54 Search Results for "Lohrey, Markus"


Document
RANDOM
Robustness for Space-Bounded Statistical Zero Knowledge

Authors: Eric Allender, Jacob Gray, Saachi Mutreja, Harsha Tirumala, and Pengxiang Wang

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)


Abstract
We show that the space-bounded Statistical Zero Knowledge classes SZK_L and NISZK_L are surprisingly robust, in that the power of the verifier and simulator can be strengthened or weakened without affecting the resulting class. Coupled with other recent characterizations of these classes [Eric Allender et al., 2023], this can be viewed as lending support to the conjecture that these classes may coincide with the non-space-bounded classes SZK and NISZK, respectively.

Cite as

Eric Allender, Jacob Gray, Saachi Mutreja, Harsha Tirumala, and Pengxiang Wang. Robustness for Space-Bounded Statistical Zero Knowledge. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 56:1-56:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{allender_et_al:LIPIcs.APPROX/RANDOM.2023.56,
  author =	{Allender, Eric and Gray, Jacob and Mutreja, Saachi and Tirumala, Harsha and Wang, Pengxiang},
  title =	{{Robustness for Space-Bounded Statistical Zero Knowledge}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{56:1--56:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.56},
  URN =		{urn:nbn:de:0030-drops-188815},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.56},
  annote =	{Keywords: Interactive Proofs}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
The Identity Problem in ℤ ≀ ℤ Is Decidable

Authors: Ruiwen Dong

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
We consider semigroup algorithmic problems in the wreath product ℤ ≀ ℤ. Our paper focuses on two decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain the neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of ℤ ≀ ℤ. We show that both problems are decidable. Our result complements the undecidability of the Semigroup Membership Problem (does a semigroup contain a given element?) in ℤ ≀ ℤ shown by Lohrey, Steinberg and Zetzsche (ICALP 2013), and contributes an important step towards solving semigroup algorithmic problems in general metabelian groups.

Cite as

Ruiwen Dong. The Identity Problem in ℤ ≀ ℤ Is Decidable. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 124:1-124:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{dong:LIPIcs.ICALP.2023.124,
  author =	{Dong, Ruiwen},
  title =	{{The Identity Problem in \mathbb{Z} ≀ \mathbb{Z} Is Decidable}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{124:1--124:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.124},
  URN =		{urn:nbn:de:0030-drops-181768},
  doi =		{10.4230/LIPIcs.ICALP.2023.124},
  annote =	{Keywords: wreath product, algorithmic group theory, identity problem, polynomial semiring, positive coefficients}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
On the Complexity of Diameter and Related Problems in Permutation Groups

Authors: Markus Lohrey and Andreas Rosowski

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
We prove that it is Π₂^𝖯-complete to verify whether the diameter of a given permutation group G = ⟨A⟩ is bounded by a unary encoded number k. This solves an open problem from a paper of Even and Goldreich, where the problem was shown to be NP-hard. Verifying whether the diameter is exactly k is complete for the class consisting of all intersections of a Π₂^𝖯-language and a Σ₂^𝖯-language. A similar result is shown for the length of a given permutation π, which is the minimal k such that π can be written as a product of at most k generators from A. Even and Goldreich proved that it is NP-complete to verify, whether the length of a given π is at most k (with k given in unary encoding). We show that it is DP-complete to verify whether the length is exactly k. Finally, we deduce from our result on the diameter that it is Π₂^𝖯-complete to check whether a given finite automaton with transitions labelled by permutations from S_n produces all permutations from S_n.

Cite as

Markus Lohrey and Andreas Rosowski. On the Complexity of Diameter and Related Problems in Permutation Groups. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 134:1-134:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{lohrey_et_al:LIPIcs.ICALP.2023.134,
  author =	{Lohrey, Markus and Rosowski, Andreas},
  title =	{{On the Complexity of Diameter and Related Problems in Permutation Groups}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{134:1--134:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.134},
  URN =		{urn:nbn:de:0030-drops-181864},
  doi =		{10.4230/LIPIcs.ICALP.2023.134},
  annote =	{Keywords: algorithms for finite groups, diameter of permutation groups, rational subsets in groups}
}
Document
Low-Latency Sliding Window Algorithms for Formal Languages

Authors: Moses Ganardi, Louis Jachiet, Markus Lohrey, and Thomas Schwentick

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)


Abstract
Low-latency sliding window algorithms for regular and context-free languages are studied, where latency refers to the worst-case time spent for a single window update or query. For every regular language L it is shown that there exists a constant-latency solution that supports adding and removing symbols independently on both ends of the window (the so-called two-way variable-size model). We prove that this result extends to all visibly pushdown languages. For deterministic 1-counter languages we present a 𝒪(log n) latency sliding window algorithm for the two-way variable-size model where n refers to the window size. We complement these results with a conditional lower bound: there exists a fixed real-time deterministic context-free language L such that, assuming the OMV (online matrix vector multiplication) conjecture, there is no sliding window algorithm for L with latency n^(1/2-ε) for any ε > 0, even in the most restricted sliding window model (one-way fixed-size model). The above mentioned results all refer to the unit-cost RAM model with logarithmic word size. For regular languages we also present a refined picture using word sizes 𝒪(1), 𝒪(log log n), and 𝒪(log n).

Cite as

Moses Ganardi, Louis Jachiet, Markus Lohrey, and Thomas Schwentick. Low-Latency Sliding Window Algorithms for Formal Languages. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ganardi_et_al:LIPIcs.FSTTCS.2022.38,
  author =	{Ganardi, Moses and Jachiet, Louis and Lohrey, Markus and Schwentick, Thomas},
  title =	{{Low-Latency Sliding Window Algorithms for Formal Languages}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{38:1--38:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.38},
  URN =		{urn:nbn:de:0030-drops-174301},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.38},
  annote =	{Keywords: Streaming algorithms, regular languages, context-free languages}
}
Document
Membership Problems in Finite Groups

Authors: Markus Lohrey, Andreas Rosowski, and Georg Zetzsche

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)


Abstract
We show that the subset sum problem, the knapsack problem and the rational subset membership problem for permutation groups are NP-complete. Concerning the knapsack problem we obtain NP-completeness for every fixed n ≥ 3, where n is the number of permutations in the knapsack equation. In other words: membership in products of three cyclic permutation groups is NP-complete. This sharpens a result of Luks [Eugene M. Luks, 1991], which states NP-completeness of the membership problem for products of three abelian permutation groups. We also consider the context-free membership problem in permutation groups and prove that it is PSPACE-complete but NP-complete for a restricted class of context-free grammars where acyclic derivation trees must have constant Horton-Strahler number. Our upper bounds hold for black box groups. The results for context-free membership problems in permutation groups yield new complexity bounds for various intersection non-emptiness problems for DFAs and a single context-free grammar.

Cite as

Markus Lohrey, Andreas Rosowski, and Georg Zetzsche. Membership Problems in Finite Groups. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2022.71,
  author =	{Lohrey, Markus and Rosowski, Andreas and Zetzsche, Georg},
  title =	{{Membership Problems in Finite Groups}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{71:1--71:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.71},
  URN =		{urn:nbn:de:0030-drops-168694},
  doi =		{10.4230/LIPIcs.MFCS.2022.71},
  annote =	{Keywords: algorithms for finite groups, intersection non-emptiness problems, knapsack problems in groups}
}
Document
Streaming Word Problems

Authors: Markus Lohrey and Lukas Lück

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)


Abstract
We study deterministic and randomized streaming algorithms for word problems of finitely generated groups. For finitely generated linear groups, metabelian groups and free solvable groups we show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems. We also show that the class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group theoretical constructions: finite extensions, direct products, free products and wreath products by free abelian groups. We contrast these results with several lower bound. An example of a finitely presented group, where the word problem has only a linear space randomized streaming algorithm, is Thompson’s group F.

Cite as

Markus Lohrey and Lukas Lück. Streaming Word Problems. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 72:1-72:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2022.72,
  author =	{Lohrey, Markus and L\"{u}ck, Lukas},
  title =	{{Streaming Word Problems}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{72:1--72:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.72},
  URN =		{urn:nbn:de:0030-drops-168707},
  doi =		{10.4230/LIPIcs.MFCS.2022.72},
  annote =	{Keywords: word problems for groups, streaming algorithms}
}
Document
Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima

Authors: J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)


Abstract
We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, "hypersuccinct trees", hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures. We apply hypersuccinct trees to obtain a universal compressed data structure for range-minimum queries. It has constant query time and the optimal worst-case space usage of 2n+o(n) bits, but the space drops to 1.736n + o(n) bits on average for random permutations of n elements, and 2lg binom{n}{r} + o(n) for arrays with r increasing runs, respectively. Both results are optimal; the former answers an open problem of Davoodi et al. (2014) and Golin et al. (2016). Compared to prior work on succinct data structures, we do not have to tailor our data structure to specific applications; hypersuccinct trees automatically adapt to the trees at hand. We show that they simultaneously achieve the optimal space usage to within lower order terms for a wide range of distributions over tree shapes, including: binary search trees (BSTs) generated by insertions in random order / Cartesian trees of random arrays, random fringe-balanced BSTs, binary trees with a given number of binary/unary/leaf nodes, random binary tries generated from memoryless sources, full binary trees, unary paths, as well as uniformly chosen weight-balanced BSTs, AVL trees, and left-leaning red-black trees.

Cite as

J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild. Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 70:1-70:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{munro_et_al:LIPIcs.ESA.2021.70,
  author =	{Munro, J. Ian and Nicholson, Patrick K. and Benkner, Louisa Seelbach and Wild, Sebastian},
  title =	{{Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{70:1--70:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.70},
  URN =		{urn:nbn:de:0030-drops-146512},
  doi =		{10.4230/LIPIcs.ESA.2021.70},
  annote =	{Keywords: analysis of algorithms, universal source code, compressed trees, succinct data structure, succinct trees, tree covering, random binary search trees, range-minimum queries}
}
Document
Subgroup Membership in GL(2,Z)

Authors: Markus Lohrey

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)


Abstract
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ⋯ p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,ℤ) can be decided in polynomial time when all matrix entries are given in binary notation.

Cite as

Markus Lohrey. Subgroup Membership in GL(2,Z). In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{lohrey:LIPIcs.STACS.2021.51,
  author =	{Lohrey, Markus},
  title =	{{Subgroup Membership in GL(2,Z)}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{51:1--51:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.51},
  URN =		{urn:nbn:de:0030-drops-136961},
  doi =		{10.4230/LIPIcs.STACS.2021.51},
  annote =	{Keywords: free groups, virtually free groups, subgroup membership, matrix groups}
}
Document
Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups

Authors: Markus Lohrey and Georg Zetzsche

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)


Abstract
We prove that the power word problem for the solvable Baumslag-Solitar groups BS(1,q) = ⟨ a,t ∣ t a t^{-1} = a^q ⟩ can be solved in TC⁰. In the power word problem, the input consists of group elements g₁, …, g_d and binary encoded integers n₁, …, n_d and it is asked whether g₁^{n₁} ⋯ g_d^{n_d} = 1 holds. Moreover, we prove that the knapsack problem for BS(1,q) is NP-complete. In the knapsack problem, the input consists of group elements g₁, …, g_d,h and it is asked whether the equation g₁^{x₁} ⋯ g_d^{x_d} = h has a solution in ℕ^d.

Cite as

Markus Lohrey and Georg Zetzsche. Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2020.67,
  author =	{Lohrey, Markus and Zetzsche, Georg},
  title =	{{Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{67:1--67:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.67},
  URN =		{urn:nbn:de:0030-drops-127364},
  doi =		{10.4230/LIPIcs.MFCS.2020.67},
  annote =	{Keywords: computational group theory, matrix problems, Baumslag-Solitar groups}
}
Document
Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems

Authors: Laurent Bartholdi, Michael Figelius, Markus Lohrey, and Armin Weiß

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)


Abstract
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk’s group and Thompson’s groups, we prove that their word problem is ALOGTIME-hard. For some of these groups (including Grigorchuk’s group and Thompson’s groups) we prove that the circuit value problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.

Cite as

Laurent Bartholdi, Michael Figelius, Markus Lohrey, and Armin Weiß. Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 29:1-29:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{bartholdi_et_al:LIPIcs.CCC.2020.29,
  author =	{Bartholdi, Laurent and Figelius, Michael and Lohrey, Markus and Wei{\ss}, Armin},
  title =	{{Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{29:1--29:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.29},
  URN =		{urn:nbn:de:0030-drops-125814},
  doi =		{10.4230/LIPIcs.CCC.2020.29},
  annote =	{Keywords: NC^1-hardness, word problem, G-programs, straight-line programs, non-solvable groups, self-similar groups, Thompson’s groups, Grigorchuk’s group}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
The Complexity of Knapsack Problems in Wreath Products

Authors: Michael Figelius, Moses Ganardi, Markus Lohrey, and Georg Zetzsche

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)


Abstract
We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable groups. For a finitely generated group we study the so-called power word problem (does a given expression u₁^{k₁} … u_d^{k_d}, where u₁, …, u_d are words over the group generators and k₁, …, k_d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u₁^{x₁} … u_d^{x_d} = v, where u₁, …, u_d,v are words over the group generators and x₁,…,x_d are variables, have a solution in the natural numbers). We prove that the power word problem for wreath products of the form G ≀ ℤ with G nilpotent and iterated wreath products of free abelian groups belongs to TC⁰. As an application of the latter, the power word problem for free solvable groups is in TC⁰. On the other hand we show that for wreath products G ≀ ℤ, where G is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is coNP-hard. For the knapsack problem we show NP-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product G ≀ ℤ, where G is uniformly efficiently non-solvable, is Σ₂^p-hard.

Cite as

Michael Figelius, Moses Ganardi, Markus Lohrey, and Georg Zetzsche. The Complexity of Knapsack Problems in Wreath Products. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 126:1-126:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{figelius_et_al:LIPIcs.ICALP.2020.126,
  author =	{Figelius, Michael and Ganardi, Moses and Lohrey, Markus and Zetzsche, Georg},
  title =	{{The Complexity of Knapsack Problems in Wreath Products}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{126:1--126:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.126},
  URN =		{urn:nbn:de:0030-drops-125339},
  doi =		{10.4230/LIPIcs.ICALP.2020.126},
  annote =	{Keywords: algorithmic group theory, knapsack, wreath product}
}
Document
Sliding Window Property Testing for Regular Languages

Authors: Moses Ganardi, Danny Hucke, Markus Lohrey, and Tatiana Starikovskaya

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)


Abstract
We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window n and a stream of symbols. At each time instant, we must decide whether the suffix of length n of the current stream ("the active window") belongs to a given regular language. Recent works [Moses Ganardi et al., 2018; Moses Ganardi et al., 2016] showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size n and provided natural language theoretic characterizations of the space complexity classes. Subsequently, [Moses Ganardi et al., 2018] extended this result to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity. In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We show that for every regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.

Cite as

Moses Ganardi, Danny Hucke, Markus Lohrey, and Tatiana Starikovskaya. Sliding Window Property Testing for Regular Languages. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{ganardi_et_al:LIPIcs.ISAAC.2019.6,
  author =	{Ganardi, Moses and Hucke, Danny and Lohrey, Markus and Starikovskaya, Tatiana},
  title =	{{Sliding Window Property Testing for Regular Languages}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{6:1--6:13},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.6},
  URN =		{urn:nbn:de:0030-drops-115023},
  doi =		{10.4230/LIPIcs.ISAAC.2019.6},
  annote =	{Keywords: Streaming algorithms, approximation algorithms, regular languages}
}
Document
Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)

Authors: Volker Diekert, Olga Kharlampovich, Markus Lohrey, and Alexei Myasnikov

Published in: Dagstuhl Reports, Volume 9, Issue 3 (2019)


Abstract
Since its early days, combinatorial group theory was deeply interwoven with computability theory. In the last 20 years we have seen many new successful interactions between group theory and computer science. On one hand, groups played an important rule in many developments in complexity theory and automata theory. On the other hand, concepts from these computer science fields as well as efficient algorithms, cryptography, and data compression led to the formulation of new questions in group theory. The Dagstuhl Seminar Algorithmic Problems in Group Theory was aimed at bringing together researchers from group theory and computer science so that they can share their expertise. This report documents the material presented during the course of the seminar.

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Volker Diekert, Olga Kharlampovich, Markus Lohrey, and Alexei Myasnikov. Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131). In Dagstuhl Reports, Volume 9, Issue 3, pp. 83-110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@Article{diekert_et_al:DagRep.9.3.83,
  author =	{Diekert, Volker and Kharlampovich, Olga and Lohrey, Markus and Myasnikov, Alexei},
  title =	{{Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)}},
  pages =	{83--110},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{9},
  number =	{3},
  editor =	{Diekert, Volker and Kharlampovich, Olga and Lohrey, Markus and Myasnikov, Alexei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.9.3.83},
  URN =		{urn:nbn:de:0030-drops-112939},
  doi =		{10.4230/DagRep.9.3.83},
  annote =	{Keywords: algorithmic group theory; generic-case complexity; circuit complexity; diophantine theories}
}
Document
The Power Word Problem

Authors: Markus Lohrey and Armin Weiß

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


Abstract
In this work we introduce a new succinct variant of the word problem in a finitely generated group G, which we call the power word problem: the input word may contain powers p^x, where p is a finite word over generators of G and x is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G). The main result of the paper states that the power word problem for a finitely generated free group F is AC^0-Turing-reducible to the word problem for F. Moreover, the following hardness result is shown: For a wreath product G Wr Z, where G is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where G is abelian: then the power word problem is shown to be in TC^0.

Cite as

Markus Lohrey and Armin Weiß. The Power Word Problem. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2019.43,
  author =	{Lohrey, Markus and Wei{\ss}, Armin},
  title =	{{The Power Word Problem}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{43:1--43:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.43},
  URN =		{urn:nbn:de:0030-drops-109871},
  doi =		{10.4230/LIPIcs.MFCS.2019.43},
  annote =	{Keywords: word problem, compressed word problem, free groups}
}
Document
Compressed Decision Problems in Hyperbolic Groups

Authors: Derek Holt, Markus Lohrey, and Saul Schleimer

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)


Abstract
We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight-line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group G, the compressed knapsack problem in G is NP-complete.

Cite as

Derek Holt, Markus Lohrey, and Saul Schleimer. Compressed Decision Problems in Hyperbolic Groups. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{holt_et_al:LIPIcs.STACS.2019.37,
  author =	{Holt, Derek and Lohrey, Markus and Schleimer, Saul},
  title =	{{Compressed Decision Problems in Hyperbolic Groups}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{37:1--37:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.37},
  URN =		{urn:nbn:de:0030-drops-102766},
  doi =		{10.4230/LIPIcs.STACS.2019.37},
  annote =	{Keywords: hyperbolic groups, algorithms for compressed words, circuit evaluation problems}
}
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