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Formalization and Execution of Linear Algebra: From Theorems to Algorithms

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Logic-Based Program Synthesis and Transformation (LOPSTR 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8901))

Abstract

In this work we present a formalization of the Rank Nullity theorem of Linear Algebra in Isabelle/HOL. The formalization is of interest because of various reasons. First, it has been carried out based on the representation of mathematical structures proposed in the HOL Multivariate Analysis library of Isabelle/HOL (which is part of the standard distribution of the proof assistant). Hence, our proof shows the adequacy of such an infrastructure for the formalization of Linear Algebra. Moreover, we enrich the proof with an additional formalization of its computational meaning; to this purpose, we choose to implement the Gauss-Jordan elimination algorithm for matrices over fields, prove it correct, and then apply the Isabelle code generation facility that permits to execute the formalized algorithm. For the algorithm to be code generated, we use again the implementation of matrices available in the HOL Multivariate Analysis library, and enrich it with some necessary features. We report on the precise modifications that we introduce to get code execution from the original representation, and on the performance of the code obtained. We present an alternative verified type refinement of vectors that outperforms the original version. This refinement performs well enough as to be applied to the computation of the rank of some biomedical digital images. Our work proves itself as a suitable basis for the formalization of numerical Linear Algebra in HOL provers that can be successfully applied for computations of real case studies.

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References

  1. Aransay, J. Divasón, J.: Rank Nullity Theorem in Linear Algebra, Archive of Formal Proofs (2013). http://afp.sourceforge.net/entries/Rank_Nullity_Theorem.shtml

  2. Aransay, J., Divasón, J.: Gauss-Jordan elimination in Isabelle/HOL (2013). http://www.unirioja.es/cu/jodivaso/Isabelle/Gauss-Jordan/

  3. Aransay, J., Divasón, J.: Performance Analysis of a Verified Linear Algebra program in SML, Taller de Programación Funcional (TPF 2013), pp. 28–35. Lars-Åke Fredlund and Laura M. Castro (eds.) (2013)

    Google Scholar 

  4. Dénès, M., Mörtberg, A., Siles, V.: A Refinement-Based Approach to Computational Algebra in Coq. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 83–98. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  5. Esparza, J., Lammich, P., Neumann, R., Nipkow, T., Schimpf, A., Smaus, J.-G.: A Fully Verified Executable LTL Model Checker. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 463–478. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  6. Formath Project: Formalisation of Mathematics. http://wiki.portal.chalmers.se/cse/pmwiki.php/ForMath

  7. Haftmann, F., Nipkow, T.: Code Generation via Higher-Order Rewrite Systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  8. Haftmann, F., Krauss, A., Kunčar, O., Nipkow, T.: Data Refinement in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 100–115. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  9. Harrison, J.: A HOL Theory of Euclidean Space. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 114–129. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  10. Harrison, J.: The HOL Light Theory of Euclidean Space. J. Autom. Reasoning 50(2), 173–190 (2013)

    Article  MATH  Google Scholar 

  11. Hales, T.C., Harrison, J., McLaughlin, S., Nipkow, T., Obua, S., Zumkeller, R.: A revision of the Proof of the Kepler Conjecture. Discrete & Computational Geometry 44(1), 1–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heras, J., Coquand, T., Mörtberg, A., Siles, V.: Computing Persistent Homology within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4). Article n. 26

    Google Scholar 

  13. Heras, J., Dénès, M., Mata, G., Mörtberg, A., Poza, M., Siles, V.: Towards a Certified Computation of Homology Groups for Digital Images. In: Ferri, M., Frosini, P., Landi, C., Cerri, A., Di Fabio, B. (eds.) CTIC 2012. LNCS, vol. 7309, pp. 49–57. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  14. Heras, J., Poza, M., Dénès, M., Rideau, L.: Incidence Simplicial Matrices Formalized in Coq/SSReflect. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) MKM 2011 and Calculemus 2011. LNCS, vol. 6824, pp. 30–44. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  15. Hölzl, J., et al.: HOL Multivariate Analysis (2013). http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate_Analysis/index.html

  16. Hölzl, J., Immler, F., Huffman, B.: Type Classes and Filters for Mathematical Analysis in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 279–294. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  17. Lammich, P., Tuerk, T.: Applying Data Refinement for Monadic Programs to Hopcroft’s Algorithm. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 166–182. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  18. Lammich, P.: Automatic Data Refinement. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 84–99. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  19. Lochbihler, A.: Light-Weight Containers for Isabelle: Efficient, Extensible, Nestable. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 116–132. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  20. Nipkow, T.: Gauss-Jordan Elimination for Matrices Represented as Functions. Archive of Formal Proofs (2011). http://afp.sourceforge.net/entries/Gauss-Jordan-Elim-Fun.shtml

  21. Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL: A proof assistant for Higher-Order Logic. Springer (2002)

    Google Scholar 

  22. Roman, S.: Advanced Linear Algebra (Third Edition). Springer (2008)

    Google Scholar 

  23. Gasner, E., Reppy, J.H. (eds.): The Standard ML Basis Library. http://www.standardml.org/Basis/

  24. Wiedijk, F.: Formalizing 100 Theorems. http://www.cs.ru.nl/~freek/100/

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Authors and Affiliations

  1. Departamento de Matemáticas y Computación, Universidad de La Rioja, Edif. Luis Vives, c. Luis de Ulloa s/n., 26004, La Rioja, Spain

    Jesús Aransay & Jose Divasón

Authors
  1. Jesús Aransay
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  2. Jose Divasón
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Correspondence to Jesús Aransay .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, Texas, USA

    Gopal Gupta

  2. Universidad Complutense de Madrid, Madrid, Spain

    Ricardo Peña

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Aransay, J., Divasón, J. (2014). Formalization and Execution of Linear Algebra: From Theorems to Algorithms. In: Gupta, G., Peña, R. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2013. Lecture Notes in Computer Science(), vol 8901. Springer, Cham. https://doi.org/10.1007/978-3-319-14125-1_1

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  • DOI: https://doi.org/10.1007/978-3-319-14125-1_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-14124-4

  • Online ISBN: 978-3-319-14125-1

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