Characterizing Algebraic Invariants by Differential Radical Invariants

We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion.

[1]  Stephen Prajna.  Barrier certificates for nonlinear model validation , 2006, Autom..

[2]  Peter Jonsson,et al.  Essential Convexity and Complexity of Semi-Algebraic Constraints , 2012, Log. Methods Comput. Sci..

[3]  André Platzer,et al.  Differential-algebraic Dynamic Logic for Differential-algebraic Programs , 2010, J. Log. Comput..

[4]  Gerardo Lafferriere,et al.  Symbolic Reachability Computation for Families of Linear Vector Fields , 2001, J. Symb. Comput..

[5]  Manfred Morari,et al.  Hybrid Systems: Computation and Control, 8th International Workshop, HSCC 2005, Zurich, Switzerland, March 9-11, 2005, Proceedings , 2005, HSCC.

[6]  Jeffrey Shallit,et al.  The Computational Complexity of Some Problems of Linear Algebra , 1996, J. Comput. Syst. Sci..

[7]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[8]  Ashish Tiwari.  Approximate Reachability for Linear Systems , 2003, HSCC.

[9]  Ashish Tiwari,et al.  Nonlinear Systems: Approximating Reach Sets , 2004, HSCC.

[10]  Arnaldo Vieira Moura,et al.  Transcendental inductive invariants generation for non-linear differential and hybrid systems , 2012, HSCC '12.

[11]  Rick Miranda,et al.  Algebraic Curves and Riemann Surfaces , 1995 .

[12]  L. Dubins.  On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents , 1957 .

[13]  Enric Rodríguez-Carbonell,et al.  An Abstract Interpretation Approach for Automatic Generation of Polynomial Invariants , 2004, SAS.

[14]  André Platzer,et al.  Logical Analysis of Hybrid Systems - Proving Theorems for Complex Dynamics , 2010 .

[15]  P. Hartman.  Ordinary Differential Equations , 1965 .

[16]  Simone Tini,et al.  Taylor approximation for hybrid systems , 2005, Inf. Comput..

[17]  Sriram Sankaranarayanan,et al.  Automatic invariant generation for hybrid systems using ideal fixed points , 2010, HSCC '10.

[18]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[19]  Roberto Giacobazzi.  Static analysis : 11th International Symposium, SAS 2004, Verona, Italy, August 26-28, 2004 : proceedings , 2004 .

[20]  Ashish Tiwari,et al.  Generating Polynomial Invariants for Hybrid Systems , 2005, HSCC.

[21]  André Platzer,et al.  Differential Dynamic Logic for Hybrid Systems , 2008, Journal of Automated Reasoning.

[22]  Robin Hartshorne,et al.  Algebraic geometry , 1977, Graduate texts in mathematics.

[23]  Rolf Neuhaus,et al.  Computation of real radicals of polynomial ideals — II , 1998 .

[24]  G. Laumon,et al.  A Series of Modern Surveys in Mathematics , 2000 .

[25]  André Platzer,et al.  Logics of Dynamical Systems , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[26]  André Platzer,et al.  The Structure of Differential Invariants and Differential Cut Elimination , 2011, Log. Methods Comput. Sci..

[27]  Sriram Sankaranarayanan,et al.  Automatic abstraction of non-linear systems using change of bases transformations , 2011, HSCC '11.

[28]  André Platzer,et al.  Characterizing Algebraic Invariants by Differential Radical Invariants (CMU-CS-13-129) , 2014 .

[29]  Henny B. Sipma,et al.  Constructing invariants for hybrid systems , 2004, Formal Methods Syst. Des..

[30]  Arnaldo Vieira Moura,et al.  Generating Invariants for Non-linear Hybrid Systems by Linear Algebraic Methods , 2010, SAS.

[31]  S. Shankar Sastry,et al.  Conflict resolution for air traffic management: a study in multiagent hybrid systems , 1998, IEEE Trans. Autom. Control..

[32]  Naijun Zhan,et al.  Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems , 2011, Math. Comput. Sci..

[33]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[34]  Naijun Zhan,et al.  Computing semi-algebraic invariants for polynomial dynamical systems , 2011, 2011 Proceedings of the Ninth ACM International Conference on Embedded Software (EMSOFT).

[35]  André Platzer,et al.  A Differential Operator Approach to Equational Differential Invariants - (Invited Paper) , 2012, ITP.

[36]  Ashish Tiwari,et al.  Abstractions for hybrid systems , 2008, Formal Methods Syst. Des..

[37]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[38]  A. Goriely.  Integrability and Nonintegrability of Dynamical Systems , 2001 .

[39]  Michael Colón,et al.  Approximating the Algebraic Relational Semantics of Imperative Programs , 2004, SAS.

[40]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[42]  Christopher J. Hillar,et al.  Most Tensor Problems Are NP-Hard , 2009, JACM.

[43]  Edmund M. Clarke,et al.  Computing differential invariants of hybrid systems as fixedpoints , 2008, Formal Methods Syst. Des..

[44]  Henny B. Sipma,et al.  Fixed Point Iteration for Computing the Time Elapse Operator , 2006, HSCC.

[45]  André Platzer,et al.  The Complete Proof Theory of Hybrid Systems , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.