Higher Observational Type Theory

Descripción del proyecto

Un lenguaje fácil de entender para la verificación formal

Las matemáticas y la informática recurren a la verificación formal para garantizar la precisión del razonamiento y la seguridad de «software» crítico. Avances recientes, como la formalización del teorema de los cuatro colores y la verificación de componentes en el navegador Chrome de Google, han demostrado la capacidad de los asistentes de prueba. Estas herramientas se basan en el lenguaje de la teoría de tipos. Si bien la teoría de tipos homotópica (HoTT, por sus siglas en inglés) ha demostrado su valía en el ámbito de la investigación científica, su adopción generalizada se ha visto limitada debido a su sintaxis compleja y a los retos conceptuales que plantea. Teniendo esto en cuenta, en el proyecto HOTT, financiado por el Consejo Europeo de Investigación, se pretende desarrollar una nueva teoría de tipos en la que el contenido homotópico surja de forma natural, simplificando así el proceso. Al basar la definición de los tipos de igualdad en el cálculo, el proyecto busca facilitar la formalización y agilizar los avances tanto en matemáticas como en la verificación de «software».

Objetivo

Recent advancements have enabled proof asistants to formally verify world-class mathematics: the liquid tensor experiment, the four colour theorem and the odd order theorem were formalised. Computer checked arguments are important for mathematicians who want to be certain their reasoning is sound, and for computer scientists to prevent bugs in safety critical software. Examples are formally verified parts of Google's Chrome web browser and verified implementations of the C and ML programming languages.

At the core of these formalisations lies type theory, upon which proof assistants are built. Type theory is both a functional programming language and a foundation of mathematics. Recently, models of type theory built on higher dimensional spaces emerged, where elements of a type are points in the space, and elements of an equality type are paths in the space. Based one these, type theory was extended to homotopy type theory (HoTT), featuring the principle that isomorphic types are equal. This moves formalisation close to actual mathematical practice where isomorphic structures are being treated as the same.

While HoTT is successful among academics, it hasn't been widely adopted. This is because type theories implementing HoTT rely on an explicit syntax for higher dimensional geometry, which is conceptually difficult and hard to use in practice. This creates a substantial barrier for formalisation, which is treated as a low-level, bureaucractic process.

Our project will develop a radically new type theory where homotopical content is emergent, rather than built-in. The idea is to define the equality type via computation. This makes HoTT explainable and conceptually simple. It also improves pragmatic aspects: with more computation, proofs become less tedious. Our theory will contribute to a new era in formalisation of mathematics and verification of software, where developing proofs in abstract, reusable ways becomes standard, accelerating progress in both areas.