An Object-oriented Interpretation of the EAT System

Lambán, Laureano; Pascual, Vico; Rubio, Julio

doi:10.1007/s00200-003-0129-1

Cited by 19 publications

(33 citation statements)

References 32 publications

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“…The method that we have followed to represent these families is based on the approach presented in [46]. Roughly speaking, the representation of a graded structure indexed on a set is achieved thanks to the introduction of an additional parameter, that ranges the elements of the index set, in each operation of the structure.…”

Section: Indexed Families Of Structuresmentioning

confidence: 99%

Modelling algebraic structures and morphisms in ACL2

Heras

Martín-Mateos

Pascual

2015

AAECC

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In this paper, we present how algebraic structures and morphisms can be modelled in the ACL2 theorem prover. Namely, we illustrate a methodology for implementing a set of tools that facilitates the formalisations related to algebraic structuresas a result, an algebraic hierarchy ranging from setoids to vector spaces has been developed. The resultant tools can be used to simplify the development of generic theories about algebraic structures. In particular, the benefits of using the tools presented in this paper, compared to a from-scratch approach, are especially relevant when working with complex mathematical structures; for example, the structures employed in Algebraic Topology. This work shows that ACL2 can be a suitable tool for formalising algebraic concepts coming, for instance, from computer algebra systems.

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Section: Indexed Families Of Structuresmentioning

confidence: 99%

Modelling algebraic structures and morphisms in ACL2

Heras

Martín-Mateos

Pascual

2015

AAECC

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“…Then, some concrete outputs of the program cannot be tested, that is, compared with any expected value. This is the reason why a project to apply formal methods to the study of Kenzo as a software system was launched some years ago [6,12]. Eventually, this research line arrived to the formalization of some parts of Algebraic Topology and Homological Algebra by using proof assistants as Isabelle/HOL [2,3] or Coq [7].…”

Section: Introductionmentioning

confidence: 98%

Formalization of a normalization theorem in simplicial topology

Lambán

Martín-Mateos

Rubio

et al. 2012

Ann Math Artif Intell

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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. This work is dedicated to our colleague and friend Mirian Andrés. She started this research but passed away at the age of only 29 due to a car accident. Mirian, the best friend for your friends, we do not forget you.

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“…In view of the obtained results, some years ago, the first author of this paper began the formal study of the programs, in order to reach a good understanding on the internal calculation processes of these software systems. In particular, our study of the data types used in EAT and Kenzo [13,6,8] shows that there are two different layers of data structures in the systems. In the first layer, one finds the usual abstract data types, like the type of integers.…”

Section: Introductionmentioning

confidence: 98%

A Parameterization Process: From a Functorial Point of View

Domínguez

Duval

2012

Int. J. Found. Comput. Sci.

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The parameterization process used in the symbolic computation systems Kenzo and EAT is studied here as a general construction in a categorical framework. This parameterization process starts from a given specification and builds a parameterized specification by adding a parameter as a new variable to some operations. Given a model of the parameterized specification, each interpretation of the parameter, called an argument, provides a model of the given specification. Moreover, under some relevant terminality assumption, this correspondence between the arguments and the models of the given specification is a bijection. It is proved in this paper that the parameterization process is provided by a functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits.

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An Object-oriented Interpretation of the EAT System

Cited by 19 publications

References 32 publications

Modelling algebraic structures and morphisms in ACL2

Modelling algebraic structures and morphisms in ACL2

Formalization of a normalization theorem in simplicial topology

A Parameterization Process: From a Functorial Point of View

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