Abstract
We introduce a new method to compute non-convex invariants of numerical programs, which includes the class of switched affine systems with affine guards. We obtain disjunctive and non-convex invariants by associating different partial execution traces with different ellipsoids. A key ingredient is the solution of non-monotone fixed points problems over the space of ellipsoids with a reduction to small size linear matrix inequalities. This allows us to analyze instances that are inaccessible in terms of expressivity or scale by earlier methods based on semi-definite programming.
- A. Adjé and P.-L. Garoche. 2015. Automatic Synthesis of Piecewise Linear Quadratic Invariants for Programs. In Proceedings of VMCAI. 99--116. Google Scholar
Digital Library
- A. Adjé, S. Gaubert, and E. Goubault. 2010. Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis. In (ESOP 2010). Number 6012 in Lecture Notes in Computer Science. Springer, 23--42. Google Scholar
Digital Library
- A. A. Ahmadi, R. M. Jungers, P. A. Parrilo, and M. Roozbehani. 2014. Joint Spectral Radius and Path-complete Graph Lyapunov Functions. SIAM J. Control and Optimization 52, 1 (2014), 687--717.Google Scholar
Cross Ref
- X. Allamigeon, S. Gaubert, E. Goubault, S. Putot, and N. Stott. 2016. A Scalable Algebraic Method to Infer Quadratic Invariants of Switched Systems. ACM Trans. Embedded Comput. Syst. 15, 4 (2016), 69:1--69:20. Google Scholar
Digital Library
- Koenraad M. R. Audenaert. 2013. Schur multiplier norms for Loewner matrices. Linear Algebra Appl. 439, 9 (2013), 2598--2608.Google Scholar
Cross Ref
- A. Ben-Tal and A. S. Nemirovskiaei. 2001. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA. Google Scholar
Digital Library
- Robert G. Bland, Donald Goldfarb, and Michael J. Todd. 1981. The ellipsoid method: A survey. Operations Research 29, 6 (1981), 1039--1091. Google Scholar
Digital Library
- A. D. Blondel and J. N. Tsitsiklis. 2000. A survey of computational complexity results in systems and control. Automatica 36 (2000), 1249--1274. Google Scholar
Digital Library
- Silvére Bonnabel and Rodolphe Sepulchre. 2010. Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31, 3 (2010), 1055--1070.Google Scholar
Cross Ref
- F. Bourdoncle. 1992. Abstract interpretation by dynamic partitioning. J. Funct. Program. 2, 4 (1992), 407--423.Google Scholar
Cross Ref
- S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. 1994. Linear Matrix Inequalities in System and Control Theory. Studies in Applied Mathematics, Vol. 15. SIAM, Philadelphia, PA.Google Scholar
- M. S. Branicky. 1998. Multiple lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automat. Control 43, 4 (Apr 1998), 475--482.Google Scholar
Cross Ref
- H. Busemann. 1950. The foundations of minkowskian geometry.Commentarii mathematici Helvetici 24 (1950), 156--187. http://eudml.org/doc/139004Google Scholar
- P. Cousot. 2005. Proving program invariance and termination by parametric abstraction, lagrangian relaxation and semidefinite programming. In VMCAI 2005, Paris, France, January 17-19, 2005, Proceedings. 1--24. Google Scholar
Digital Library
- P. Cousot and R. Cousot. 1977. Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In Proceedings of the 4th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL’77). Google Scholar
Digital Library
- P. Cousot and N. Halbwachs. 1978. Automatic discovery of linear restraints among variables of a program. In Proceedings of POPL’78. ACM, 84--96. Google Scholar
Digital Library
- E. de Klerk and F. Vallentin. 2016. On the turing model complexity of interior point methods for semidefinite programming. SIAM J. Optim. 26, 3 (2016), 1944--1961.Google Scholar
Cross Ref
- A. Deutsch. 1990. On determining lifetime and aliasing of dynamically allocated data in higher-order functional specifications. In Conference Record of the Seventeenth Annual ACM Symposium on Principles of Programming Languages, San Francisco, California, USA, January 1990. 157--168. Google Scholar
Digital Library
- J. Feret. 2004. Static analysis of digital filters. In Proceedings of ESOP’04. 33--48.Google Scholar
Cross Ref
- E. Feron and F. Alegre. 2008. Control software analysis, part I Open-loop properties. CoRR abs/0809.4812 (2008).Google Scholar
- T. Martin Gawlitza, H. Seidl, A. Adjé, S. Gaubert, and E. Goubault. 2012. Abstract interpretation meets convex optimization. J. Symb. Comput. 47, 12 (2012), 1416--1446. Google Scholar
Digital Library
- R. Giacobazzi and F. Ranzato. 1998. Optimal domains for disjunctive abstract interpretation. Science of Computer Programming 32, 1-3 (1998), 177--210. 6th European Symposium on Programming. Google Scholar
Digital Library
- E. Goubault and S. Putot. 2009. A zonotopic framework for functional abstractions. CoRR abs/0910.1763 (2009).Google Scholar
- M. Stingl J. Fiala, M. Koċvara. 2013. PENLAB: A MATLAB solver for nonlinear semidefinite optimization. (2013).Google Scholar
- B. Jeannet, N. Halbwachs, and P. Raymond. 1999. Dynamic partitioning in analyses of numerical properties. In Static Analysis, 6th International Symposium, SAS’99, Venice, Italy, September 22-24, 1999, Proceedings. 39--50. Google Scholar
Digital Library
- R. V. Kadison. 1951. Order properties of bounded self-adjoint operators. Proc. Amer. Math. Soc. 2, 3 (1951), 505--510. http://www.jstor.org/stable/2031784.Google Scholar
Cross Ref
- J. Löfberg. 2004. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference. Taipei, Taiwan.Google Scholar
Cross Ref
- M. Martel. 2003. Improving the static analysis of loops by dynamic partitioning techniques. In (SCAM 2003), 26-27 September 2003, Amsterdam, The Netherlands. 13--21.Google Scholar
Cross Ref
- L. Mauborgne and X. Rival. 2005. Trace partitioning in abstract interpretation based static analyzers. In European Symposium on Programming (ESOP’05) (Lecture Notes in Computer Science), M. Sagiv (Ed.), Vol. 3444. Springer-Verlag, 5--20. Google Scholar
Digital Library
- A. Miné. 2004. Weakly Relational Numerical Abstract Domains. Ph.D. Dissertation. École Polytechnique, Palaiseau, France.Google Scholar
- A. Miné. 2006. Symbolic methods to enhance the precision of numerical abstract domains. In VMCAI 2006, Charleston, SC, USA, January 8-10, 2006, Proceedings. 348--363. Google Scholar
Digital Library
- M. Müller-Olm and H. Seidl. 2004. Computing polynomial program invariants. Inf. Process. Lett. 91, 5 (2004), 233--244. Google Scholar
Digital Library
- P. Nilsson, U. Boscain, M. Sigalotti, and J. Newling. 2013. Invariant sets of defocused switched systems. In Conference of Decision and Control.Google Scholar
- M. Oulamara and A. J. Venet. 2015. CAV 2015, San Francisco, CA, USA, July 18-24, 2015, Proceedings, Part I. Springer International Publishing, Cham, Chapter Abstract Interpretation with Higher-Dimensional Ellipsoids and Conic Extrapolation, 415--430.Google Scholar
- E. Rodríguez-Carbonell and D. Kapur. 2007. Automatic generation of polynomial invariants of bounded degree using abstract interpretation. Sci. Comput. Program. 64, 1 (2007), 54--75. Google Scholar
Digital Library
- P. Roux and P.-L. Garoche. 2013. Integrating policy iterations in abstract interpreters. In ATVA (Lecture Notes in Computer Science), D. Van Hung and M. Ogawa (Eds.), Vol. 8172. Springer, 240--254.Google Scholar
- P. Roux, R. Jobredeaux, P.-L. Garoche, and E. Feron. 2012. A generic ellipsoid abstract domain for linear time invariant systems. In Proceedings of HSCC. 105--114. Google Scholar
Digital Library
- Pierre Roux, Yuen-Lam Voronin, and Sriram Sankaranarayanan. 2016. Validating Numerical Semidefinite Programming Solvers for Polynomial Invariants. Springer Berlin Heidelberg, Berlin, Heidelberg, 424--446.Google Scholar
- S. Sankaranarayanan, H. B. Sipma, and Z. Manna. 2005. Scalable analysis of linear systems using mathematical programming. In The Sixth International Conference on Verification, Model Checking and Abstract Interpretation (VMCAI’05) (LNCS), Vol. 3385. 25--41. Google Scholar
Digital Library
- P. Sotin, B. Jeannet, F. Védrine, and E. Goubault. 2011. Policy Iteration within Logico-Numerical Abstract Domains. 290--305. Google Scholar
Digital Library
- R. H. Tütüncü, K. C. Toh, and M. J. Todd. 2003. Solving semidefinite-quadratic-linear programs using SDPT3. Mathematical Programming 95, 2 (2003), 189--217.Google Scholar
Cross Ref
- A. Venet. 1996. Abstract cofibered domains: Application to the alias analysis of untyped programs. In Static Analysis, Third International Symposium, SAS’96, Aachen, Germany, September 24-26, 1996, Proceedings. 366--382. Google Scholar
Digital Library
- A. Venet. 2002. Nonuniform alias analysis of recursive data structures and arrays. In Static Analysis, 9th International Symposium, SAS 2002, Madrid, Spain, September 17-20, 2002, Proceedings. 36--51. Google Scholar
Digital Library
Index Terms
A Fast Method to Compute Disjunctive Quadratic Invariants of Numerical Programs
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