Dynamical and Arithmetical Model Checking

Descripción del proyecto

Una aproximación interdisciplinar al estudio de los sistemas dinámicos discretos

Los sistemas dinámicos discretos son modelos matemáticos utilizados en muchos campos, como la informática y la biología, para estudiar cómo cambian los sistemas a lo largo del tiempo. Aunque parecen sencillas de describir, dan lugar a fascinantes problemas abiertos. Uno de estos ejemplos es el problema de Skolem, que se pregunta si un sistema dinámico llega alguna vez a un hiperplano dado. El equipo del proyecto DynAMiCs, financiado con fondos europeos, combinará herramientas de distintas áreas de las matemáticas y la informática, como la teoría de números y la lógica, para estudiar sistemas dinámicos discretos. En el proyecto se combinarán los conocimientos de teoría de números, la dinámica simbólica y la lógica matemática.

Objetivo

Discrete dynamical systems pervade the quantitative sciences, and lie at the heart of central computational challenges in a wide variety of areas, from program analysis and computer-aided verification to neural networks and theoretical biology. Such systems are typically simple to describe, yet give rise to a rich algorithmic and mathematical theory that is replete with easily stated and compelling open problems. One such example is the famous Skolem Problem: does the orbit of a given linear dynamical system ever hit a given hyperplane? The decidability of this question is a longstanding open problem going back nearly a century!

The main goal DynAMiCs is to achieve major advances in the algorithmic theory of discrete linear dynamical systems and related formalisms. Whilst our motivation and outlook originate from computer science and automated verification, we approach these challenging problems from both mathematical and computational angles, combining tools and techniques from number theory (e.g. Diophantine approximation, transcendence theory, continued fractions), symbolic dynamics (e.g. word combinatorics, subshifts, numeration systems), and mathematical logic (e.g. monadic second-order logic, Presburger arithmetic, automata theory). One of our leading objectives is to substantially broaden the classes of dynamical systems and properties that can be algorithmically handled via model checking. We aim to achieve these ambitious advances by pursuing new approaches to central open problems in linear systems, such as the Skolem Problem, building on recent breakthroughs in these areas (some of which from the PIs and their groups).

We take on this challenge with a team of PIs consisting of international leaders in algorithmic verification, symbolic dynamics, and analytic number theory. We believe our complementary expertise places us in a unique position to achieve major breakthroughs in longstanding fundamental problems in the area of discrete dynamical systems.