Differential Equation Axiomatization The Impressive Power of - PowerPoint PPT Presentation

Differential Equation Axiomatization The Impressive Power of Differential Ghosts Andr´ e Platzer Yong Kiam Tan 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 1 / 18

Outline Differential Dynamic Logic 1 Proofs for Differential Equations 2 Differential Invariants / Cuts / Ghosts Completeness for Differential Equation Invariants 3 Darboux are Differential Ghosts Derived Semialgebraic Invariants Real Induction Derived Local Progress Completeness for Invariants Summary 4 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 1 / 18

Hybrid Systems: Example Robot Control Challenge (Hybrid Systems) Fixed law describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 1.5 (control decisions) 1.0 0.5 Continuous dynamics 0.0 0 1 2 3 4 5 6 (differential equations) 7 x 3.0 v 2 a 6 m 2.5 5 1 2.0 4 1.5 5 t 0 3 1 2 3 4 1.0 2 � 1 0.5 1 5 t 5 t 0.0 0 � 2 0 1 2 3 4 0 1 2 3 4 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 2 / 18

Contributions: Differential Equation Axiomatization Classical approach: 1 � Given ODE 2 � Solve ODE 3 � Analyze solution Descriptive power of ODEs: ODE much easier than its solution � Analyzing ODEs via their solutions undoes their descriptive power! describe ODE analyze ODE Poincar´ e 1881 � describe solution analyze solution 1 Now: Logical foundations of differential equation invariants 2 Identify axioms for differential equations 3 Completeness for differential equation invariants 4 Uniformly substitutable axioms, not infinite axiom schemata 5 Decide invariance by proof Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 3 / 18

Outline Differential Dynamic Logic 1 Proofs for Differential Equations 2 Differential Invariants / Cuts / Ghosts Completeness for Differential Equation Invariants 3 Darboux are Differential Ghosts Derived Semialgebraic Invariants Real Induction Derived Local Progress Completeness for Invariants Summary 4 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 3 / 18

Hybrid Systems = Differential Equations + Discrete Concept (Differential Dynamic Logic) (JAR’08,LICS’12) u 2 ≤ v 2 +9 2 → [ u ′ = − v + u 4(1 − u 2 − v 2 ) , v ′ = u + v 4(1 − u 2 − v 2 )] u 2 ≤ v 2 +9 2 u 2 + v 2 = 1 → [ u ′ = − v + u 4(1 − u 2 − v 2 ) , v ′ = u + v 4(1 − u 2 − v 2 )] u 2 + v 2 = 1 v 1 0 - 1 u - 2 0 2 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 4 / 18

Outline Differential Dynamic Logic 1 Proofs for Differential Equations 2 Differential Invariants / Cuts / Ghosts Completeness for Differential Equation Invariants 3 Darboux are Differential Ghosts Derived Semialgebraic Invariants Real Induction Derived Local Progress Completeness for Invariants Summary 4 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 4 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost y ′ = g ( x , y ) x x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations Differential Invariant Differential Cut Differential Ghost y ′ = g ( x , y ) x inv x ′ = f ( x ) 0 t x C x x P w w w w w Q Q Q Q u u u r r r 0 0 0 x ′ = f ( x ) & Q x ′ = f ( x ) & Q x ′ = f ( x ) & Q Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 5 / 18

Differential Invariants for Differential Equations x P Differential Invariant w w Q ⊢ [ x ′ := f ( x )]( P ) ′ Q u P ⊢ [ x ′ = f ( x ) & Q ] P r 0 x ′ = f ( x ) & Q C x Differential Cut w w P ⊢ [ x ′ = f ( x ) & Q ] C P ⊢ [ x ′ = f ( x ) & Q ∧ C ] P Q Q u P ⊢ [ x ′ = f ( x ) & Q ] P r 0 x ′ = f ( x ) & Q x Differential Ghost w G ⊢ [ x ′ = f ( x ) , y ′ = g ( x , y ) & Q ] G P ↔ ∃ y G Q u P ⊢ [ x ′ = f ( x ) & Q ] P r 0 x ′ = f ( x ) & Q deductive power adds DI ≺ DC ≺ DG JLogComput’10,LMCS’12, LICS’12,JAR’17,LICS’18 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 6 / 18

Differential Invariants for Differential Equations x P Differential Invariant w w Q ⊢ [ x ′ := f ( x )]( P ) ′ Q u P ⊢ [ x ′ = f ( x ) & Q ] P r 0 x ′ = f ( x ) & Q C x Differential Cut w w P ⊢ [ x ′ = f ( x ) & Q ] C P ⊢ [ x ′ = f ( x ) & Q ∧ C ] P Q Q u P ⊢ [ x ′ = f ( x ) & Q ] P r 0 x ′ = f ( x ) & Q x Differential Ghost w G ⊢ [ x ′ = f ( x ) , y ′ = g ( x , y ) & Q ] G P ↔ ∃ y G Q u P ⊢ [ x ′ = f ( x ) & Q ] P r 0 x ′ = f ( x ) & Q if new y ′ = g ( x , y ) has long enough solution JLogComput’10,LMCS’12, LICS’12,JAR’17,LICS’18 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 6 / 18

Outline Differential Dynamic Logic 1 Proofs for Differential Equations 2 Differential Invariants / Cuts / Ghosts Completeness for Differential Equation Invariants 3 Darboux are Differential Ghosts Derived Semialgebraic Invariants Real Induction Derived Local Progress Completeness for Invariants Summary 4 Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 6 / 18

Differential Equation Axiomatization Theorem (Algebraic Completeness) (LICS’18) dL calculus is a sound & complete axiomatization of algebraic invariants of polynomial differential equations. They are decidable by DI,DC,DG Theorem (Semialgebraic Completeness) (LICS’18) dL calculus with RI is a sound & complete axiomatization of semialgebraic invariants of differential equations. They are decidable in dL Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 7 / 18

ODE Axiomatization: Derived Darboux Rules Gaston Darboux 1878 Darboux equalities are DG Q ⊢ p ′ = gp ( g ∈ R [ x ]) p = 0 ⊢ [ x ′ = f ( x ) & Q ] p = 0 Definable p ′ for Lie-derivative w.r.t. ODE Andr´ e Platzer, Yong Kiam Tan (CMU) Differential Equation Axiomatization LICS’18 8 / 18