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DuaLL Report Summary

Project ID: 670624
Funded under: H2020-EU.1.1.

Periodic Reporting for period 1 - DuaLL (Duality in Formal Languages and Logic - a unifying approach to complexity and semantics)

Reporting period: 2015-09-01 to 2017-02-28

Summary of the context and overall objectives of the project

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. The goals of this project in this direction are 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’ 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 modelling 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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Several papers have been written which provide foundations for the project. These provide, among other, a full account of the duality theoretic underpinnings of syntactic recognisers and the ensuing Eilenberg-Reiterman theory for regular languages and of the theory of completions in the abstract setting of Pervin spaces and its application to obtaining characterisations by inequalities for lattices of languages (regular or not).
Specific progress on the individual objectives have also been achieved. In particular, under Objective 1, a proof of concept, establishing the feasibility of equational characterisations, has been obtained. In a series of papers, accepted at ICALP 2016 and LICS 2017, respectively, significant progress on Objective 1 has been made in the form of constructions on topo-algebraic recognisers which mirror the action of quantifiers, such as classical first-order quantifiers and modular quantifiers.
Under Objective 2, contributions include refinements of known results concerning minimisation of automata thanks to a better categorical understanding, the definition of new categorical constructions that are suitable for the description of minimisable classes of recognisers, and, as a central application, a class of automata that can be named “hybrid-set-vector automata”, that generalises both the class of finite deterministic automata, and the class of automata weighted over a field, while possessing minimal recognisers for a language. This class of automata is an important contribution to automata theory in itself, and has been obtained by design thanks to a cross fertilisation between category theory and automata theory.
Under Objective 3, contributions include a general form of the Isbell duality for certain type refinement systems, an embedding of the classical categorical semantics structures known as hyperdoctrines in a larger bifibration with the bicategory of distributors (or profunctors) as basis. In addition, a new notion of higher-order parity automaton which is designed to recognise languages of infinitary simply-typed λ-terms has been introduced.

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. In particular, 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 are scheduled including an invited talk by Thomas Colcombet at LICS 2017 and a 3 hour tutorial at Logic Colloquium 2017 by Mai Gehrke.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The majority of the results described above move significantly beyond the state of the art, however, it is too early to assess the impact of these results in a larger sense.
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