Incidence Simplicial Matrices Formalized in Coq/SSReflect

Abstract

Simplicial complexes are at the heart of Computational Algebraic Topology, since they give a concrete, combinatorial description of otherwise rather abstract objects which makes many important topological computations possible. The whole theory has many applications such as coding theory, robotics or digital image analysis. In this paper we present a formalization in the COQ theorem prover of simplicial complexes and their incidence matrices as well as the main theorem that gives meaning to the definition of homology groups and is a first step towards their computation.

Neural Networks in Imandra: Matrix Representation as a Verification Choice

“…The first choice was taken in [8] (in the context of dependent types in Coq), in [12] (in the context of refinement types of F * ) and in [7] (for sparse matrix encodings in Haskell). The difference between the first and second approaches was discussed in [20] (in Agda, but with no neural network application in mind).…”

“…type 'a vector = 'a list type 'a matrix = 'a vector list It is possible to extend this formalisation by using dependent [8] or refinement [12] types to check the matrix size, e.g. when performing matrix multiplication.…”

Computing persistent homology within Coq/SSReflect

“…In previous work, see [28,27], we have formalized the notions presented in subsections 2.1 and 2.2. However, for the sake of clarity of the exposition we include the main definitions and results which have been developed previously.…”

“…A facet of a simplicial complex K is a maximal simplex with respect to the subset order ⊆ among the simplices of K. To construct the simplicial complex associated with a sequence of facets, F , we generate all the faces of the simplices of F ; subsequently, if we perform the set union of all the faces we obtain the simplicial complex associated with F . This procedure have been implemented, and its correctness have been proved, using Coq in [28]. In the case of the diabolo complex of Figure 1 its facets are: {(2, 3), (3,4), (3,5), (4, 5), (0, 1, 2)}.…”

Towards a Certified Computation of Homology Groups for Digital Images

“…The correctness of the programs in charge of both the construction of a simplicial complex from an image and the generation of the boundary matrices associated with a simplicial complex have been formally proved using proof assistant tools as can be seen in [21] and [14] respectively. Then, there only remains the verification of the third point, the computation of homology groups from the boundary matrices.…”