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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 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and a robustness property related to Solovay tests. Our main work connects our property to the growth of the initial segment complexity for measures μ; the latter is defined as a μ-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We briefly discuss K-triviality for measures, which means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Löf randomness of a measure implies Martin-Löf absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures.

André Nies and Frank Stephan. Randomness and Initial Segment Complexity for Probability Measures. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 55:1-55:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{nies_et_al:LIPIcs.STACS.2020.55, author = {Nies, Andr\'{e} and Stephan, Frank}, title = {{Randomness and Initial Segment Complexity for Probability Measures}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {55:1--55:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.55}, URN = {urn:nbn:de:0030-drops-119168}, doi = {10.4230/LIPIcs.STACS.2020.55}, annote = {Keywords: algorithmic randomness, probability measure on Cantor space, Kolmogorov complexity, statistical superposition, quantum states} }

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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 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

XSAT is defined as the following: Given a propositional formula in conjunctive normal form, can one find an assignment to variables such that there is exactly only 1 literal that is true in every clause, while the other literals are false. The decision problem XSAT is known to be NP-complete. Crescenzi and Rossi [Pierluigi Crescenzi and Gianluca Rossi, 2002] introduced the variant where one searches for a pair of two solutions of an X3SAT instance with maximal Hamming Distance among them, that is, one wants to identify the largest number k such that there are two solutions of the instance with Hamming Distance k. Dahllöf [Vilhelm Dahllöf, 2005; Vilhelm Dahllöf, 2006] provided an algorithm using branch and bound method for Max Hamming Distance XSAT in O(1.8348^n); Fu, Zhou and Yin [Linlu Fu and Minghao Yin, 2012] worked on a more specific problem, the Max Hamming Distance X3SAT, and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an exact exponential algorithm to solve the Max Hamming Distance XSAT problem in O(1.4983^n) time. Like all of them, we will use the branch and bound technique alongside a newly defined measure to improve the analysis of the algorithm.

Gordon Hoi and Frank Stephan. Measure and Conquer for Max Hamming Distance XSAT. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 15:1-15:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hoi_et_al:LIPIcs.ISAAC.2019.15, author = {Hoi, Gordon and Stephan, Frank}, title = {{Measure and Conquer for Max Hamming Distance XSAT}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.15}, URN = {urn:nbn:de:0030-drops-115119}, doi = {10.4230/LIPIcs.ISAAC.2019.15}, annote = {Keywords: XSAT, Measure and Conquer, DPLL, Exponential Time 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 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

An infinite bit sequence is called recursively random if no computable strategy betting along the sequence has unbounded capital. It is well-known that the property of recursive randomness is closed under computable permutations. We investigate analogous statements for randomness notions defined by betting strategies that are computable within resource bounds. Suppose that S is a polynomial time computable permutation of the set of strings over the unary alphabet (identified with the set of natural numbers). If the inverse of S is not polynomially bounded, it is not hard to build a polynomial time random bit sequence Z such that Z o S is not polynomial time random. So one
should only consider permutations S satisfying the extra condition
that the inverse is polynomially bounded. Now the closure depends on additional assumptions in complexity theory.
Our first main result, Theorem 4, shows that if BPP contains a superpolynomial deterministic time class, then polynomial time randomness is not preserved by some permutation S such that in fact both S and its inverse are in P. Our second result, Theorem 11, shows that polynomial space randomness is preserved by polynomial time permutations with polynomially bounded inverse, so if P = PSPACE then polynomial time randomness is preserved.

André Nies and Frank Stephan. Closure of Resource-Bounded Randomness Notions Under Polynomial-Time Permutations. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 51:1-51:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{nies_et_al:LIPIcs.STACS.2018.51, author = {Nies, Andr\'{e} and Stephan, Frank}, title = {{Closure of Resource-Bounded Randomness Notions Under Polynomial-Time Permutations}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {51:1--51:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.51}, URN = {urn:nbn:de:0030-drops-84938}, doi = {10.4230/LIPIcs.STACS.2018.51}, annote = {Keywords: Computational complexity, Randomness via resource-bounded betting strategies, Martingales, Closure under permutations} }

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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:** Dagstuhl Seminar Proceedings, Volume 4421, Algebraic Methods in Computational Complexity (2005)

A well-studied problem in fault diagnosis is to identify the set of all good processors in a given set $\{p_1,p_2,\ldots,p_n\}$
of processors via asking some processors $p_i$ to test whether processor $p_j$ is good or faulty. Mathematically, the set $C$ of the indices of good processors forms an isolated clique in the graph with the edges $E = \{(i,j):$ if you ask $p_i$
to test $p_j$ then $p_i$ states that ``$p_j$ is good''$\}$; where $C$ is an isolated clique iff it holds for every $i \in C$ and $j \neq i$ that $(i,j) \in E$ iff $j \in C$.
In the present work, the classical setting of fault diagnosis is modified by no longer requiring that $C$ contains at least $\frac{n+1}{2}$ of the $n$ nodes of the graph. Instead, one is given a lower bound $a$ on the size of $C$ and the number
$n$ of nodes and one has to find a list of up to $n/a$ candidates containing all isolated cliques of size $a$ or more where the number of queries whether a given edge is in $E$ is as small as possible.
It is shown that the number of queries necessary differs at most by $n$ for the case of directed and undirected graphs. Furthermore,
for directed graphs the lower bound $n^2/(2a-2)-3n$ and the upper
bound $2n^2/a$ are established. For some constant values of $a$, better bounds are given. In the case of parallel queries, the number of rounds is at least $n/(a-1)-6$ and at most $O(\log(a)n/a)$.

William Gasarch and Frank Stephan. Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 4421, pp. 1-16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2005)

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@InProceedings{gasarch_et_al:DagSemProc.04421.3, author = {Gasarch, William and Stephan, Frank}, title = {{Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults}}, booktitle = {Algebraic Methods in Computational Complexity}, pages = {1--16}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {4421}, editor = {Harry Buhrman and Lance Fortnow and Thomas Thierauf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04421.3}, URN = {urn:nbn:de:0030-drops-1066}, doi = {10.4230/DagSemProc.04421.3}, annote = {Keywords: Isolated Cliques , Query-Complexity , Fault Diagnosis} }

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