Differential Game Logic Andr´ e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 1 / 26
Outline CPS Applications 1 Differential Game Logic 2 Differential Hybrid Games Denotational Semantics Determinacy Proofs for CPS 3 Axiomatization Soundness and Completeness Corollaries Separating Axioms Expressiveness 4 Summary 5 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 1 / 26
Can you trust a computer to control physics? Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 2 / 26
Can you trust a computer to control physics? Rationale 1 Safety guarantees require analytic foundations. 2 Foundations revolutionized digital computer science & our society. 3 Need even stronger foundations when software reaches out into our physical world. How can we provide people with cyber-physical systems they can bet their lives on? — Jeannette Wing Cyber-physical Systems CPS combine cyber capabilities with physical capabilities to solve problems that neither part could solve alone. Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 2 / 26
Outline CPS Applications 1 Differential Game Logic 2 Differential Hybrid Games Denotational Semantics Determinacy Proofs for CPS 3 Axiomatization Soundness and Completeness Corollaries Separating Axioms Expressiveness 4 Summary 5 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 2 / 26
CPS Analysis: Robot Control Challenge (Hybrid Systems) Fixed rule describing state evolution with both Discrete dynamics (control decisions) Continuous dynamics (differential equations) a 1.0 v p 0.2 8 0.8 10 t 2 4 6 8 6 0.6 � 0.2 p x 4 0.4 � 0.4 � 0.6 0.2 2 p y � 0.8 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 3 / 26
CPS Analysis: Robot Control Challenge (Hybrid Systems) Fixed rule describing state evolution with both Discrete dynamics (control decisions) Continuous dynamics (differential equations) a d Ω 1.0 d x 0.2 0.5 10 t 2 4 6 8 0.5 10 t 2 4 6 8 � 0.2 � 0.5 10 t � 0.4 2 4 6 8 d y � 0.6 � 1.0 � 0.5 � 0.8 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 3 / 26
CPS Analysis: Robot Control Challenge (Games) Game rules describing play evolution with both Angelic choices (player ⋄ Angel) Demonic choices (player ⋄ Demon) 8 rmbl0skZ 0,0 7 ZpZ0ZpZ0 6 0Zpo0ZpZ ⋄ \ ⋄ Tr Pl 2,1 5 o0ZPo0Zp Trash 1,2 0,0 4 PZPZPZ0O 1,2 3 Z0Z0ZPZ0 Plant 0,0 2,1 2 0O0J0ZPZ 1 SNAQZBMR 3,1 a b c d e f g h Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 4 / 26
CPS Analysis: Robot Control Challenge (Hybrid Games) Game rules describing play evolution with Discrete dynamics (control decisions) Continuous dynamics (differential equations) Adversarial dynamics (Angel ⋄ vs. Demon ⋄ ) a 1.2 v 7 p 0.4 6 1.0 0.2 5 0.8 10 t 4 2 4 6 8 0.6 p x � 0.2 3 0.4 � 0.4 2 0.2 1 � 0.6 p y 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 5 / 26
CPS Analysis: Robot Control Challenge (Hybrid Games) Game rules describing play evolution with Discrete dynamics (control decisions) Continuous dynamics (differential equations) Adversarial dynamics (Angel ⋄ vs. Demon ⋄ ) a d Ω 1.0 d x 0.4 0.5 0.2 0.5 10 t 10 t d y 2 4 6 8 2 4 6 8 � 0.2 � 0.5 10 t 2 4 6 8 � 0.4 � 1.0 � 0.6 � 0.5 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 5 / 26
CPS Analysis: RoboCup Soccer Challenge (Hybrid Games) Game rules describing play evolution with Discrete dynamics (control decisions) Continuous dynamics (differential equations) Adversarial dynamics (Angel ⋄ vs. Demon ⋄ ) a d Ω 1.0 d x 0.4 0.5 0.2 0.5 10 t 10 t d y 2 4 6 8 2 4 6 8 � 0.2 � 0.5 10 t 2 4 6 8 � 0.4 � 1.0 � 0.6 � 0.5 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 6 / 26
Contributions Logical foundations for hybrid games 1 Compositional programming language for hybrid games 2 Compositional logic and proof calculus for winning strategy existence 3 Hybrid games determined 4 Winning region computations terminate after ≥ ω CK iterations 1 5 Separate truth ( ∃ winning strategy) vs. proof (winning certificate) vs. proof search (automatic construction) 6 Sound & relatively complete 7 Expressiveness 8 Fragments quite successful in applications 9 Generalizations in logic enable more applications Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 7 / 26
Outline CPS Applications 1 Differential Game Logic 2 Differential Hybrid Games Denotational Semantics Determinacy Proofs for CPS 3 Axiomatization Soundness and Completeness Corollaries Separating Axioms Expressiveness 4 Summary 5 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 7 / 26
Differential Game Logic dG L : Syntax Definition (Hybrid game a ) x := f ( x ) | ? Q | x ′ = f ( x ) | a ∪ b | a ; b | a ∗ | a d Definition (dG L Formula P ) p ( e 1 , . . . , e n ) | e 1 ≥ e 2 | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � a � P | [ a ] P TOCL’15 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 8 / 26
Differential Game Logic dG L : Syntax Discrete Differential Test Choice Seq. Repeat Assign Equation Game Game Game Game Definition (Hybrid game a ) x := f ( x ) | ? Q | x ′ = f ( x ) | a ∪ b | a ; b | a ∗ | a d Definition (dG L Formula P ) p ( e 1 , . . . , e n ) | e 1 ≥ e 2 | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � a � P | [ a ] P All Some Reals Reals TOCL’15 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 8 / 26
Differential Game Logic dG L : Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game a ) x := f ( x ) | ? Q | x ′ = f ( x ) | a ∪ b | a ; b | a ∗ | a d Definition (dG L Formula P ) p ( e 1 , . . . , e n ) | e 1 ≥ e 2 | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � a � P | [ a ] P All Some Reals Reals TOCL’15 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 8 / 26
Differential Game Logic dG L : Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game a ) x := f ( x ) | ? Q | x ′ = f ( x ) | a ∪ b | a ; b | a ∗ | a d Definition (dG L Formula P ) p ( e 1 , . . . , e n ) | e 1 ≥ e 2 | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � a � P | [ a ] P All Some Angel Reals Reals Wins TOCL’15 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 8 / 26
Differential Game Logic dG L : Syntax Discrete Differential Test Choice Seq. Repeat Dual Assign Equation Game Game Game Game Game Definition (Hybrid game a ) x := f ( x ) | ? Q | x ′ = f ( x ) | a ∪ b | a ; b | a ∗ | a d Definition (dG L Formula P ) p ( e 1 , . . . , e n ) | e 1 ≥ e 2 | ¬ P | P ∧ Q | ∀ x P | ∃ x P | � a � P | [ a ] P All Some Angel Demon Reals Reals Wins Wins TOCL’15 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 8 / 26
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 ) a else b ≡ (? Q ; a ) ∪ (? ¬ Q ; b ) while ( Q ) a ≡ (? Q ; a ) ∗ ; ? ¬ Q a ∩ b ≡ ( a d ∪ b d ) d a × ≡ (( a d ) ∗ ) d ( x ′ = f ( x ) & Q ) d �≡ x ′ = f ( x ) & Q ( x := f ( x )) d ≡ x := f ( x ) ? Q d �≡ ? Q Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 9 / 26
Simple Examples � ( x := x + 1; ( x ′ = x 2 ) d ∪ x := x − 1) ∗ � (0 ≤ x < 1) � ( x := x + 1; ( x ′ = x 2 ) d ∪ ( x := x − 1 ∩ x := x − 2)) ∗ � (0 ≤ x < 1) ( 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 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 10 / 26
Simple Examples � � ( x := x + 1; ( x ′ = x 2 ) d ∪ x := x − 1) ∗ � (0 ≤ x < 1) � ( x := x + 1; ( x ′ = x 2 ) d ∪ ( x := x − 1 ∩ x := x − 2)) ∗ � (0 ≤ x < 1) ( 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 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 10 / 26
Simple Examples � � ( x := x + 1; ( x ′ = x 2 ) d ∪ x := x − 1) ∗ � (0 ≤ x < 1) � � ( x := x + 1; ( x ′ = x 2 ) d ∪ ( x := x − 1 ∩ x := x − 2)) ∗ � (0 ≤ x < 1) ( 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 Andr´ e Platzer (CMU) Differential Game Logic TOCL’15 10 / 26
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