-Demon vs. Angel-Factory Environment) � (? true ∪ (?( x < e x ∧ y < e y ∧ eff 1 = 1 ); v x := v x + c x ; eff 1 := 0 ) // belt ∪ (?( e x ≤ x ∧ y ≤ f y ∧ eff 2 = 1 ); v y := v y + c y ; eff 2 := 0 ) ); ( r x , r y ) f y ( v x , v y ) � ∗ e y x b ( l x , l y ) e x f x André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 26 / 24
Robot Factory Automation ( RF ) Example (Robot-Demon vs. Angel-Factory Environment) � (? true ∪ (?( x < e x ∧ y < e y ∧ eff 1 = 1 ); v x := v x + c x ; eff 1 := 0 ) // belt ∪ (?( e x ≤ x ∧ y ≤ f y ∧ eff 2 = 1 ); v y := v y + c y ; eff 2 := 0 ) ); ( a x := ∗ ; ?( − A ≤ a x ≤ A ); a y := ∗ ; ?( − A ≤ a y ≤ A ); // “independent” robot acceleration t s := 0 ) d ; ( r x , r y ) f y ( v x , v y ) � ∗ e y x b ( l x , l y ) e x f x André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 26 / 24
Robot Factory Automation ( RF ) Example (Robot-Demon vs. Angel-Factory Environment) � (? true ∪ (?( x < e x ∧ y < e y ∧ eff 1 = 1 ); v x := v x + c x ; eff 1 := 0 ) // belt ∪ (?( e x ≤ x ∧ y ≤ f y ∧ eff 2 = 1 ); v y := v y + c y ; eff 2 := 0 ) ); ( a x := ∗ ; ?( − A ≤ a x ≤ A ); a y := ∗ ; ?( − A ≤ a y ≤ A ); // “independent” robot acceleration t s := 0 ) d ; ( x ′ = v x , y ′ = v y , v ′ y = a y , t ′ = 1 , t ′ x = a x , v ′ s = 1 & t s ≤ ε ); ( r x , r y ) f y ( v x , v y ) � ∗ e y x b ( l x , l y ) e x f x André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 26 / 24
Robot Factory Automation ( RF ) Example (Robot-Demon vs. Angel-Factory Environment) � (? true ∪ (?( x < e x ∧ y < e y ∧ eff 1 = 1 ); v x := v x + c x ; eff 1 := 0 ) // belt ∪ (?( e x ≤ x ∧ y ≤ f y ∧ eff 2 = 1 ); v y := v y + c y ; eff 2 := 0 ) ); ( a x := ∗ ; ?( − A ≤ a x ≤ A ); a y := ∗ ; ?( − A ≤ a y ≤ A ); // “independent” robot acceleration t s := 0 ) d ; ( x ′ = v x , y ′ = v y , v ′ y = a y , t ′ = 1 , t ′ x = a x , v ′ � s = 1 & t s ≤ ε ); ∩ (?( a x v x ≤ 0 ∧ a y v y ≤ 0 ) d ; // brake if v x = 0 then a x := 0 fi ; // per direction: no time lock if v y = 0 then a y := 0 fi ; ( r x , r y ) ( x ′ = v x , y ′ = v y , v ′ y = a y , t ′ = 1 , t ′ x = a x , v ′ s = 1 f y ( v x , v y ) �� ∗ & t s ≤ ε ∧ a x v x ≤ 0 ∧ a y v y ≤ 0 )) e y x b ( l x , l y ) e x f x André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 26 / 24
Robot Factory Automation ( RF ) Proposition (Robot stays in � ) Controllability Assumptions ) | = ( x = y = 0 ∧ v x = v y = 0 ∧ → [ RF ]( x ∈ [ l x , r x ] ∧ y ∈ [ l y , r y ]) Proposition (Stays in � and leaves on time) RF | x : RF projected to the x-axis Controllability Assumptions ) | = ( x = 0 ∧ v x = 0 ∧ → [ RF | x ]( x ∈ [ l x , r x ] ∧ ( t ≥ ε → x ≥ x b )) ( r x , r y ) f y ( v x , v y ) e y x b ( l x , l y ) e x f x André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 27 / 24
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 . André Platzer. Logics of dynamical systems. In LICS [12], pages 13–24. doi:10.1109/LICS.2012.13 . André Platzer. Logic & proofs for cyber-physical systems. In Nicola Olivetti and Ashish Tiwari, editors, IJCAR , volume 9706 of LNCS , pages 15–21, Berlin, 2016. Springer. doi:10.1007/978-3-319-40229-1_3 . André Platzer (CMU) LFCPS/14: Hybrid Systems & Games LFCPS/14 27 / 24
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