Logic of Hybrid Games Andr´ e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA http://symbolaris.com/ 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) Logic of Hybrid Games 1 / 26
Outline Hybrid Systems Applications 1 Differential Game Logic 2 Operational Semantics Denotational Semantics Determinacy Strategic Closure Ordinals Proofs for Hybrid Systems 3 Axiomatization Soundness and Completeness Corollaries Summary 4 Andr´ e Platzer (CMU) Logic of Hybrid Games 1 / 26
Can you trust a computer to control physics? Andr´ e Platzer (CMU) Logic of Hybrid Games 2 / 26
Hybrid Systems 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) Logic of Hybrid Games 3 / 26
Hybrid Systems 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) Logic of Hybrid Games 3 / 26
Hybrid Systems 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) Logic of Hybrid Games 4 / 26
Hybrid Systems Analysis: Robot Control Challenge (Hybrid Games) Game rules describing play evolution with Discrete dynamics (control decisions) Continuous dynamics (differential equations) Angel/demon choices 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) Logic of Hybrid Games 5 / 26
Hybrid Systems Analysis: Robot Control Challenge (Hybrid Games) Game rules describing play evolution with Discrete dynamics (control decisions) Continuous dynamics (differential equations) Angel/demon choices 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) Logic of Hybrid Games 5 / 26
Family of Differential Dynamic Logics e o n c t t e i n r c u s o i u d s l a stochastic i r a s r nondet e v d a Andr´ e Platzer (CMU) Logic of Hybrid Games 6 / 26
Family of Differential Dynamic Logics differential dynamic logic [ α ] φ φ d L = DL + HP α e o n c t t e i n r c u s o i u d s differential game logic stochastic differential DL dG L = GL + HG Sd L = DL + SHP l a stochastic i r a s r nondet e v d � α � φ � α � φ a φ φ quantified differential DL Qd L = FOL + DL + QHP Andr´ e Platzer (CMU) Logic of Hybrid Games 6 / 26
Successful Hybrid Systems Proofs 0 * [SB := ((amax / b + 1) * ep * v + (v ^ 2 - d ^ 2) / (2 * b) + ((amax / b + 1) * amax * ep ^ 2) / 2)] far 1 [do := d] [state := brake] [?v <= vdes] [?v >= vdes] 2 10 13 neg [mo := m] 8 [a := *] [a := *] 3 11 14 [m := *] [?a >= 0 & a <= amax] [?a <= 0 & a >= -b] 4 12 15 [d := *] cor 5 24 [vdes := *] [?m - z <= SB | state = brake] [?m - z >= SB & state != brake] * 6 17 [?d >= 0 & do ^ 2 - d ^ 2 <= 2 * b * (m - mo) & vdes >= 0] [a := -b] 19 7 18 17 28 [t := 0] rec fsa 21 [{z‘ = v, v‘ = a, t‘ = 1, v >= 0 & t <= ep}] 22 31 � � y t i x e c c c � � � e n t � r y x x y � z ¯ ϑ y 2 y � ̟ ω e x c x 2 � d x x 1 y 1 Andr´ e Platzer (CMU) Logic of Hybrid Games 7 / 26
Successful Hybrid Systems Proofs ( r x , r y ) f y ( v x , v y ) e y x b ( l x , l y ) e x f x Andr´ e Platzer (CMU) Logic of Hybrid Games 7 / 26
Successful Hybrid Systems Proofs z x � i � d � i � x � l � 2 minr � i � y � � i r n x � j � i m p x � k � disc � i � x � i � c x � m � x D Virtual fixture boundary d 0.3 0.2 1 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1.0 0.1 - 1 0.0 5 10 15 20 � 0.1 � 0.1 � 0.2 � 0.3 � 0.2 Andr´ e Platzer (CMU) Logic of Hybrid Games 7 / 26 � 0.3
Differential Game Logic dG L : Syntax Definition (Hybrid game α ) x := θ | ? H | x ′ = θ & H | α ∪ β | α ; β | α ∗ | α d Definition (dG L Formula φ ) p ( θ 1 , . . . , θ n ) | θ 1 ≥ θ 2 | ¬ φ | φ ∧ ψ | ∀ x φ | ∃ x φ | � α � φ | [ α ] φ Andr´ e Platzer (CMU) Logic of Hybrid Games 8 / 26
Differential Game Logic dG L : Syntax Discrete Differential Test Choice Seq. Repeat Assign Equation Game Game Game Game Definition (Hybrid game α ) x := θ | ? H | x ′ = θ & H | α ∪ β | α ; β | α ∗ | α d Definition (dG L Formula φ ) p ( θ 1 , . . . , θ n ) | θ 1 ≥ θ 2 | ¬ φ | φ ∧ ψ | ∀ x φ | ∃ x φ | � α � φ | [ α ] φ All Some Reals Reals Andr´ e Platzer (CMU) Logic of Hybrid Games 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 α ) x := θ | ? H | x ′ = θ & H | α ∪ β | α ; β | α ∗ | α d Definition (dG L Formula φ ) p ( θ 1 , . . . , θ n ) | θ 1 ≥ θ 2 | ¬ φ | φ ∧ ψ | ∀ x φ | ∃ x φ | � α � φ | [ α ] φ All Some Reals Reals Andr´ e Platzer (CMU) Logic of Hybrid Games 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 α ) x := θ | ? H | x ′ = θ & H | α ∪ β | α ; β | α ∗ | α d Definition (dG L Formula φ ) p ( θ 1 , . . . , θ n ) | θ 1 ≥ θ 2 | ¬ φ | φ ∧ ψ | ∀ x φ | ∃ x φ | � α � φ | [ α ] φ All Some Angel Reals Reals Wins Andr´ e Platzer (CMU) Logic of Hybrid Games 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 α ) x := θ | ? H | x ′ = θ & H | α ∪ β | α ; β | α ∗ | α d Definition (dG L Formula φ ) p ( θ 1 , . . . , θ n ) | θ 1 ≥ θ 2 | ¬ φ | φ ∧ ψ | ∀ x φ | ∃ x φ | � α � φ | [ α ] φ All Some Angel Demon Reals Reals Wins Wins Andr´ e Platzer (CMU) Logic of Hybrid Games 8 / 26
Definable Game Operators if ( H ) α else β ≡ (? H ; α ) ∪ (? ¬ H ; β ) while ( H ) α ≡ (? H ; α ) ∗ ; ? ¬ H α ∩ β ≡ ( α d ∪ β d ) d α × ≡ (( α d ) ∗ ) d ( x ′ = θ & H ) d �≡ x ′ = θ & H ( x := θ ) d ≡ x := θ Andr´ e Platzer (CMU) Logic of Hybrid Games 9 / 26
More Operators Repeat α as long as both Angel and Demon want to repeat: α ∗∧× ≡ ( c := 0 ∩ c := 1); (? c � = 0; α ; ( c := 0 ∩ c := 1)) ∗ ≡ (( c := 0 ∩ c := 1); ? c � = 0; α ) ∗ Andr´ e Platzer (CMU) Logic of Hybrid Games 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) �� ( ω := 1 ∪ ω := − 1 ∪ ω := 0); ( ̺ := 1 ∩ ̺ := − 1 ∩ ̺ := 0); ( x ′′ = ω x ′ ⊥ , y ′′ = ̺ y ′ ⊥ ) d � ∗ � � x − y � ≤ 1 Andr´ e Platzer (CMU) Logic of Hybrid Games 11 / 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) �� ( ω := 1 ∪ ω := − 1 ∪ ω := 0); ( ̺ := 1 ∩ ̺ := − 1 ∩ ̺ := 0); ( x ′′ = ω x ′ ⊥ , y ′′ = ̺ y ′ ⊥ ) d � ∗ � � x − y � ≤ 1 Andr´ e Platzer (CMU) Logic of Hybrid Games 11 / 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) �� ( ω := 1 ∪ ω := − 1 ∪ ω := 0); ( ̺ := 1 ∩ ̺ := − 1 ∩ ̺ := 0); ( x ′′ = ω x ′ ⊥ , y ′′ = ̺ y ′ ⊥ ) d � ∗ � � x − y � ≤ 1 Andr´ e Platzer (CMU) Logic of Hybrid Games 11 / 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) �� � ( ω := 1 ∪ ω := − 1 ∪ ω := 0); ( ̺ := 1 ∩ ̺ := − 1 ∩ ̺ := 0); ( x ′′ = ω x ′ ⊥ , y ′′ = ̺ y ′ ⊥ ) d � ∗ � � x − y � ≤ 1 Andr´ e Platzer (CMU) Logic of Hybrid Games 11 / 26
Differential Game Logic: Operational Semantics Definition (Hybrid game α : operational semantics) x := θ s x := θ s [ [ θ ] ] s x Andr´ e Platzer (CMU) Logic of Hybrid Games 12 / 26
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