Abstract
This work is aimed at developing an efficient data structure for representing symbolically convex polyhedra. We introduce an original data structure, the Decomposed Convex Polyhedron (DCP), that is closed under intersection and linear transformations, and allows to check inclusion, equality, and emptiness. The main feature of DCPs lies in their ability to represent concisely polyhedra that can be expressed as combinations of simpler sets, which can overcome combinatorial explosion in high dimensional spaces. DCPs also have the advantage of being reducible into a canonical form, which makes them efficient for representing simple sets constructed by long sequences of manipulations, such as those handled by state-space exploration tools. Their practical efficiency has been evaluated with the help of a prototype implementation, with promising results.
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Notes
- 1.
They are also known as NNC (Not Necessarily Closed) polyhedra, or copolyhedra.
- 2.
The test cases are available at http://www.montefiore.ulg.ac.be/~boigelot/research/atva2018-case-study.tgz.
References
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)
Avis, D.: A revised implementation of the reverse search vertex enumeration algorithm. Polytopes – Combinatorics and Computation, pp. 177–198. Birkhäuser, Basel (2000)
Bachem, A., Grötschel, M.: Characterizations of adjacency of faces of polyhedra. Mathematical Programming at Oberwolfach, pp. 1–22. Springer, Berlin (1981)
Bagnara, R., Hill, P.M., Zaffanella, E.: The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Sci. Comput. Program. 72(1–2), 3–21 (2008)
Bagnara, R., Hill, P.M., Zaffanella, E.: Applications of polyhedral computations to the analysis and verification of hardware and software systems. Theor. Comput. Sci. 410(46), 4672–4691 (2009)
Barrett, C., Stump, A., Tinelli, C.: The SMT-LIB Standard: Version 2.0. In: Proceedings of the SMT’10 (2010)
Boigelot, B., Herbreteau, F., Mainz, I.: Acceleration of affine hybrid transformations. In: Cassez, F., Raskin, J.-F. (eds.) ATVA 2014. LNCS, vol. 8837, pp. 31–46. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11936-6_4
Boigelot, B., Jodogne, S., Wolper, P.: An effective decision procedure for linear arithmetic over the integers and reals. ACM Trans. Comput. Log. 6(3), 614–633 (2005)
Bournez, O., Maler, O., Pnueli, A.: Orthogonal polyhedra: Representation and computation. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC’1999. LNCS, vol. 1569, pp. 46–60. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48983-5_8
Bouton, T., Caminha B. de Oliveira, D., Déharbe, D., Fontaine, P.: veriT: an open, trustable and efficient SMT-solver. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 151–156. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02959-2_12
Chernikova, N.: Algorithm for finding a general formula for the non-negative solutions of a system of linear inequalities. USSR Comput. Math. Math. Phys. 5(2), 228–233 (1965)
Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the POPL’77. pp. 238–252. ACM Press (1977)
Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Proceedings of the POPL’78. pp. 84–96. ACM (1978)
Degbomont, J.F.: Implicit Real-Vector Automata. Ph.D. thesis, Université de Liège (2013)
Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. Int. J. Softw. Tools Technol. Transf. 10(3), 263–279 (2008)
Fukuda, K.: cdd. https://www.inf.ethz.ch/personal/fukudak/cdd_home/
Singh, G., Püschel, M., Vechev, M.: Fast polyhedra abstract domain. In: Proceedings of the POPL’17, pp. 46–59. ACM (2017)
Halbwachs, N., Proy, Y.-E., Raymond, P.: Verification of linear hybrid systems by means of convex approximations. In: Le Charlier, B. (ed.) SAS 1994. LNCS, vol. 864, pp. 223–237. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58485-4_43
Halbwachs, N., Proy, Y.E., Roumanoff, P.: Verification of real-time systems using linear relation analysis. Form. Methods Syst. Des. 11(2), 157–185 (1997)
Jeannet, B., Miné, A.: Apron: A library of numerical abstract domains for static analysis. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 661–667. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02658-4_52
Le Verge, H., Wilde, D.: PolyLib. http://www.irisa.fr/polylib/
Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The Double Description Method, pp. 51–74. Princeton University Press, Princeton (1953)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1999)
Acknowledgment
The authors wish to thank Pascal Fontaine and Laurent Poirrier for their precious help in obtaining relevant benchmarks.
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Boigelot, B., Mainz, I. (2018). Efficient Symbolic Representation of Convex Polyhedra in High-Dimensional Spaces. In: Lahiri, S., Wang, C. (eds) Automated Technology for Verification and Analysis. ATVA 2018. Lecture Notes in Computer Science(), vol 11138. Springer, Cham. https://doi.org/10.1007/978-3-030-01090-4_17
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