A Formal Proof of the Expressiveness of Deep Learning

International audienceDeep learning has had a profound impact on computer science in recent years, with applications to image recognition, language processing, bioinformatics, and more. Recently , Cohen et al. provided theoretical evidence for the superiority of deep learning over shallow learning. We formalized their mathematical proof using Isabelle/HOL. The Isabelle development simplifies and generalizes the original proof, while working around the limitations of the HOL type system. To support the formalization, we developed reusable libraries of formalized mathematics, including results about the matrix rank, the Borel measure, and multivariate polynomials as well as a library for tensor analysis

Algebraically Closed Fields in Isabelle/HOL

A fundamental theorem states that every field admits an algebraically closed extension. Despite its central importance, this theorem has never before been formalised in a proof assistant. We fill this gap by documenting its formalisation in Isabelle/HOL, describing the difficulties that impeded this development and their solutions.ERC Advanced Grant ALEXANDRIA (Project GA 742178

A modular, efficient formalisation of real algebraic numbers

This paper presents a construction of the real algebraic numbers with executable arithmetic operations in Isabelle/HOL. Instead of verified resultants, arithmetic operations on real algebraic numbers are based on a decision procedure to decide the sign of a bivariate polynomial (with rational coefficients) at a real algebraic point. The modular design allows the safe use of fast external code. This work can be the basis for decision procedures that rely on real algebraic numbers.The CSC Cambridge International Scholarship is generously funding Wenda Li’s Ph.D. course.This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2854065.285407

A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms

International audienceWe generalize the Knuth-Bendix order (KBO) to higher-order terms without λ-abstraction. The restriction of this new order to first-order terms coincides with the traditional KBO. The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness. Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order superposition calculus