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Four approaches to automated reasoning with differential algebraic structures
 AISC 2004, LNAI
, 2004
"... Abstract. While implementing a proof for the Basic Perturbation Lemma (a central result in Homological Algebra) in the theorem prover Isabelle one faces problems such as the implementation of algebraic structures, partial functions in a logic of total functions, or the level of abstraction in formal ..."
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Abstract. While implementing a proof for the Basic Perturbation Lemma (a central result in Homological Algebra) in the theorem prover Isabelle one faces problems such as the implementation of algebraic structures, partial functions in a logic of total functions, or the level of abstraction in formal proofs. Different approaches aiming at solving these problems will be evaluated and classified according to features such as the degree of mechanization obtained or the direct correspondence to the mathematical proofs. From this study, an environment for further developments in Homological Algebra will be proposed. 1
Deduction and Computation in Algebraic Topology
 In Proceedings IDEIA 2002, Universidad de Sevilla
"... In this paper, a project to develop a computeraided proof of the Basic Perturbation Lemma is presented. This Perturbation Lemma is one of the central results in algorithmic algebraic topology and to obtain a mechanised proof of it, would be a first step to increase the reliability of several sy ..."
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In this paper, a project to develop a computeraided proof of the Basic Perturbation Lemma is presented. This Perturbation Lemma is one of the central results in algorithmic algebraic topology and to obtain a mechanised proof of it, would be a first step to increase the reliability of several symbolic computation systems in this area. Techniques to encode the necessary algebraic structures in the theorem prover Isabelle are described, and a sequence of high level lemmas designed to reach the proof is included.
Towards a Higher Reasoning Level in Formalized Homological Algebra
 11th Symposium on the Integration of Symbolic Computation and Mechanised Reasoning (Calculemus) (Rome, Italy) (Thérèse Hardin and Renaud Rioboo, eds.), Aracne Editrice, 2003, http://www4.in.tum.de/˜ballarin/publications/calculemus2003.pdf
, 2003
"... We present a possible solution to some problems to mechanize proofs in Homological Algebra: how to deal with partial functions in a logic of total functions and how to get a level of abstraction that allows the prover to work with morphisms in an equational way. ..."
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We present a possible solution to some problems to mechanize proofs in Homological Algebra: how to deal with partial functions in a logic of total functions and how to get a level of abstraction that allows the prover to work with morphisms in an equational way.
Computing all maps into a sphere ∗
, 2011
"... We present an algorithm for computing [X, Y], i.e., all homotopy classes of continuous maps X → Y, where X, Y are topological spaces given as finite simplicial complexes, Y is (d − 1)connected for some d ≥ 2 (for example, Y can be the ddimensional sphere Sd), and dim X ≤ 2d − 2. These conditions o ..."
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We present an algorithm for computing [X, Y], i.e., all homotopy classes of continuous maps X → Y, where X, Y are topological spaces given as finite simplicial complexes, Y is (d − 1)connected for some d ≥ 2 (for example, Y can be the ddimensional sphere Sd), and dim X ≤ 2d − 2. These conditions on X, Y guarantee that [X, Y] has a natural structure of a finitely generated Abelian group, and the algorithm finds generators and relations for it. We combine several tools and ideas from homotopy theory (such as Postnikov systems, simplicial sets, and obstruction theory) with algorithmic tools from effective algebraic topology (objects with effective homology). We hope that a further extension of the methods developed here will yield an algorithm for computing, in some cases of interest, the Z2index, which is a quantity playing a prominent role in Borsuk–Ulam style applications of topology in combinatorics and geometry, e.g., in topological lower bounds for the chromatic number of a graph. In a certain range of dimensions, deciding the embeddability of a simplicial complex into R d also amounts to a Z2index computation. This is the main motivation of our work. We believe that investigating the computational complexity of questions in homotopy theory and similar * The research by M. Č. and L. V. was supported by a Czech
An iterative algorithm for homology computation on simplicial shapes
, 2011
"... We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its subcomponents. The proposed algorithm retrieves the comple ..."
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We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its subcomponents. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators. To the best of our knowledge, this is the first algorithm based on the constructive MayerVietoris sequence, which relates the homology of a topological space to the homologies of its subspaces, i.e. the subcomponents of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the subcomponents and increases the efficiency of the algorithm.