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

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en ACL2 titulado Verifying the bridge between simplicial topology and algebra: the Eilenberg-Zilber algorithm.

Sus autores son

Su resumen es

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.

El artículo se ha publicado en Logic Journal of the IGPL.

Los códigos de las correspondientes teorías en ACL2 se encuentran aquí.

Formalization of basic linear algebra

PorJosé A. Alonso

Se ha publicado una tesis de máster de razonamiento formalizado en Isabelle/HOL titulada Formalization of basic linear algebra.

Su autor es Bhavisha M. Talsania (de la California State University, San Marcos).

Su resumen es

This thesis discusses the formalization of basic results from linear algebra up to the following theorems, using the computer proof checker Isabelle:

  • Theorem 1 (Basis Size Theorem): Suppose that V is a finite dimensional vector space. Then any two bases have the same number of elements.
  • Theorem 2 (Basis Characterization Theorem): Suppose that V is a vector space of dimension n. Then any spanning set has at least n elements, and any spanning set with exactly n elements is a basis. In addition, any linearly independent set of vectors has at most n elements, and any set of linearly independent vectors with exactly n elements is a basis.
  • Theorem 3 (Rank-Nullity Theorem): Let L : V1 → V2 be a linear map between finite dimensional vector spaces. Then the nullity of L plus the rank of L equals the dimension of V1 : dim(V1 ) = rank(L) + null(L).

Chapter 1 provides a discussion of informal and formal proofs and an introduction to the thesis topic. Chapter 2 provides a brief overview of the proof assistant Isabelle and it’s layout and syntax. Chapter 3 to Chapter 13 go into detail of the formalized proofs done in Isabelle. In Chapter 14, we end with remarks and conclusion of the thesis.

The Königsberg bridge problem and the friendship theorem

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Isabelle/HOL titulado The Königsberg bridge problem and the friendship theorem.

Su autor es Wenda Li (de la Universidad de Cambridge).

El artículo se ha publicado el 17 de julio en el Archive of Formal Proofs.

Su resumen es

This development provides a formalization of undirected graphs and simple graphs, which are based on Benedikt Nordhoff and Peter Lammich’s simple formalization of labelled directed graphs in the archive. Then, with our formalization of graphs, we show both necessary and sufficient conditions for Eulerian trails and circuits as well as the fact that the Königsberg Bridge Problem does not have a solution. In addition, we show the Friendship Theorem in simple graphs.

Formal verification of a proof procedure for the description logic ALC

PorJosé A. Alonso

Se ha publicado un artñiculo de razonamiento formalizado en Isabelle/HOL titulado Formal verification of a proof procedure for the description logic ALC.

Sus autores son

  • Mohamed Chaabani (LIMOSE, Universidad de Boumerdes, Argelia),
  • Mohamed Mezghiche (LIMOSE, Universidad de Boumerdes, Argelia) y
  • Martin Strecker (IRIT (Institut de Recherche en Informatique de Toulouse), Francia)

El trabajo se presentó en el SCSS 2012 (Fourth International Symposium on
Symbolic Computation in Software Science), cuyas actas se publicaron la semana pasada.

Su resumen es

Description Logics (DLs) are a family of languages used for the representation and reasoning on the knowledge of an application domain, in a structured and formal manner. In order to achieve this objective, several provers, such as RACER and FaCT++, have been implemented, but these provers themselves have not been yet certified. In order to ensure the soundness of derivations in these DLs, it is necessary to formally verify the deductions applied by these reasoners. Formal methods offer powerful tools for the specification and verification of proof procedures, among them there are methods for proving properties such as soundness, completeness and termination of a proof procedure. In this paper, we present the definition of a proof procedure for the Description Logic ALC, based on a semantic tableau method. We ensure validity of our prover by proving its soundness, completeness and termination properties using Isabelle proof assistant. The proof proceeds in two phases, first by establishing these properties on an abstract level, and then by instantiating them for an implementation based on lists.

En el 2007 presentamos una formalización análoga en PVS: A formally verified prover for the ALC description logic.

Exámenes de programación funcional con Haskell

PorJosé A. Alonso

He publicado el libro Exámenes de programación funcional con Haskell (2009-2013) en el que recopilo los exámenes de la asignatura de Informática (de primero del Grado en Matemáticas) desde el curso 2009-10 hasta el actual. El libro contiene 74 exámenes y 475 ejercicios.

Este libro es el complemento de los anteriores: