Periodic Reporting for period 4 - DuaLL (Duality in Formal Languages and Logic - a unifying approach to complexity and semantics)
Reporting period: 2020-03-01 to 2021-08-31
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. The goals of this project in this direction are to:
- (Objective 1) 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.
- (Objective 2) 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:
- (Objective 3) 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’ first-order logic duality, thus unifying topics in semantics and formal languages.
This project may provide new tools in the search for lower bounds in complexity theory, which is an important issue in that it informs us of the limits and nature of computational processes. It may also provide tools for semantic modeling of computational processes with binding of variables in a modular setting, which is important for the development of flexible and sophisticated modern computer programs. Finally, since the mathematical tools behind these two developments are one and the same, it may contribute to the unification of the semantic and the algorithmic study of the foundations of computer science.
Apart from the standard dissemination of research results, additional effort has been made to disseminate the results and the nature of the project to as wide an audience as possible. Thus, the project quicked off with a workshop on Duality Theory at the Dagstuhl center. Also, an invited tutorial at the Logic in Computer Science (LICS) conference in July 2016 provided an introduction and overview of the duality theoretic components of the project and how they connect with other topics in semantics. Further, two papers, that provide an invitation and introduction to Objective 1 and Objective 2, respectively, have appeared as invited columns (on complexity and semantics, respectively) of the ACM SIGLOG News of April 2017. Further broad dissemination activities include invited talks by J.-É. Pin at RAMiCS 2017, by Thomas Colcombet at LICS 2017 and a 3 hour tutorial at Logic Colloquium 2017, a whole BLAST 2021 special session and invited talk, and an invited talk at CT 2021 by Mai Gehrke. In addition, Colcombet, Gehrke, and Petrisan all participated as long term visitors to the special trimester at Simons Institute in Berkeley on Logic and Computation, which focused on the same novel interdisciplinary edge between semantics and complexity theory as does this project.