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Case Study: Lattice Cover Graph Construction

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Abstract

In this chapter, the correctness by construction approach is applied to an algorithmic problem that lies well off the beaten track of classical text book examples. The algorithm has been in the public domain since about 2000, but was only clearly explained and its correctness shown in 2010 [26]. The algorithm has also been shown to be considerably more efficient than its rivals.

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Notes

  1. 1.

    It is trivial to adapt the algorithm for cases where \(\neg (X \subset {\top }_{\mathcal{L}})\). Two cases then need to be handled. If \(X \supset {\top }_{\mathcal{L}}\) then X simply becomes the new top of the enlarged SICL. If \(\neg (X \subset {\top }_{\mathcal{L}}\wedge X \supset {\top }_{\mathcal{L}})\) then insert \(X \cap {\top }_{\mathcal{L}}\) into \(\mathcal{L}\), create a supremum for X and \({\top }_{\mathcal{L}}\) and join X and \({\top }_{\mathcal{L}}\) to their supremum.

  2. 2.

    Of course, we assume that \(\vert \mathcal{X}\vert < \infty \). However, the algorithm specifications above are not inherently limited to dealing with finite sets. They could, in principle, be implemented on a computer that had some way of representing infinitely large sets and which could carry out the required set operations (intersection, membership, union) on those infinite sets.

  3. 3.

    Note: Concept Explorer’s author has requested that users cite his Russian text, [49].

  4. 4.

    Algorithms were implemented in C++ on the same codesbase. Tests were performed on a PentiumTM 4–2 GHz computer with 1 Gigabyte RAM running under Windows XPTM.

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Authors and Affiliations

  1. Department of Computer Science, University of Pretoria, Pretoria, South Africa

    Derrick G. Kourie

  2. FASTAR Group, Information Science, Stellenbosch University, Stellenbosch, South Africa

    Bruce W. Watson

Authors
  1. Derrick G. Kourie
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  2. Bruce W. Watson
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© 2012 Springer-Verlag Berlin Heidelberg

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Kourie, D.G., Watson, B.W. (2012). Case Study: Lattice Cover Graph Construction. In: The Correctness-by-Construction Approach to Programming. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27919-5_6

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  • DOI: https://doi.org/10.1007/978-3-642-27919-5_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27918-8

  • Online ISBN: 978-3-642-27919-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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