Characterizing Algebraic Invariants by Differential Radical - PowerPoint PPT Presentation

Characterizing Algebraic Invariants by Differential Radical Invariants Khalil Ghorbal Carnegie Mellon university Joint work with Andr´ e Platzer CMACS AVACS November 21st, 2013 K. Ghorbal (CMU) CMACS AVACS 1 CMACS 1 / 31

Introduction Context: ODE in Computer Science/Formal Verification Goal. Automated Formal Reasoning about Ordinary Differential Equations. Formal Reasoning: Global Properties of All solutions. Applications to the Formal Verification of Hybrid Systems • Reachability Analysis • Proof Rules • Synthesis Useful in many other fields: Control Theory, Stability Analysis, Numerical Integration, Integrability of ODE. K. Ghorbal (CMU) CMACS AVACS 2 CMACS 2 / 31

Introduction Algebraic Differential Equations Example x ι = (1 , 0 , 0 , 1) x 1 = − x 2 ˙ x 2 = x 1 ˙ x 3 = x 2 ˙ x 4 = x 3 x 4 ˙ 4 Formally, we study the Initial Value Problem: dx i ( t ) x i = p i ( x ) , 1 ≤ i ≤ n , x (0) = x ι ∈ R n . = ˙ dt ⊕ Parameters are allowed ⊕ Many analytic functions can be encoded (sin, cos, ln, . . . ) ⊕ / ⊖ The initial value ( x ι ) are not restricted ⊖ Evolution domain abstracted (still sound) K. Ghorbal (CMU) CMACS AVACS 3 CMACS 3 / 31

Introduction Approach Algebraic Invariant Expression ∀ t , h ( x ( t )) = 0 , for all x ( t ) solution of the Initial Value Problem. Tools • Classical Algebraic Geometry: Polynomial Ring, Ideals, Varieties • Symbolic Linear Algebra K. Ghorbal (CMU) CMACS AVACS 4 CMACS 4 / 31

Time Abstraction Outline Introduction 1 Time Abstraction 2 Characterization of Invariant Expressions 3 Automated Generation 4 Conclusion 5 K. Ghorbal (CMU) CMACS AVACS 5 CMACS 5 / 31

Time Abstraction Orbits Definition O ( x ι ) def = { x ( t ) | t ∈ U t } ⊆ R n U t domain of definition of the maximal solution of the Initial Value Problem (˙ x = p ( x ) , x (0) = x ι ). Example Solar System Galileo Orbit K. Ghorbal (CMU) CMACS AVACS 6 CMACS 6 / 31

Time Abstraction Orbits: Issues Example 1.0 1.0 0.5 0.5 � 1.0 � 0.5 0.5 1.0 � 1.0 � 0.5 0.5 1.0 � 0.5 � 0.5 � 1.0 � 1.0 Cornu Spiral Lissajous Curve Solutions → Exact Orbit • Computation issues • Decidability issues Idea: Time Abstraction � K. Ghorbal (CMU) CMACS AVACS 7 CMACS 7 / 31

Time Abstraction Affine Varieties and Ideals Roots of h 2 Polynomials 1 h def = x 4 1 + x 2 2 − 2 0 What about the polynomials ph ? � 1 � 2 � 2 � 1 0 1 2 Ideal : stable set of polynomials under external multiplication I = � h 1 , . . . , h r � def = { � r i =1 g i h i | g 1 , . . . , g r ∈ R [ x ] } Affine Variety : common roots of all polynomials in I = { x ∈ R n | ∀ h ∈ I , h ( x ) = 0 } V ( I ) def K. Ghorbal (CMU) CMACS AVACS 8 CMACS 8 / 31

Time Abstraction Affine Varieties and Ideals Roots of h 2 Polynomials 1 h def = x 4 1 + x 2 2 − 2 0 What about the polynomials ph ? � 1 � 2 � 2 � 1 0 1 2 Ideal : stable set of polynomials under external multiplication I = � h 1 , . . . , h r � def = { � r i =1 g i h i | g 1 , . . . , g r ∈ R [ x ] } Affine Variety : common roots of all polynomials in I = { x ∈ R n | ∀ h ∈ I , h ( x ) = 0 } V ( I ) def K. Ghorbal (CMU) CMACS AVACS 8 CMACS 8 / 31

Time Abstraction Variety Embedding of Orbits Zariski Closure Vanishing Ideal : all polynomials that vanish on O ( x ι ) I ( O ( x ι )) def = { h ∈ R [ x ] | ∀ x ∈ O ( x ι ) , h ( x ) = 0 } Closure: Sound Abstraction O ( x ι ) def O ( x ι ) ⊆ ¯ = V ( I ( O ( x ι ))) Orbit − → Vanishing Ideal − → Closure ⊇ Orbit Closure is the smallest variety that contains Orbit . Example x = x � x ( t ) = x ι e t � O ( x ι ) = [ x ι , ∞ [ � I = � 0 � � ¯ ˙ O ( x ι ) = R K. Ghorbal (CMU) CMACS AVACS 9 CMACS 9 / 31

Time Abstraction Example: Variety Embedding Zariski Closure (Intuition) K. Ghorbal (CMU) CMACS AVACS 10 CMACS 10 / 31

Characterization of Invariant Expressions Outline Introduction 1 Time Abstraction 2 Characterization of Invariant Expressions 3 Automated Generation 4 Conclusion 5 K. Ghorbal (CMU) CMACS AVACS 11 CMACS 11 / 31

Characterization of Invariant Expressions Hold on ... Sound Abstraction Orbit ⊆ Closure Goal Explicit Characterization of the Vanishing Ideal I ( O ( x ι )) K. Ghorbal (CMU) CMACS AVACS 12 CMACS 12 / 31

Characterization of Invariant Expressions Lie Derivation Lie derivative along a vector field n ∂ h L p ( h ) def � = p i ( x ) ∂ x i i =1 Properties • Algebraic differentiation • Applies to the polynomial h (not the function t �→ h ( x ( t ))) • Corresponds to the time derivative when the solution is substituted back The Vanishing Ideal is a Differential Ideal L p ( h ) ∈ I ( O ( x ι )) for all h ∈ I ( O ( x ι )). K. Ghorbal (CMU) CMACS AVACS 13 CMACS 13 / 31

Characterization of Invariant Expressions Differential Radical Invariants Theorem h ∈ I ( O ( x ι )) if and only if there exists a finite integer N s.t. L ( N ) ( h ) ∈ � L (0) p ( h ) , . . . , L ( N − 1) ( h ) � ⊆ I ( O ( x ι )) ( ı ) p p L (0) p ( h )( x ι ) = 0 , . . . , L ( N − 1) ( h )( x ι ) = 0 . ( ıı ) p Proof Sketch “ ⇒ ” Ascending Chain Condition on ideals ( R [ x ] is Notherian ) “ ⇐ ” (Global) Cauchy-Lipschitz Theorem K. Ghorbal (CMU) CMACS AVACS 14 CMACS 14 / 31

Characterization of Invariant Expressions Special Case: Invariant (Algebraic) Functions N = 1 and L p ( h ) ∈ � 0 � • L p ( h ) = 0 ∧ h ( x ι ) = 0 − → h = 0 Example 2 2 2 1 1 1 0 0 0 � 1 � 1 � 1 � 2 � 2 � 2 � 2 � 1 0 1 2 � 2 � 1 0 1 2 � 2 � 1 0 1 2 Vector Field ˙ x 1 = − x 2 , Roots of Roots of L p ( h ): h def = x 2 1 + x 2 x 2 = x 1 , x ι = (1 , 0) ˙ 2 − 1 Whole Space K. Ghorbal (CMU) CMACS AVACS 15 CMACS 15 / 31

Characterization of Invariant Expressions Special Case ( N = 1) Darboux Invariants a.k.a. λ -Invariant, Exponential Invariants, P -Consecution, • L p ( h ) = ph ∧ h ( x ι ) = 0 − → h = 0 x 1 = − x 1 + 2 x 2 h = ( x ι 2 − x ι 1 x ι 2 2 ) x 1 − x ι 1 ( x 2 − x 1 x 2 ˙ 2 ) 1 x 2 x 2 = x 2 ˙ L p ( h ) = ( − 1 + 2 x 1 x 2 ) h Example 10 10 1.0 5 5 0.5 0 0.0 0 � 0.5 � 5 � 5 � 1.0 � 10 � 10 � 10 � 5 0 5 10 � 1.0 � 0.5 0.0 0.5 1.0 � 10 � 5 0 5 10 Roots of L p ( h ) Vector Field Roots of h K. Ghorbal (CMU) CMACS AVACS 16 CMACS 16 / 31

Characterization of Invariant Expressions Decidability Corollary It is decidable whether a polynomial h with real algebraic coefficients is an algebraic invariant of an algebraic differential system with real algebraic coefficients and real algebraic initial values. Related Work Generalizes the decidability of invariant functions [A. Platzer ITP’12] K. Ghorbal (CMU) CMACS AVACS 17 CMACS 17 / 31

Characterization of Invariant Expressions Sound Approximation of the Closure ¯ O ( x ι ) Differential Radical Ideals def = � L ( i ) J j p ( h j ) � 0 ≤ i ≤ N − 1 Underapproximation of I ( O ( x ι )) � J j = I ( O ( x ι )) , ℑ finite j ∈ℑ Overapproximation of ¯ O ( x ι ) ¯ � O ( x ι ) ⊆ V ( J i ) 1 ≤ i ≤ r K. Ghorbal (CMU) CMACS AVACS 18 CMACS 18 / 31

Characterization of Invariant Expressions Example System x 1 = − x 2 ˙ x 2 = x 1 ˙ x 3 = x 2 ˙ x 4 = x 3 x 4 ˙ 4 Differential Radical Invariants h 1 = x 3 − x 2 x 4 and h 2 = x 2 4 − x 2 3 − 1 Roots of h 1 Roots of h 2 Orbit K. Ghorbal (CMU) CMACS AVACS 19 CMACS 19 / 31

Characterization of Invariant Expressions Example: cont’d Overapproximation of ¯ O ( x ι ) K. Ghorbal (CMU) CMACS AVACS 20 CMACS 20 / 31

Automated Generation Outline Introduction 1 Time Abstraction 2 Characterization of Invariant Expressions 3 Automated Generation 4 Conclusion 5 K. Ghorbal (CMU) CMACS AVACS 21 CMACS 21 / 31

Automated Generation So ... Sound Abstraction Orbit ⊆ Closure Characterization of I ( O ( x ι )) Explicit Characterization of I ( O ( x ι )) by Differential Radical Invariants Goal Automate the generation of Differential Radical Invariants K. Ghorbal (CMU) CMACS AVACS 22 CMACS 22 / 31

Automated Generation Matrix Representation: Intuition invariant of degree 1 x 1 = a 1 x 1 + a 2 x 2 ˙ h = α 1 x 1 + α 2 x 2 + α 3 x 0 x 2 = b 1 x 1 + b 2 x 2 ˙ L p ( h ) = α 1 ( a 1 x 1 + a 2 x 2 ) + α 2 ( b 1 x 1 + b 2 x 2 ) L p ( h ) ∈ � h � if and only if ∃ β ∈ R s.t. L p ( h ) = β h     ( − a 1 + β ) α 1 + ( − b 1 ) α 2 = 0 − a 1 + β − b 1 0 α 1  = 0 ( − a 2 ) α 1 + ( − b 2 + β ) α 2 = 0 ↔ − a 2 − b 2 + β 0 α 2    ( β ) α 3 = 0 0 0 β α 3 K. Ghorbal (CMU) CMACS AVACS 23 CMACS 23 / 31

Automated Generation Matrix Representation Explicit Ideal Membership N − 1 L ( N ) ( h ) ∈ � L (0) p ( h ) , . . . , L ( N − 1) ( h ) � ↔ L ( N ) g i L ( i ) � ( h ) = p ( h ) p p p i =0 � n + d � Polynomial ↔ Coefficients (up to monomial order) d h ↔ α = ( α 1 , α 2 , . . . , α r ) ↔ β i = ( β 1 , β 2 , . . . , β s i ) g i Matrix Representation N − 1 L ( N ) g i L ( i ) � ( h ) = p ( h ) ↔ M ( β ) α = 0 p i =0 α lies in the Kernel of M ( β ) def = { α ∈ R r | M ( β ) α = 0 } K. Ghorbal (CMU) CMACS AVACS 24 CMACS 24 / 31