ACL2 Verification of Simplicial Degeneracy Programs in the Kenzo System

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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“…This constraint has not been an obstacle for us to effectively use ACL2 to study first order fragments of Kenzo [10,18].…”

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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“…So it is natural to use ACL2, a theorem prover intimately linked to this language. Even if ACL2 is not suitable to model all Kenzo features (Kenzo uses higher-order functional programming, while ACL2 is a first-order tool; this explains the role of Isabelle/HOL and Coq in the global project), it is superior to any other tool to formalize the actual Kenzo source code (for an example of such a formalization, see [9]). …”

Section: Introductionmentioning

confidence: 99%

Certified symbolic manipulation

Lambán

Martín-Mateos

Rubio

et al. 2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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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.

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“…Kenzo, Axiom [38], Maxima [52], Reduce [32] and Weyl [66]). In addition, since the language is the same, ACL2 formalisations are closer to actual CAS code than formalisations in other ITPs-this might require some code-transformations and proofs by successive refinements, as for instance applied in [50].…”

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confidence: 98%

“…loops were replaced by tail-recursive functions). The work presented in [2,35,36,50] did not concern algebraic structures, but ACL2 has also been used to formalise constructions (the Normalisation theorem and the Eilenberg-Zilberg theorem [59]) involving algebraic structures implemented in Kenzo, see [44,47]. In contrast to the work in Isabelle and Coq, the algebraic structures involved in the ACL2 formalisations were developed from scratch instead of using, as a basis, a previously developed algebraic hierarchy.…”

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confidence: 99%

“…The ACL2 theorem prover is oriented to prove properties of programs written in (a subset) of Common Lisp; and, it has been successfully employed to study some critical fragments of actual Kenzo code [2,35,36,50]-due to the restrictions of the ACL2 language, some well-known and safe transformations were required (e.g. loops were replaced by tail-recursive functions).…”

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confidence: 99%

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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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ACL2 Verification of Simplicial Degeneracy Programs in the Kenzo System

Cited by 8 publications

References 8 publications

Formalization of a normalization theorem in simplicial topology

Formalization of a normalization theorem in simplicial topology

Certified symbolic manipulation

Modelling algebraic structures and morphisms in ACL2

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