Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal 1 Andrew Sogokon 2 e Platzer 1 Andr´ 1. Carnegie Mellon University 2. University of Edinburgh SAS, Munich, Germany September 11th, 2014 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 1 / 24
Introduction Problem: Checking the Invariance of Algebraic Sets Ordinary Differential Equation ˙ x yz = = f y ˙ − xz z ˙ − xy Algebraic Sets S = { ( x , y , z ) | 3 x 2 + 3 y 2 − 2 x 2 y 2 + 3 z 2 − 2 x 2 z 2 − 2 y 2 z 2 = 0 } � �� � p ( x , y , z ) K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 2 / 24
Introduction Motivations • Theorem Proving with Hybrid Systems • Stability and Safety Analysis of Dynamical Systems • Qualitative Analysis of Differential Equations K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 3 / 24
Introduction Related and Previous Work • Invariance of algebraic sets is decidable • 2 procedures are available: Liu et al. [Liu Zhan Zhao 2011] Differential Radical Characterization [TACAS’14] In this talk We build on top of our previous work [TACAS’14]: • New efficient procedure for algebraic sets • New proof strategies exploiting differential cuts K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 4 / 24
Introduction Related and Previous Work • Invariance of algebraic sets is decidable • 2 procedures are available: Liu et al. [Liu Zhan Zhao 2011] Differential Radical Characterization [TACAS’14] In this talk We build on top of our previous work [TACAS’14]: • New efficient procedure for algebraic sets • New proof strategies exploiting differential cuts K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 4 / 24
Introduction Abstracting Orbits Using Algebraic Sets 1.0 Concrete Domain 0.5 The trajectory of the solution of an x 2 0.0 Initial Value Problem (˙ x = f , x 0 ). � 0.5 Abstract Domain � 1.0 Algebraic Sets. � 1.0 � 0.5 0.0 0.5 1.0 x 1 Problem: Checking soundness Checking the soundness of the abstraction: does a given algebraic set overapproximate the trajectory of the solution ? K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 5 / 24
Efficient Procedure for Algebraic Sets Outline Introduction 1 Efficient Procedure for Algebraic Sets 2 Alternative Lightweight Approach 3 Conclusion 4 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 5 / 24
Efficient Procedure for Algebraic Sets Notation for“ p = 0 is invariant for f ” ( p = 0) → [˙ x = f ]( p = 0) ≡ Zero set of p is an invariant algebraic set for f ≡ Starting with x 0 s.t p ( x 0 ) = 0: for all t > 0, x ( t ) solution of the IVP (˙ x = f , x (0) = x 0 ) is a zero of p N.B. Treating ˙ x = f as a program, one can think of the top formula as representing the Hoare triple { p = 0 } ˙ x = f { p = 0 } . K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 6 / 24
Efficient Procedure for Algebraic Sets Notation for“ p = 0 is invariant for f ” ( p = 0) → [˙ x = f ]( p = 0) ≡ Zero set of p is an invariant algebraic set for f ≡ Starting with x 0 s.t p ( x 0 ) = 0: for all t > 0, x ( t ) solution of the IVP (˙ x = f , x (0) = x 0 ) is a zero of p N.B. Treating ˙ x = f as a program, one can think of the top formula as representing the Hoare triple { p = 0 } ˙ x = f { p = 0 } . K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 6 / 24
Efficient Procedure for Algebraic Sets Notation for“ p = 0 is invariant for f ” ( p = 0) → [˙ x = f ]( p = 0) ≡ Zero set of p is an invariant algebraic set for f ≡ Starting with x 0 s.t p ( x 0 ) = 0: for all t > 0, x ( t ) solution of the IVP (˙ x = f , x (0) = x 0 ) is a zero of p N.B. Treating ˙ x = f as a program, one can think of the top formula as representing the Hoare triple { p = 0 } ˙ x = f { p = 0 } . K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 6 / 24
Efficient Procedure for Algebraic Sets Some Useful Definitions Lie Derivative along a vector field ˙ x = f n n ∂ p ∂ p f i = dp ( x ( t )) � � D ( p ) def = x i = ˙ ∂ x i ∂ x i dt i =1 i =1 Higher-order Lie derivatives: D ( k +1) ( p ) = D ( D ( k ) ( p )) Ideal Membership ∃ λ i ∈ R [ x ] : p = λ 1 q 1 + · · · + λ r q r ↔ p ∈ � q 1 , . . . , q r � Ideal membership can be checked effectively using Gr¨ obner bases. K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 7 / 24
Efficient Procedure for Algebraic Sets Differential Radical Characterization [TACAS’14] D ( N p ) ( p ) ∈ � p , . . . , D ( N p − 1) ( p ) � ∧ p = 0 → D ( N p − 1) ( p ) = 0 . . . D (3) ( p ) ∈ � p , D ( p ) , D (2) ( p ) � ∧ p = 0 → D (2) ( p ) = 0 D (2) ( p ) ∈ � p , D ( p ) � ∧ p = 0 → D ( p ) = 0 D ( p ) ∈ � p � ( ∃ λ ∈ R [ x ] : D ( p ) = λ p ) ( p = 0) → [˙ x = f ]( p = 0) • order N p is finite : unknown a priori and computed on the fly • < N p ideal membership problems: D ( i +1) ( p ) ∈ � p , . . . , D ( i ) ( p ) � • < N p − 1 quantifier elimination problems: p = 0 → D ( i ) ( p ) = 0 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 8 / 24
Efficient Procedure for Algebraic Sets Differential Radical Characterization [TACAS’14] D ( N p ) ( p ) ∈ � p , . . . , D ( N p − 1) ( p ) � ∧ p = 0 → D ( N p − 1) ( p ) = 0 . . . D (3) ( p ) ∈ � p , D ( p ) , D (2) ( p ) � ∧ p = 0 → D (2) ( p ) = 0 D (2) ( p ) ∈ � p , D ( p ) � ∧ p = 0 → D ( p ) = 0 ✗ D ( p ) ∈ � p � ( ∃ λ ∈ R [ x ] : D ( p ) = λ p ) ( p = 0) → [˙ x = f ]( p = 0) • order N p is finite : unknown a priori and computed on the fly • < N p ideal membership problems: D ( i +1) ( p ) ∈ � p , . . . , D ( i ) ( p ) � • < N p − 1 quantifier elimination problems: p = 0 → D ( i ) ( p ) = 0 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 8 / 24
Efficient Procedure for Algebraic Sets Differential Radical Characterization [TACAS’14] D ( N p ) ( p ) ∈ � p , . . . , D ( N p − 1) ( p ) � ∧ p = 0 → D ( N p − 1) ( p ) = 0 . . . D (3) ( p ) ∈ � p , D ( p ) , D (2) ( p ) � ∧ p = 0 → D (2) ( p ) = 0 ✗ D (2) ( p ) ∈ � p , D ( p ) � ∧ p = 0 → D ( p ) = 0 ✓ ✗ D ( p ) ∈ � p � ( ∃ λ ∈ R [ x ] : D ( p ) = λ p ) ( p = 0) → [˙ x = f ]( p = 0) • order N p is finite : unknown a priori and computed on the fly • < N p ideal membership problems: D ( i +1) ( p ) ∈ � p , . . . , D ( i ) ( p ) � • < N p − 1 quantifier elimination problems: p = 0 → D ( i ) ( p ) = 0 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 8 / 24
Efficient Procedure for Algebraic Sets Differential Radical Characterization [TACAS’14] ✓ D ( N p ) ( p ) ∈ � p , . . . , D ( N p − 1) ( p ) � ∧ p = 0 → D ( N p − 1) ( p ) = 0 ✓ . . . ✗ D (3) ( p ) ∈ � p , D ( p ) , D (2) ( p ) � ∧ p = 0 → D (2) ( p ) = 0 ✓ ✗ D (2) ( p ) ∈ � p , D ( p ) � ∧ p = 0 → D ( p ) = 0 ✓ ✗ D ( p ) ∈ � p � ( ∃ λ ∈ R [ x ] : D ( p ) = λ p ) ( p = 0) → [˙ x = f ]( p = 0) • order N p is finite : unknown a priori and computed on the fly • < N p ideal membership problems: D ( i +1) ( p ) ∈ � p , . . . , D ( i ) ( p ) � • < N p − 1 quantifier elimination problems: p = 0 → D ( i ) ( p ) = 0 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 8 / 24
Efficient Procedure for Algebraic Sets Differential Radical Characterization [TACAS’14] ✓ D ( N p ) ( p ) ∈ � p , . . . , D ( N p − 1) ( p ) � ∧ p = 0 → D ( N p − 1) ( p ) = 0 ✓ . . . ✗ D (3) ( p ) ∈ � p , D ( p ) , D (2) ( p ) � ∧ p = 0 → D (2) ( p ) = 0 ✓ ✗ D (2) ( p ) ∈ � p , D ( p ) � ∧ p = 0 → D ( p ) = 0 ✓ ✗ D ( p ) ∈ � p � ( ∃ λ ∈ R [ x ] : D ( p ) = λ p ) ( p = 0) → [˙ x = f ]( p = 0) • order N p is finite : unknown a priori and computed on the fly • < N p ideal membership problems: D ( i +1) ( p ) ∈ � p , . . . , D ( i ) ( p ) � • < N p − 1 quantifier elimination problems: p = 0 → D ( i ) ( p ) = 0 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 8 / 24
Efficient Procedure for Algebraic Sets Differential Radical Characterization [TACAS’14] ✓ D ( N p ) ( p ) ∈ � p , . . . , D ( N p − 1) ( p ) � ∧ p = 0 → D ( N p − 1) ( p ) = 0 ✓ . . . ✗ D (3) ( p ) ∈ � p , D ( p ) , D (2) ( p ) � ∧ p = 0 → D (2) ( p ) = 0 ✓ ✗ D (2) ( p ) ∈ � p , D ( p ) � ∧ p = 0 → D ( p ) = 0 ✓ ✗ D ( p ) ∈ � p � ( ∃ λ ∈ R [ x ] : D ( p ) = λ p ) ( p = 0) → [˙ x = f ]( p = 0) • order N p is finite : unknown a priori and computed on the fly • < N p ideal membership problems: D ( i +1) ( p ) ∈ � p , . . . , D ( i ) ( p ) � • < N p − 1 quantifier elimination problems: p = 0 → D ( i ) ( p ) = 0 K. Ghorbal, A. Sogokon, A. Platzer Invariance of Conjunctive Equations SAS 2014 8 / 24
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