Skip to main content
SpringerLink
Log in

Polyhedral Analysis Using Parametric Objectives

  • Conference paper
Static Analysis (SAS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7460))

Included in the following conference series:

Abstract

The abstract domain of polyhedra lies at the heart of many program analysis techniques. However, its operations can be expensive, precluding their application to polyhedra that involve many variables. This paper describes a new approach to computing polyhedral domain operations. The core of this approach is an algorithm to calculate variable elimination (projection) based on parametric linear programming. The algorithm enumerates only non-redundant inequalities of the projection space, hence permits anytime approximation of the output.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amenta, N., Ziegler, G.: Shadows and Slices of Polytopes. In: Symposium on Computational Geometry, pp. 10–19. ACM Press (1996)

    Google Scholar 

  2. Avis, D.: lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes – Combinatorics and Computation, pp. 177–198. Brikhäuser, Basel (2000)

    Chapter  Google Scholar 

  3. 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. Science of Computer Programming 72(1-2), 3–21 (2008)

    Article  MathSciNet  Google Scholar 

  4. Boyd, S., Vandenberghe, S.: Convex Optimization. Cambridge University Press (2004)

    Google Scholar 

  5. Burger, E.: Über Homogene Lineare Ungleichungssysteme. Zeitschrift für Angewandte Mathematik und Mechanik 36, 135–139 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chernikova, N.V.: Algorithm for Discovering the Set of All the Solutions of a Linear Programming Problem. Computational Mathematics and Mathematical Physics 8(6), 1387–1395 (1968)

    MATH  Google Scholar 

  7. Chvátal, V.: Linear Programming. W.H. Freeman and Company (1983)

    Google Scholar 

  8. Clarisó, R., Cortadella, J.: The Octahedron Abstract Domain. Science of Computer Programming 64(1), 115–139 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colón, M.A., Sankaranarayanan, S.: Generalizing the Template Polyhedral Domain. In: Barthe, G. (ed.) ESOP 2011. LNCS, vol. 6602, pp. 176–195. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  10. Cousot, P., Halbwachs, N.: Automatic Discovery of Linear Restraints among Variables of a Program. In: Principles of Programming Languages, pp. 84–97. ACM Press (1978)

    Google Scholar 

  11. Dantzig, G.B., Orchard-Hays, W.: The Product Form for the Inverse in the Simplex Method. Mathematical Tables and other Aids to Computation 8(46), 64–67 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  12. Duffin, R.J.: On Fourier’s Analysis of Linear Inequality Systems. Mathematical Programming Studies 1, 71–95 (1974)

    Article  MathSciNet  Google Scholar 

  13. Flexeder, A., Müller-Olm, M., Petter, M., Seidl, H.: Fast Interprocedural Linear Two-Variable Equalities. ACM Transactions on Programming Languages and Systems 33(6) (2011)

    Google Scholar 

  14. Fulara, J., Durnoga, K., Jakubczyk, K., Shubert, A.: Relational Abstract Domain of Weighted Hexagons. Electronic Notes in Theoretical Computer Science 267(1), 59–72 (2010)

    Article  Google Scholar 

  15. Gass, S., Saaty, T.: The Computational Algorithm for the Parametric Objective Function. Naval Research Logistics Quarterly 2(1-2), 39–45 (1955)

    Article  MathSciNet  Google Scholar 

  16. Howe, J.M., King, A.: Logahedra: A New Weakly Relational Domain. In: Liu, Z., Ravn, A.P. (eds.) ATVA 2009. LNCS, vol. 5799, pp. 306–320. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  17. Imbert, J.-L.: Fourier’s Elimination: Which to Choose?. In: First Workshop on Principles and Practice of Constraint Programming, pp. 117–129 (1993)

    Google Scholar 

  18. Jarvis, R.A.: On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters 2(1), 18–21 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: On Polyhedral Projection and Parametric Programming. Journal of Optimization Theory and Applications 138(2), 207–220 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jones, C.N., Maciejowski, J.M.: Reverse Search for Parametric Linear Programming. In: IEEE Conference on Decision and Control, pp. 1504–1509 (2006)

    Google Scholar 

  21. Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V.: Generating All Vertices of a Polyhedron is Hard. Discrete and Computational Geometry 39, 174–190 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kohler, D.A.: Projections of Convex Polyhedral Sets. Technical Report 67-29, Operations Research Centre, University of California, Berkeley (1967)

    Google Scholar 

  23. Lassez, J.-L., Huynh, T., McAloon, K.: Simplification and Elimination of Redundant Linear Arithmetic Constraints. In: Benhamou, F., Colmerauer, A. (eds.) Constraint Logic Programming, pp. 73–87. MIT Press (1993)

    Google Scholar 

  24. Le Verge, H.: A Note on Chernikova’s algorithm. Technical Report 1662, Institut de Recherche en Informatique, Campus Universitaire de Beaulieu, France (1992)

    Google Scholar 

  25. Miné, A.: A New Numerical Abstract Domain Based on Difference-Bound Matrices. In: Danvy, O., Filinski, A. (eds.) PADO 2001. LNCS, vol. 2053, pp. 155–172. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  26. Miné, A.: The Octagon Abstract Domain. Higher-Order and Symbolic Computation 19(1), 31–100 (2006)

    Article  MATH  Google Scholar 

  27. Motzkin, T.S.: Beiträge zur Theorie der Linearen Ungleichungen. PhD thesis, Universität Zurich (1936)

    Google Scholar 

  28. Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The Double Description Method. In: Annals of Mathematics Studies, vol. 2, pp. 51–73. Princeton University Press (1953)

    Google Scholar 

  29. Ponce, J., Sullivan, S., Sudsang, A., Boissonnat, J.-D., Merlet, J.-P.: On Computing Four-Finger Equilibrium and Force-Closure Grasps of Polyhedral Objects. International Journal of Robotics Research 16(2), 11–35 (1997)

    Google Scholar 

  30. Sankaranarayanan, S., Colón, M.A., Sipma, H., Manna, Z.: Efficient Strongly Relational Polyhedral Analysis. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 111–125. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  31. Seidl, H., Flexeder, A., Petter, M.: Interprocedurally Analysing Linear Inequality Relations. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 284–299. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  32. Simon, A., King, A.: Exploiting Sparsity in Polyhedral Analysis. In: Hankin, C., Siveroni, I. (eds.) SAS 2005. LNCS, vol. 3672, pp. 336–351. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  33. Simon, A., King, A., Howe, J.M.: Two Variables per Linear Inequality as an Abstract Domain. In: Leuschel, M. (ed.) LOPSTR 2002. LNCS, vol. 2664, pp. 71–89. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  34. Todd, M.J.: The Many Facets of Linear Programming. Mathematical Programming 91(3), 417–436 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wilde, D.K.: A Library for Doing Polyhedral Operations. Technical Report 785, Institut de Recherche en Informatique, Campus Universitaire de Beaulieu, France (1993)

    Google Scholar 

  36. Ziegler, G.M.: Lectures on Polytopes. Springer (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Informatics, City University London, EC1V 0HB, UK

    Jacob M. Howe

  2. School of Computing, University of Kent, CT2 7NF, UK

    Andy King

Authors
  1. Jacob M. Howe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andy King
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département d’Informatique, École Normale Supérieure, 45, rue d’Ulm, 75005, Paris, France

    Antoine Miné

  2. Department of Computing and Information Sciences, Kansas State University, 234 Nichols Hall, 66506, Manhattan, KS, USA

    David Schmidt

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Howe, J.M., King, A. (2012). Polyhedral Analysis Using Parametric Objectives. In: Miné, A., Schmidt, D. (eds) Static Analysis. SAS 2012. Lecture Notes in Computer Science, vol 7460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33125-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33125-1_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33124-4

  • Online ISBN: 978-3-642-33125-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation