Applying ACL2 to the Formalization of Algebraic Topology: Simplicial Polynomials

In this paper we present a complete formalization, using the ACL2 theorem prover, of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. The interest of this work stems from three sources. First, the normalization theorem is the basis for some design decisions in the Kenzo computer algebra system, a program for computing in Algebraic Topology. Second, our proof of the theorem is new and shows the correctness of some formulas found experimentally, giving explicit expressions for the above-mentioned homotopy equivalence. And third, it demonstrates that the ACL2 theorem prover can be effectively used to formalize mathematics, even in areas where higher-order tools could be thought to be more appropriate.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384

Certified Symbolic Manipulation: Bivariate Simplicial Polynomials

Certified symbolic manipulation is an emerging new field where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In this paper, we focus on algebraic algorithms implemented in the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study is that of bivariate simplicial polynomials, a data structure used to help the proof of properties in Simplicial Topology. Simplicial polynomials can be computationally interpreted in two ways. As symbolic expressions, they can be handled algorithmically, increasing the automation in ACL2 proofs. As representations of functional operators, they help proving properties of categorical morphisms. As an application of this second view, we present the definition in ACL2 of some morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have proved the ACL2 implementations are correct and tested that they get the same results as Kenzo does.Ministerio de Ciencia e Innovación MTM2009-13842Unión Europea nr. 243847 (ForMath

Verifying the bridge between simplicial topology and algebra: the Eilenberg–Zilber algorithm

The Eilenberg–Zilber algorithm is one of the central components of the computer algebra system called Kenzo, devoted to computing in Algebraic Topology. In this article we report on a complete formal proof of the underlying Eilenberg–Zilber theorem, using the ACL2 theorem prover. As our formalization is executable, we are able to compare the results of the certified programme with those of Kenzo on some universal examples. Since the results coincide, the reliability of Kenzo is reinforced. This is a new step in our long-term project towards certified programming for Algebraic Topology.Ministerio de Ciencia e Innovación MTM2009-13842European Union’s 7th Framework Programme [243847] (ForMath)

Formalization of a normalization theorem in simplicial topology

In this paper we present a complete formalization of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. Even if the Normalization Theorem is usually stated as a higher-order result (with a Category Theory flavor) we manage to give a first-order proof of it. To this aim it is instrumental the introduction of an algebraic data structure called simplicial polynomial. As a demonstration of the validity of our techniques we developed a formal proof in the ACL2 theorem prover.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384

Towards a verifiable topology of data

Ministerio de Economía y Competitividad TIN2013-41086-PMinisterio de Economía y Competitividad MTM2014-5415

On bornologies, locales and toposes of M -sets

Let M be the monoid of all endomaps of a non-empty set N, the locale of all ideals of M, and let be the topos of all M-sets. The core of this paper is formed by a locale B, a subtopos and two theorems, where B is the locale of all bornologies defined on subsets of N and is the topos of j-sheaves for a topology j : . The first theorem shows a morphism of locales B with nucleus j which induces an isomorphism of locales between B and the sublocale j . The second theorem, which generalizes the first one, gives an equivalence between the category of Kolmogorov bornological spaces and bounded maps, and the full subcategory ' formed by all j-sheaves which are separated for the double negation topology of . © 2002 Elsevier Science B.V. All rights reserved

A tensor-hom adjunction in a topos related to vector topologies and bornologies

In this paper, = {} is the one-point compactification of the discrete space of natural numbers, is the monoid of continuous maps f : such that f () = , and M is the topos of -sets. We define two sheaf subtoposes C and B of M and construct a tensor-hom adjunction between certain categories of modules in C and B. Finally, we prove that this construction induces an adjunction between adequate categories of topological and bornological real vector spaces. © 2000 Elsevier Science B.V. All rights reserved

Object oriented institutions to specify symbolic computation systems

The specification of the data structures used in EAT, a software system for symbolic computation in algebraic topology, is based on an operation that defines a link among different specification frameworks like hidden algebras and coalgebras. In this paper, this operation is extended using the notion of institution, giving rise to three institution encodings. These morphisms define a commutative diagram which shows three possible views of the same construction, placing it in an equational algebraic institution, in a hidden institution or in a coalgebraic institution. Moreover, these morphisms can be used to obtain a new description of the final objects of the categories of algebras in these frameworks, which are suitable abstract models for the EAT data structures. Thus, our main contribution is a formalization allowing us to encode a family of data structures by means of a single algebra (which can be described as a coproduct on the image of the institution morphisms). With this aim, new particular definitions of hidden and coalgebraic institutions are presented

Topología simplicial en ACL2

Ministerio de Educación y Ciencia MTM2009-1384