Document

**Published in:** LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

In this paper we study the class EqP of automatic equivalence structures of the form ?=(D, E) where the domain D is a regular language of polynomial growth and E is an equivalence relation on D. Our goal is to investigate the following two foundational problems (in the theory of automatic structures) aimed for the class EqP. The first is to find algebraic characterizations of structures from EqP, and the second is to investigate the isomorphism problem for the class EqP. We provide full solutions to these two problems. First, we produce a characterization of structures from EqP through multivariate polynomials. Second, we present two contrasting results. On the one hand, we prove that the isomorphism problem for structures from the class EqP is undecidable. On the other hand, we prove that the isomorphism problem is decidable for structures from EqP with domains of quadratic growth.

Moses Ganardi and Bakhadyr Khoussainov. Automatic Equivalence Structures of Polynomial Growth. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{ganardi_et_al:LIPIcs.CSL.2020.21, author = {Ganardi, Moses and Khoussainov, Bakhadyr}, title = {{Automatic Equivalence Structures of Polynomial Growth}}, booktitle = {28th EACSL Annual Conference on Computer Science Logic (CSL 2020)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-132-0}, ISSN = {1868-8969}, year = {2020}, volume = {152}, editor = {Fern\'{a}ndez, Maribel and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.21}, URN = {urn:nbn:de:0030-drops-116645}, doi = {10.4230/LIPIcs.CSL.2020.21}, annote = {Keywords: automatic structures, polynomial growth, isomorphism problem} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup (G,+) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of (G,+); second, what learnability properties can one extract from G and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of (G,+) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for G with respect to any generating sequence for G is not even semi-decidable, one can build a generating sequence beta such that the word problem for G with respect to beta is co-recursively enumerable (assuming that the set of generators of G is limit-recursive). In regard to the second question, it is proven that there is a generating sequence beta for G such that every non-trivial finitely generated subgroup of G is recursively enumerable and the class of all such subgroups of G is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of G is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of G cannot be syntactically identified in the limit with respect to any generating sequence for G. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, and Frank Stephan. Random Subgroups of Rationals. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{gao_et_al:LIPIcs.MFCS.2019.25, author = {Gao, Ziyuan and Jain, Sanjay and Khoussainov, Bakhadyr and Li, Wei and Melnikov, Alexander and Seidel, Karen and Stephan, Frank}, title = {{Random Subgroups of Rationals}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.25}, URN = {urn:nbn:de:0030-drops-109693}, doi = {10.4230/LIPIcs.MFCS.2019.25}, annote = {Keywords: Martin-L\"{o}f randomness, subgroups of rationals, finitely generated subgroups of rationals, learning in the limit, behaviourally correct learning} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 7441, Algorithmic-Logical Theory of Infinite Structures (2008)

From 28.10. to 02.11.2007, the Dagstuhl Seminar 07441 ``Algorithmic-Logical Theory of Infinite Structures'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Rod Downey, Bakhadyr Khoussainov, Dietrich Kuske, Markus Lohrey, and Moshe Y. Vardi. 07441 Abstracts Collection – Algorithmic-Logical Theory of Infinite Structures. In Algorithmic-Logical Theory of Infinite Structures. Dagstuhl Seminar Proceedings, Volume 7441, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

Copy BibTex To Clipboard

@InProceedings{downey_et_al:DagSemProc.07441.1, author = {Downey, Rod and Khoussainov, Bakhadyr and Kuske, Dietrich and Lohrey, Markus and Vardi, Moshe Y.}, title = {{07441 Abstracts Collection – Algorithmic-Logical Theory of Infinite Structures}}, booktitle = {Algorithmic-Logical Theory of Infinite Structures}, pages = {1--13}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7441}, editor = {Rod Downey and Bakhadyr Khoussainov and Dietrich Kuske and Markus Lohrey and Moshe Y. Vardi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07441.1}, URN = {urn:nbn:de:0030-drops-14122}, doi = {10.4230/DagSemProc.07441.1}, annote = {Keywords: Theories of infinite structures , computable model theory and automatic structures , model checking infinite systems} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 7441, Algorithmic-Logical Theory of Infinite Structures (2008)

One of the important research fields of theoretical and applied
computer science and mathematics is the study of algorithmic, logical
and model theoretic properties of structures and their
interactions. By a structure we mean typical objects that arise in
computer science and mathematics such as data structures, programs,
transition systems, graphs, large databases, XML documents, algebraic
systems including groups, integers, fields, Boolean algebras and so
on.

Rod Downey, Bakhadyr Khoussainov, Dietrich Kuske, Markus Lohrey, and Moshe Y. Vardi. 07441 Summary – Algorithmic-Logical Theory of Infinite Structures. In Algorithmic-Logical Theory of Infinite Structures. Dagstuhl Seminar Proceedings, Volume 7441, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

Copy BibTex To Clipboard

@InProceedings{downey_et_al:DagSemProc.07441.2, author = {Downey, Rod and Khoussainov, Bakhadyr and Kuske, Dietrich and Lohrey, Markus and Vardi, Moshe Y.}, title = {{07441 Summary – Algorithmic-Logical Theory of Infinite Structures}}, booktitle = {Algorithmic-Logical Theory of Infinite Structures}, pages = {1--2}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7441}, editor = {Rod Downey and Bakhadyr Khoussainov and Dietrich Kuske and Markus Lohrey and Moshe Y. Vardi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07441.2}, URN = {urn:nbn:de:0030-drops-14111}, doi = {10.4230/DagSemProc.07441.2}, annote = {Keywords: Theories of infinite structures , computable model theory and automatic structures , model checking infinite systems} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail