Laureano Lambán

An Object-oriented Interpretation of the EAT System

Abstract.In a previous paper we characterized, in the Category Theory setting, a class of implementations of Abstract Data Types, which has been suggested by the way of programming in the EAT system. (EAT, Effective Algebraic Topology, is one of Sergeraert’s systems for effective homology and homotopy computation.) This characterization was established using classical tools, in an unrelated way to the current mainstream topics in the field of Algebraic Specifications. Looking for a connection with these topics, we have found, rather unexpectedly, that our approach is related to some object-oriented formalisms, namely hidden specifications and the coalgebraic view. In this paper, we explore these relations making explicit the implicit object-oriented features of the EAT system and generalizing the data structure analysis we had previously done.

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.

Hidden Specification of a Functional System

This paper is devoted to the formal study of the data structures appearing in a symbolic computation system, namely the EAT system. One of the main features of the EAT system is that it intensively uses functional programming techniques. This implies that some formalisms for the algebraic specification of systems must be adapted to this functional setting. Specifically, this work deals with hidden and coalgebraic methodologies through an institutional framework. As a byproduct, the new concept of coalgebraic institution associated to an institution is introduced. Then, the problem of modeling functorial relationships between data structures is tackled, giving a hidden specification for this aspect of the EAT system and proving the existence of final objects in convenient categories, which accurately model the EAT way of working.

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.

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. It this paper 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 programs with those of Kenzo on some universal examples. Since the results coincide, the reliability of Kenzo is so reinforced. This is a new step in our long term project towards certified programming for Algebraic Topology.

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.

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 M be the topos of all M -sets. The core of this paper is formed by a locale B, a subtopos B ,→ M and two theorems, where B is the locale of all bornologies de3ned on subsets of N and B is the topos of j-sheaves for a topology j : � → � . The 3rst theoremshows a morphism of locales B → � with nucleus j which induces an isomorphism of locales between

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

Abstract In this paper, N = N ∪{∞} is the one-point compactification of the discrete space of natural numbers, M is the monoid of continuous maps f : N → N such that f (∞)=∞, and M is the topos of M -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.

Using Abstract Stobjs in ACL2 to Compute Matrix Normal Forms.

We present here an application of abstract single threaded objects (abstract stobjs) in the ACL2 theorem prover, to define a formally verified algorithm that given a matrix with elements in the ring of integers, computes an equivalent matrix in column echelon form. Abstract stobjs allow us to define a sound logical interface between matrices defined as lists of lists, convenient for reasoning but inefficient, and matrices represented as unidimensional stobjs arrays, which implement accesses and (destructive) updates in constant time. Also, by means of the abstract stobjs mechanism, we use a more convenient logical representation of the transformation matrix, as a sequence of elemental transformations. Although we describe here a particular normalization algorithm, we think this approach could be useful to obtain formally verified and efficient executable implementations of a number of matrix normal form algorithms.

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.