Designing and proving correct a convex hull algorithm with hypermaps in Coq

Brun, Christophe; Dufourd, Jean-François; Magaud, Nicolas

doi:10.1016/j.comgeo.2010.06.006

Cited by 19 publications

(13 citation statements)

References 11 publications

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“…Our framework is a sound basis for subsequent software developments with triangulations and Flip in computational geometry and geometric modeling, for instance in the way of [3,12,13,8] where hypermaps are represented by linked lists. The functional, side-effect-free approach in this formal description has been very useful for the proofs.…”

Section: Resultsmentioning

confidence: 99%

Formal Study of Plane Delaunay Triangulation

Dufourd

Bertot

2010

Lecture Notes in Computer Science

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This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set.

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Section: Resultsmentioning

confidence: 99%

Formal Study of Plane Delaunay Triangulation

Dufourd

Bertot

2010

Lecture Notes in Computer Science

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“…Some formal proofs about combinatorial maps or variants have already been carried out in the domain of computational geometry. Dufourd et al have developed a large Coq library specifying hypermaps used to prove some classical results such as Euler formula for polyhedra [Duf08], the Jordan curve theorem [Duf09], and also some algorithms such as convex hull [BDM12] and image segmentation [Duf07]. In these papers, a combinatorial map or hypermap is represented by an inductive type with some constraints.…”

Section: Map Formalizationmentioning

confidence: 99%

Tests and proofs for custom data generators

Dubois

Giorgetti

2018

Form. Asp. Comput.

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We address automated testing and interactive proving of properties involving complex data structures with constraints, like the ones studied in enumerative combinatorics, e.g., permutations and maps. In this paper we show testing techniques to check properties of custom data generators for these structures. We focus on random property-based testing and bounded exhaustive testing, to find counterexamples for false conjectures in the Coq proof assistant. For random testing we rely on the existing Coq plugin QuickChick and its toolbox to write random generators. For bounded exhaustive testing, we use logic programming to generate all the data up to a given size. We also propose an extension of QuickChick with bounded exhaustive testing based on generators developed inside Coq, but also on correct-by-construction generators developed with Why3. These tools are applied to an original Coq formalization of the combinatorial structures of permutations and rooted maps, together with some operations on them and properties about them. Recursive generators are defined for each combinatorial family. They are used for debugging properties which are finally proved in Coq. This large case study is also a contribution in enumerative combinatorics.

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“…Meikle and Fleuriot [15] formalized an imperative algorithm and verified it using Hoare logic in Isabelle/HOL. Brun et al [4] verify an algorithm based on hypermaps to compute the convex hull.…”

Section: Related Workmentioning

confidence: 99%

A Verified Algorithm for Geometric Zonotope/Hyperplane Intersection

Immler

2015

Proceedings of the 2015 Conference on Certified Programs and Proofs

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To perform rigorous numerical computations, one can use a generalization of interval arithmetic, namely affine arithmetic (AA), which works with zonotopes instead of intervals. Zonotopes are also widely used for reachability analysis of continuous or hybrid systems, where an important operation is the geometric intersection of zonotopes with hyperplanes.We have implemented a functional algorithm to compute the zonotope/hyperplane intersection and verified it in Isabelle/HOL. The algorithm is similar to convex hull computations, our verification is therefore inspired by Knuth's axioms for an orientation predicate of points in the plane, which have been successfully used to verify convex hull algorithms. The interesting fact is that we combine a mixture of different fields: a discrete geometrical algorithm to perform operations on the continuous sets represented by zonotopes.

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Designing and proving correct a convex hull algorithm with hypermaps in Coq

Cited by 19 publications

References 11 publications

Formal Study of Plane Delaunay Triangulation

Formal Study of Plane Delaunay Triangulation

Tests and proofs for custom data generators

A Verified Algorithm for Geometric Zonotope/Hyperplane Intersection

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