A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal 1 Andrew Sogokon 2 e Platzer 1 Andr´ 1. Carnegie Mellon University 2. University of Edinburgh VMCAI, Mumbai, India January 13th, 2015 K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 1 / 24
Introduction Problem: Checking the Invariance of Algebraic Sets Ordinary Differential Equation x 2 � ˙ � � � x y = = p − x + (1 − x 2 ) y ˙ y x 1 (Real) Algebraic Sets V R ( h ) = { ( x , y ) ∈ R | x 2 + y 2 − 1 = 0 } � �� � h ( x , y )=0 K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 2 / 24
Introduction Context Motivations • Theorem Proving for Hybrid Systems • Stability and Safety Analysis of Dynamical Systems • Qualitative Analysis of Differential Equations Current Status • Invariance of algebraic sets is decidable • A decision procedure exists • Many sufficient conditions are known K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 3 / 24
Introduction Contributions Hierarchy of the different proof rules • Compare the deductive power of 7 proof rules, 2 of which are novel • Subclasses of algebraic sets characterized by each proof rule Assess the deductive power versus efficiency trade-off • Deductive power increase � computational cost increase ? • What is the practical efficiency of those proof rules ? K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 4 / 24
Proof rules Outline Introduction 1 Proof rules 2 Deductive Hierarchy 3 Practical performance analysis 4 Square-free Reduction 5 Conclusion 6 K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 4 / 24
Proof rules Definitions ∇ h := ( ∂ h , . . . , ∂ h Gradient ) ∂ x 1 ∂ x n D p ( h ) := dh ( x ( t )) Lie Derivation = �∇ h , p � (˙ x = p ) dt Singular Locus � x ∈ R n | ∂ h = 0 ∧ · · · ∧ ∂ h � SL( h ) := { x ∈ R n | ∇ h = 0 } = = 0 ∂ x 1 ∂ x n h ( x ) = 0 ( x ∈ V R ( h )) is singular if x ∈ SL( h ), regular otherwise. K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 5 / 24
Proof rules Lie’s Criterion [Platzer, ITP 2012] Necessary and sufficient for smooth invariant manifolds (Lie, 1893). (Lie) h = 0 → ( D p ( h ) = 0 ∧ ∇ h � = 0 ) ( h = 0) → [˙ x = p ] ( h = 0) x 2 x 2 x 1 x 1 h = 0 non-smooth ✗ h = 0 smooth ✓ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 6 / 24
Proof rules Extensions of Lie Contribution Handling certain singularities (points where ∇ h = 0 ) No flow in the problem variables at singularities on the variety � � �� (Lie ◦ ) h = 0 → D p ( h ) = 0 ∧ ∇ h = 0 → p = 0 ( h = 0) → [˙ x = p ] ( h = 0) Flow at singularities on the variety is directed into the variety � � (Lie ∗ ) h = 0 → D p ( h ) = 0 ∧ ( ∇ h = 0 → h ( x + λ p ) = 0) . ( h = 0) → [˙ x = p ] ( h = 0) K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 7 / 24
Proof rules Extensions of Lie Contribution Handling certain singularities (points where ∇ h = 0 ) No flow in the problem variables at singularities on the variety � � �� (Lie ◦ ) h = 0 → D p ( h ) = 0 ∧ ∇ h = 0 → p = 0 ( h = 0) → [˙ x = p ] ( h = 0) Flow at singularities on the variety is directed into the variety � � (Lie ∗ ) h = 0 → D p ( h ) = 0 ∧ ( ∇ h = 0 → h ( x + λ p ) = 0) . ( h = 0) → [˙ x = p ] ( h = 0) K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 7 / 24
Proof rules Extensions of Lie Contribution Handling certain singularities (points where ∇ h = 0 ) No flow in the problem variables at singularities on the variety � � �� (Lie ◦ ) h = 0 → D p ( h ) = 0 ∧ ∇ h = 0 → p = 0 ( h = 0) → [˙ x = p ] ( h = 0) Flow at singularities on the variety is directed into the variety � � (Lie ∗ ) h = 0 → D p ( h ) = 0 ∧ ( ∇ h = 0 → h ( x + λ p ) = 0) . ( h = 0) → [˙ x = p ] ( h = 0) K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 7 / 24
Proof rules Extensions of Lie: Lie ◦ Contribution Handling certain singularities (points where ∇ h = 0 ) x 2 x 1 Lie ◦ ✓ Lie ∗ ✓ Lie ✗ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 8 / 24
Proof rules Extensions of Lie: Lie ∗ Contribution Handling certain singularities (points where ∇ h = 0 ) Lie ◦ ✗ Lie ∗ ✓ Lie ✗ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 9 / 24
Proof rules Differential Invariant (DI = ) [Platzer, J. Log. Comput. 2010] Necessary and sufficient for conserved quantities (integrals of motion). D p ( h ) = 0 (DI = ) ( h = 0) → [˙ x = p ] ( h = 0) x 2 x 2 x 1 x 1 h conserved ✓ h not conserved ✗ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 10 / 24
Proof rules Extensions of DI = [Sankaranarayanan et al., FMSD 2008] Continuous consecutions (C-c) and polynomial consecutions (P-c) are Darboux polynomials (Darboux, 1878). ∃ λ ∈ R , D p ( h ) = λ h (C-c) x = p ] ( h = 0) , ( h = 0) → [˙ (P-c) ∃ λ ∈ R [ x ] , D p ( h ) = λ h x = p ] ( h = 0) . ( h = 0) → [˙ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 11 / 24
Proof rules Extensions of DI = [Sankaranarayanan et al., FMSD 2008] � � x 2 , − x 2 h = x 4 2 + 2 x 1 x 3 2 + 6 x 2 2 + 2 x 1 x 2 + x 2 p = (3 1 − 4 2 + x 1 x 2 + 3) , 1 + 3 , D p ( h ) = (6 x 1 − 4 x 2 ) h � �� � λ x 2 x 1 DI = ✗ C-c ✗ P-c ✓ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 12 / 24
Proof rules Differential Radical Invariants (DRI) [G. et al., TACAS 2014, SAS 2014] Necessary and sufficient for invariant varieties. (DRI) h = 0 → � N − 1 i =0 D ( i ) p ( h ) = 0 ( h = 0) → [˙ x = p ] ( h = 0) x 2 x 2 x 2 x 1 x 1 x 1 ✓ ✓ ✓ K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 13 / 24
Proof rules How to compare these different proof rules ? { DI = , C-c , P-c , Lie , Lie ◦ , Lie ∗ , DRI } For some classes of problems, the premises of the proof rules lead to decision procedures . Natural questions: • Given two decision procedures, which is more practical? • Are any of these proof rules redundant? To answer these, we perform • Theoretical comparison • Empirical performance analysis K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 14 / 24
Proof rules How to compare these different proof rules ? { DI = , C-c , P-c , Lie , Lie ◦ , Lie ∗ , DRI } For some classes of problems, the premises of the proof rules lead to decision procedures . Natural questions: • Given two decision procedures, which is more practical? • Are any of these proof rules redundant? To answer these, we perform • Theoretical comparison • Empirical performance analysis K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 14 / 24
Proof rules How to compare these different proof rules ? { DI = , C-c , P-c , Lie , Lie ◦ , Lie ∗ , DRI } For some classes of problems, the premises of the proof rules lead to decision procedures . Natural questions: • Given two decision procedures, which is more practical? • Are any of these proof rules redundant? To answer these, we perform • Theoretical comparison • Empirical performance analysis K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 14 / 24
Proof rules How to compare these different proof rules ? { DI = , C-c , P-c , Lie , Lie ◦ , Lie ∗ , DRI } For some classes of problems, the premises of the proof rules lead to decision procedures . Natural questions: • Given two decision procedures, which is more practical? • Are any of these proof rules redundant? To answer these, we perform • Theoretical comparison • Empirical performance analysis K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 14 / 24
Deductive Hierarchy Outline Introduction 1 Proof rules 2 Deductive Hierarchy 3 Practical performance analysis 4 Square-free Reduction 5 Conclusion 6 K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 14 / 24
Deductive Hierarchy Order Relation A B (R A ) (R B ) → [˙ → [˙ ( h = 0) − x = p ]( h = 0) ( h = 0) − x = p ]( h = 0) Partial Order R A � R B if and only if A = ⇒ B . • R A ∼ R B (R A � R B and R A � R B ) Equivalence . • R A ≺ R B (R A � R B and R A � � R B ) Strict increase of deductive power K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 15 / 24
Deductive Hierarchy Hasse Diagram: Deductive Hierarchy DRI P-c Lie ∗ Lie ◦ C-c DI = Lie K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 16 / 24
Deductive Hierarchy Hasse Diagram: Deductive Hierarchy DRI P-c Lie ∗ Lie ◦ C-c DI = Lie Lie-based Darboux-based K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 16 / 24
Practical performance analysis Outline Introduction 1 Proof rules 2 Deductive Hierarchy 3 Practical performance analysis 4 Square-free Reduction 5 Conclusion 6 K. Ghorbal, A. Sogokon, A. Platzer A Hierarchy of Proof Rules VMCAI 2015 16 / 24
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