Local-to-global Expansion and PCPs

Descripción del proyecto

Desarrollar construcciones de pruebas comprobables probabilísticamente más sencillas y eficientes

Las pruebas comprobables probabilísticamente (PCP) son un concepto fundamental en la informática teórica moderna, ya que demuestran un comportamiento de local a global robusto. Esta teoría está estrechamente relacionada con la dureza de la aproximación y tiene aplicaciones importantes en criptografía. El equipo del proyecto LEAP, financiado con fondos europeos, pretende desarrollar construcciones de PCP más sencillas y eficientes. También mejorará nuestra comprensión de las codificaciones robustas con características locales a globales aprovechando la teoría emergente de la expansión de alta dimensión (HDX, por sus siglas en inglés). Esta teoría interdisciplinar permite inferir sobre la estructura global a partir de las propiedades de los enlaces locales y tiene aplicaciones potenciales en la comprobación de las propiedades y la construcción de códigos comprobables localmente. En el proyecto se explorará la teoría de la HDX, se construirán PCP nuevos, se investigará la inaproximabilidad y se crearán nuevos códigos de corrección de errores.

Objetivo

This proposal aims to delve further into the study of Probabilistically Checkable Proofs (PCPs), a cornerstone of modern theoretical computer science, that exhibits some of the most powerful local to global behavior. Any NP proof can be written in a format that is locally testable, meaning that local pieces of the proof imply very rich global structure. This theory is strongly tied to hardness of approximation and has significant applications in cryptography. A major goal is to develop simpler and more efficient PCP constructions, with better parameters, as well as to deepen our understanding of robust encodings with local to global features, such as PCPs.

The main methodology of this exploration will be through harnessing the power of the beautiful emerging theory of high-dimensional expansion (HDX). Generalizing the concept of expander graphs to higher dimensions, HDX emerge as an exciting cross-disciplinary theory with roots in group theory, number theory, algebraic topology, combinatorics, and theoretical computer science. The HDX theory is characterized by a local-to-global principle, which allows inferences about the global structure based on local link properties. This principle bears significant potential for areas like property testing and it has already shown its power with the construction of C^3 locally testable codes. As these are very closely connected to PCPs, we seek to harness HDX towards advancing our understanding of more general local-to-global encodings, and potentially paving the way for novel PCP constructions.

The research directions outlined in this proposal cover a wide array of goals, including the exploration of the HDX theory, the construction of new PCPs, inapproximability, and the creation of novel error-correcting codes. These directions seek to bridge various areas of mathematics and computer science, promising to contribute significantly to the field and open up new horizons for further research and applications.