## Periodic Reporting for period 2 - DFLOW (Duality For Logic On Words)

Okres sprawozdawczy: 2018-01-01 do 2018-12-31

The mathematical theory of Stone duality underlies a deep connection between syntax and semantics in logic and theoretical computer science, and allows for powerful applications in both of these research fields. The overall objective of this project was to apply the topological methods provided by Stone duality to the study of the expressive power of logic in describing formal languages of both finite and infinite words. During the outgoing phase of the project, the researcher has performed an in-depth study of profinite semigroups and their applications in formal language theory, through the lens of Stone duality and finite model theory. During the return phase, the project has had an impact on the European scientific activity in the fields of mathematics and theoretical computer science, specifically in profinite semigroup theory and formal language theory. As the applications of research in computer science are still developing at an extremely high pace in 21st century Europe, the need for a firmly grounded mathematical, theoretical foundation is quickly increasing as well. The research in this project was of a foundational nature, and has made significant contributions to the research field of computer science as a whole. In terms of foundational research in computer science, this project facilitated a knowledge transfer from the USA to the EU.

During the outgoing phase of the project, the researcher has been working on several mathematical problems in collaboration with Steinberg, who was the host during the outgoing phase of the project. Moreover, the researcher has continued to work with several researchers at European universities, during conferences, mutual research visits, and through remote online collaborations.

What follows is a summary of the six sub-projects which the researcher has worked on during the project, with a focus on the main results obtained so far.

In collaboration with Steinberg, the host during the outgoing phase of the project:

1. A detailed study of the free pro-aperiodic monoid using in particular the model-theoretic notion of saturated models.

Result: a paper in the international computer science conference STACS 2017 and a paper accepted for publication in Israel Journal of Mathematics, expected to appear in 2019.

2. New proofs of theorems concerning pointlike sets and separation problems for varieties of semigroups determined by groups.

Result: two articles, one published in the Canadian Mathematical Bulletin, another currently under review to a mathematics journal.

In collaboration with researchers at European universities, working remotely during the outgoing phase of the project:

3. An extension of earlier results on the connection between model companions and monadic second order logic joint with Ghilardi in Milan.

Result: a paper in the international computer science conference LICS 2016. An extended version is currently being prepared for submission to a mathematics journal.

4. A duality theorem for sheaf representations, joint with Gehrke in Paris/Nice.

Result: a paper in the Journal of Pure and Applied Algebra.

In collaboration with one researcher at a European university (Metcalfe) and one researcher at a US university (Tsinakis):

5. A study of uniform interpolation through the notion of compact congruences.

Result: a paper in the Annals of Pure and Applied Logic.

In collaboration with researchers at European universities during the return phase of the project:

6. A positive result on interpolation in predicate Gödel logic.

Result: a preprint on arXiv, to be expanded to a full article.

7. A new research direction on automata for first-order logic on trees.

Result: an article in preparation with Venema and a graduate student.

What follows is a summary of the six sub-projects which the researcher has worked on during the project, with a focus on the main results obtained so far.

In collaboration with Steinberg, the host during the outgoing phase of the project:

1. A detailed study of the free pro-aperiodic monoid using in particular the model-theoretic notion of saturated models.

Result: a paper in the international computer science conference STACS 2017 and a paper accepted for publication in Israel Journal of Mathematics, expected to appear in 2019.

2. New proofs of theorems concerning pointlike sets and separation problems for varieties of semigroups determined by groups.

Result: two articles, one published in the Canadian Mathematical Bulletin, another currently under review to a mathematics journal.

In collaboration with researchers at European universities, working remotely during the outgoing phase of the project:

3. An extension of earlier results on the connection between model companions and monadic second order logic joint with Ghilardi in Milan.

Result: a paper in the international computer science conference LICS 2016. An extended version is currently being prepared for submission to a mathematics journal.

4. A duality theorem for sheaf representations, joint with Gehrke in Paris/Nice.

Result: a paper in the Journal of Pure and Applied Algebra.

In collaboration with one researcher at a European university (Metcalfe) and one researcher at a US university (Tsinakis):

5. A study of uniform interpolation through the notion of compact congruences.

Result: a paper in the Annals of Pure and Applied Logic.

In collaboration with researchers at European universities during the return phase of the project:

6. A positive result on interpolation in predicate Gödel logic.

Result: a preprint on arXiv, to be expanded to a full article.

7. A new research direction on automata for first-order logic on trees.

Result: an article in preparation with Venema and a graduate student.

The results described above are all going beyond the state of the art in the research directions that they address, as also witnessed by the fact that two of the results were accepted in highly competitive international conferences on theoretical computer science, which allowed the researcher to present them to the wider research community, and that two other results appeared in peer-reviewed high-quality specialized mathematics journals.

The researcher made significant progress on the research in both finite and profinite semigroup theory in the collaboration with Steinberg during the outgoing phase. Steinberg is an international expert on semigroup theory, and he has helped the researcher to acquire new skills and knowledge in the highly technical field of semigroup theory. One important implication of this is that the researcher will be able to transfer this knowledge to the scientific community in Europe during the return phase of the project. This knowledge transfer will take place through conferences, local seminars, and research visits within Europe. Moreover, the results with European collaborators mentioned above will have a positive impact on both the researcher's career opportunities and on the European scientific activity on logic in theoretical computer science. The research in this project is of a fundamental mathematical nature, with the potential for applications in computer science. In particular, the decidability results that were obtained in collaboration with Steinberg are mathematical descriptions of algorithms which could be implemented in the future.

During the return phase, the project saw a continuation of the international collaborations established during the outgoing phase of the project, by working remotely and by organizing research visits at the host institution. Moreover, the researcher has set up new scientific collaborations at the host institution during the return phase, where he benefited from the local expertise in coalgebra and modal logic, which are likely to lead to new research publications in the near future. The researcher has successfully disseminated the results at several conferences and international meetings in Europe: the researcher has become a frequently invited speaker at several such meetings. Finally, the researcher has spent considerable time and effort on his future career development, grant acquisition, and networking.

The researcher made significant progress on the research in both finite and profinite semigroup theory in the collaboration with Steinberg during the outgoing phase. Steinberg is an international expert on semigroup theory, and he has helped the researcher to acquire new skills and knowledge in the highly technical field of semigroup theory. One important implication of this is that the researcher will be able to transfer this knowledge to the scientific community in Europe during the return phase of the project. This knowledge transfer will take place through conferences, local seminars, and research visits within Europe. Moreover, the results with European collaborators mentioned above will have a positive impact on both the researcher's career opportunities and on the European scientific activity on logic in theoretical computer science. The research in this project is of a fundamental mathematical nature, with the potential for applications in computer science. In particular, the decidability results that were obtained in collaboration with Steinberg are mathematical descriptions of algorithms which could be implemented in the future.

During the return phase, the project saw a continuation of the international collaborations established during the outgoing phase of the project, by working remotely and by organizing research visits at the host institution. Moreover, the researcher has set up new scientific collaborations at the host institution during the return phase, where he benefited from the local expertise in coalgebra and modal logic, which are likely to lead to new research publications in the near future. The researcher has successfully disseminated the results at several conferences and international meetings in Europe: the researcher has become a frequently invited speaker at several such meetings. Finally, the researcher has spent considerable time and effort on his future career development, grant acquisition, and networking.