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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

This paper studies the recursion-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings α and β such that α is strictly quasi-isometrically reducible to β, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka.

Karen Frilya Celine, Ziyuan Gao, Sanjay Jain, Ryan Lou, Frank Stephan, and Guohua Wu. Quasi-Isometric Reductions Between Infinite Strings. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{celine_et_al:LIPIcs.MFCS.2024.37, author = {Celine, Karen Frilya and Gao, Ziyuan and Jain, Sanjay and Lou, Ryan and Stephan, Frank and Wu, Guohua}, title = {{Quasi-Isometric Reductions Between Infinite Strings}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {37:1--37:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.37}, URN = {urn:nbn:de:0030-drops-205931}, doi = {10.4230/LIPIcs.MFCS.2024.37}, annote = {Keywords: Quasi-isometry, recursion theory, infinite strings} }

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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary and n the number of states of the finite automata considered. The following main results are obtained:
1) Equality and inclusion of NFAs can be decided within time 2^O((n log n)^{1/3}); previous upper bound 2^O((n log n)^{1/2}) was by Chrobak (1986) via DFA conversion.
2) The state complexity of operations of UFAs (unambiguous finite automata) increases for complementation and union at most by quasipolynomial; however, for concatenation of two n-state UFAs, the worst case is an UFA of at least 2^Ω(n^{1/6}) states. Previously the upper bounds for complementation and union were exponential-type and this lower bound for concatenation is new.

Wojciech Czerwiński, Maciej Dębski, Tomasz Gogasz, Gordon Hoi, Sanjay Jain, Michał Skrzypczak, Frank Stephan, and Christopher Tan. Languages Given by Finite Automata over the Unary Alphabet. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{czerwinski_et_al:LIPIcs.FSTTCS.2023.22, author = {Czerwi\'{n}ski, Wojciech and D\k{e}bski, Maciej and Gogasz, Tomasz and Hoi, Gordon and Jain, Sanjay and Skrzypczak, Micha{\l} and Stephan, Frank and Tan, Christopher}, title = {{Languages Given by Finite Automata over the Unary Alphabet}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {22:1--22:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.22}, URN = {urn:nbn:de:0030-drops-193959}, doi = {10.4230/LIPIcs.FSTTCS.2023.22}, annote = {Keywords: Nondeterministic Finite Automata, Unambiguous Finite Automata, Upper Bounds on Runtime, Conditional Lower Bounds, Languages over the Unary Alphabet} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

X3SAT is the problem of whether one can satisfy a given set of clauses with up to three literals such that in every clause, exactly one literal is true and the others are false. A related question is to determine the maximal Hamming distance between two solutions of the instance. Dahllöf provided an algorithm for Maximum Hamming Distance XSAT, which is more complicated than the same problem for X3SAT, with a runtime of O(1.8348^n); Fu, Zhou and Yin considered Maximum Hamming Distance for X3SAT and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an algorithm in O(1.3298^n) time to solve the Max Hamming Distance X3SAT problem; the algorithm actually counts for each k the number of pairs of solutions which have Hamming Distance k.

Gordon Hoi, Sanjay Jain, and Frank Stephan. A Fast Exponential Time Algorithm for Max Hamming Distance X3SAT. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hoi_et_al:LIPIcs.FSTTCS.2019.17, author = {Hoi, Gordon and Jain, Sanjay and Stephan, Frank}, title = {{A Fast Exponential Time Algorithm for Max Hamming Distance X3SAT}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {17:1--17:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.17}, URN = {urn:nbn:de:0030-drops-115799}, doi = {10.4230/LIPIcs.FSTTCS.2019.17}, annote = {Keywords: X3SAT Problem, Maximum Hamming Distance of Solutions, Exponential Time Algorithms, DPLL Algorithms} }

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**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)

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@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} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

The present work investigates inductive inference from the perspective
of reverse mathematics. Reverse mathematics is a framework which relates
the proof strength of theorems and axioms throughout many areas of
mathematics in an interdisciplinary way. The present work looks at
basic notions of learnability including Angluin's tell-tale condition and its variants for learning in the limit and for conservative learning. Furthermore, the more general criterion of partial learning is investigated. These notions are studied in the reverse mathematics context for uniformly and weakly represented families of languages. The results are stated in terms of axioms referring to domination and induction strength.

Rupert Hölzl, Sanjay Jain, and Frank Stephan. Inductive Inference and Reverse Mathematics. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 420-433, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{holzl_et_al:LIPIcs.STACS.2015.420, author = {H\"{o}lzl, Rupert and Jain, Sanjay and Stephan, Frank}, title = {{Inductive Inference and Reverse Mathematics}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {420--433}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.420}, URN = {urn:nbn:de:0030-drops-49324}, doi = {10.4230/LIPIcs.STACS.2015.420}, annote = {Keywords: reverse mathematics, recursion theory, inductive inference, learning from positive data} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Within the frameworks of learning in the limit of indexed classes of
recursive languages from positive data and automatic learning in the
limit of indexed classes of regular languages (with automatically
computable sets of indices), we study the problem of minimizing the
maximum number of mind changes F_M(n) by a learner M on all languages
with indices not exceeding n. For inductive inference of recursive
languages, we establish two conditions under which F_M(n) can be made
smaller than any recursive unbounded non-decreasing function. We also
establish how F_M(n) is affected if at least one of these two
conditions does not hold. In the case of automatic learning, some
partial results addressing speeding up the function F_M(n) are obtained.

Sanjay Jain and Efim Kinber. Mind Change Speed-up for Learning Languages from Positive Data. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 350-361, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{jain_et_al:LIPIcs.STACS.2012.350, author = {Jain, Sanjay and Kinber, Efim}, title = {{Mind Change Speed-up for Learning Languages from Positive Data}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {350--361}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.350}, URN = {urn:nbn:de:0030-drops-33936}, doi = {10.4230/LIPIcs.STACS.2012.350}, annote = {Keywords: Algorithmic and automatic learning, mind changes, speedup} }

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