Reseña: A survey of interactive theorem proving

PorJosé A. Alonso

Se ha publicado un artículo sobre razonamiento formalizado titulado A survey of interactive theorem proving.

Su autor es Filip Marić (del Automated Reasoning GrOup (ARGO) en la Universidad de Belgrado, Serbia).

Su resumen es

Fully formally verified mathematics and software are long-standing aims that became practically realizable with modern computer tools. Reasoning can be reduced to several basic logical principles, and performed using specialized software, with significant automation. Although full automation is not possible, three main paradigms are represented in formal reasoning tools: (i) decision procedures for special classes of problems, (ii) complete, but potentially unterminating proof search, (iii) checking of proof-sketches given by a human user while automatically constructing simpler proof steps. In this paper, we present a survey of the third approach, embodied in modern interactive theorem provers (ITP), also called proof-assistants. These tools have been successfully developed for more than 40 years, and the current state-of-the-art tools have reached maturity needed to perform real-world large-scale formalizations of mathematics (e.g., Four-Color Theorem, Prime Number Theorem, and Feith-Thompson’s Odd Order theorem) and software correctness (e.g., substantial portions of operating systems and compilers have been verified). We discuss history of ITP, its logical foundations, main features of state-of-the-art systems, and give some details about the most prominent results in the field. We also summarize main results of the researchers from Serbia and personal results of the author.

Este artículo puede servir de lectura complementaria en los cursos de Razonamiento automático, Razonamiento asistido por ordenador y Lógica computacional y teoría de modelos.

Reseña: Formalisation in constructive type theory of Stoughton’s substitution for the lambda calculus

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Agda sobre metalógica titulado Formalisation in constructive type theory of Stoughton’s substitution for the lambda calculus.

Sus autores son Álvaro Tasistro, Ernesto Copello y Nora Szasz (del Grupo de Computación Teórica (Compute) en la Universidad ORT, Uruguay).

Su resumen es

In Substitution revisited, Alley Stoughton proposed a notion of (simultaneous) substitution for the Lambda calculus as formulated in its original syntax – i.e. with only one sort of symbols (names) for variables – and without identifying α-convertible terms. According to such formulation, the action of substitution on terms is defined by simple structural recursion and an interesting theory arises concerning the connection to α-conversion.

In this paper we present a formalisation of Stoughton’s work in Constructive Type Theory using the language Agda, which reaches up to the Substitution Lemma for α-conversion. The development has been quite inexpensive e.g. in labour cost, and we are able to formulate some improvements over the original presentation. For instance, our definition of α-conversion is just syntax directed and we prove it to be an equivalence relation in an easy way, whereas in Substitution revisited the latter was included as part of the definition and then proven to be equivalent to an only nearly structural definition as corollary of a lengthier development. As a result of this work we are inclined to assert that Stoughton’s is the right way to formulate the Lambda calculus in its original, conventional syntax and that it is a formulation amenable to fully formal treatment.

El trabajo se ha publicado en Electronic Notes in Theoretical Computer Science.

El código de las correspondientes teorías en Agda literario se encuentra aquí.

Reseña: Hofstadter’s problem for curious readers

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Coq titulado Hofstadter’s problem for curious readers.

Su autor es Pierre Letouzey (del grupo PPS (Preuves, Programmes et Systèmes) de la Universidad de París VII Denis Diderot, Francia).

Su resumen es

This document summarizes the proofs made during a Coq development in Summer 2015. This development investigates the function G introduced by Hofstadter in his famous “Gödel, Escher, Bach” book as well as a related infinite tree. The left/right flipped variant of this G tree has also been studied here, following Hofstadter’s “problem for the curious reader”. The initial G function is refered as sequence A005206 in OEIS, while the flipped version is the sequence A123070.

The detailed and machine-checked proofs can be found in the files of
this development and can be re-checked by running Coq version 8.4 on it. No prior knowledge of Coq is assumed here, on the contrary this document has rather been a “Coq-to-English” translation exercise for the author. Nonetheless, some proofs given in this document are still quite sketchy: in this case, the interested reader is encouraged to consult the Coq files given as references.

El código de las correspondientes teorías en Coq se encuentra aquí.

Este artículo puede servir de lectura complementaria en los cursos de Razonamiento automático, Razonamiento asistido por ordenador y Lógica computacional y teoría de modelos.

Reseña: Deriving comparators and show functions in Isabelle/HOL

PorJosé A. Alonso

Se ha publicado un artículo de automatización del razonamiento en Isabelle/HOL titulado Deriving comparators and show functions in Isabelle/HOL.

Sus autores son Christian Sternagel y René Thiemann (del Computational Logic Research Group en la Universidad de Innsbruck, Austria).

Su resumen es

We present an Isabelle/HOL development that allows for the automatic generation of certain operations for user-defined datatypes. Since the operations are defined within the logic, they are applicable for code generation. Triggered by the demand to provide readable error messages as well as to access efficient data structures like sorted trees in generated code, we provide show functions that compute the string representation of a given value, comparators that yield linear orders, and hash functions. Moreover, large parts of the employed machinery should be reusable for other operations like read functions, etc.

In contrast to similar mechanisms, like Haskell’s “deriving,” we do not only generate definitions, but also prove some desired properties, e.g., that a comparator indeed orders values linearly. This is achieved by a collection of tactics that discharge the corresponding proof obligations automatically.

El trabajo se presentó el 25 de agosto en el ITP 2015 (The 6th conference on Interactive Theorem Proving).

El código de las correspondientes teorías en Isabelle/HOL se encuentra aquí.

Este artículo puede servir de lectura complementaria en los cursos de Razonamiento automático, Razonamiento asistido por ordenador y Lógica computacional y teoría de modelos.

Reseña: Case of (quite) painless dependently typed programming: Fully certified merge sort in Agda

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Agda titulado Case of (quite) painless dependently typed programming: Fully certified merge sort in Agda.

Sus autores son Ernesto Copello, Álvaro Tasistro y Bruno Bianchi (del Grupo de Computación Teórica (Compute) en la Universidad ORT, Uruguay).

Su resumen es

We present a full certification of merge sort in the language Agda. It features: termination warrant without explicit proof, no proof cost to ensure that the output is sorted, and a succinct proof that the output is a permutation of the input.

El trabajo se ha presentado en The Brazilian Symposium on Programming Languages (SBLP).

El código de las correspondientes teorías en Agda se encuentra aquí.

Este artículo puede servir de lectura complementaria en los cursos de Razonamiento automático, Razonamiento asistido por ordenador y Lógica computacional y teoría de modelos.