Description du projet
Les dualités entre structure algébrique et topologique éclairent les langages formels et la logique
Les dualités sont comme deux façons différentes de voir la même chose. Il existe de nombreuses dualités entre les structures algébriques et topologiques et le fait de passer d’un point de vue à l’autre a été à l’origine de nombreuses percées en mathématiques. Le projet DuaLL, financé par le CER, utilisera ces dualités comme outil pour faire avancer des sujets dans le domaine de l’informatique théorique. L’équipe se concentrera sur la recherche d’extensions robustes de la théorie des langages réguliers et sur le développement de méthodes de théorie de la dualité pour les logiques à sémantique catégorielle. Les chercheurs utiliseront des outils issus de la dualité de Stone, notamment les extensions canoniques de Jonsson-Tarski et l’algèbre profinie, ainsi que de l’algèbre universelle et de la théorie des catégories.
Objectif
Dualities between algebraic and topological structure are pervasive in mathematics, and toggling back and forth between them has often been associated with important breakthroughs. The main objective of this project is to bring this important tool to bear on a number of subjects in theoretical computer science thereby advancing, systematising, and unifying them.
One subject of focus is the search for robust extensions of the theory of regular languages. A powerful technical tool for classifying regular languages and proving decidability results is Eilenberg-Reiterman theory, which assigns classes of finite monoids or single profinite algebras to classes of languages. Recent results by the PI and her co-authors show that the theory may be seen as a special case of Stone duality for Boolean algebras with operators. We want to:
- Develop an Eilenberg-Reiterman theory beyond regular languages with the goal of obtaining new tools and separation results for Boolean circuit classes, an active area in the search for lower bounds in complexity theory.
-Systematise and advance the search for robust generalisations of regularity to other structures such as infinite words, finite and infinite trees, cost functions, and words with data.
The second subject of focus is the development of duality theoretic methods for logics with categorical semantics. We want to approach the problem incrementally:
- View duality for categorical semantics through a spectrum of intermediate cases going from regular languages over varying alphabets, Ghilardi-Zawadowski duality for finitely presented Heyting algebras, and the Bodirsky-Pinsker topological Birkhoff theorem to Makkai's, Awodey and Forssell's, and Coumans' recent work on first-order logic duality, thus unifying topics in semantics and formal languages.
Our main tools come from Stone duality in various forms including the Jonsson-Tarski canonical extensions and profinite algebra, and from universal algebra and category theory.
Champ scientifique
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ERC-ADG - Advanced GrantInstitution d’accueil
75794 Paris
France