Theory of Computing Systems最新文献

{"title":"Subgroup Membership in GL(2,Z)","authors":"Markus Lohrey","doi":"10.1007/s00224-023-10122-2","DOIUrl":"https://doi.org/10.1007/s00224-023-10122-2","url":null,"abstract":"Abstract It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all group elements are represented by so-called power words, i.e., words of the form $$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:msubsup> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:msubsup> <mml:mo>⋯</mml:mo> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msubsup> </mml:mrow> </mml:math> . Here the $$p_i$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> are explicit words over the generating set of the group and all $$z_i$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $$textsf{GL}(2,mathbb {Z})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be decided in polynomial time when elements of $$textsf{GL}(2,mathbb {Z})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are represented by matrices with binary encoded integers. For the same input representation, it also shown that one can compute in polynomial time the index of a given finitely generated subgroup of $$textsf{GL}(2,mathbb {Z})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135806947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reachability in Two-Parametric Timed Automata with one Parameter is EXPSPACE-Complete","authors":"Stefan Göller, Mathieu Hilaire","doi":"10.1007/s00224-023-10121-3","DOIUrl":"https://doi.org/10.1007/s00224-023-10121-3","url":null,"abstract":"Abstract Parametric timed automata (PTA) have been introduced by Alur, Henzinger, and Vardi as an extension of timed automata in which clocks can be compared against parameters. The reachability problem asks for the existence of an assignment of the parameters to the non-negative integers such that reachability holds in the underlying timed automaton. The reachability problem for PTA is long known to be undecidable, already over three parametric clocks. A few years ago, Bundala and Ouaknine proved that for PTA over two parametric clocks and one parameter the reachability problem is decidable and also showed a lower bound for the complexity class P S P A C E N E X P . Our main result is that the reachability problem for two-parametric timed automata with one parameter is E X P S P A C E -complete. Our contribution is two-fold. For the E X P S P A C E lower bound, inspired by [13, 14], we make use of deep results from complexity theory, namely a serializability characterization of E X P S P A C E (in turn based on Barrington’s Theorem) and a logspace translation of numbers in Chinese remainder representation to binary representation due to Chiu, Davida, and Litow. It is shown that with small PTA over two parametric clocks and one parameter one can simulate serializability computations. For the E X P S P A C E upper bound, we first give a careful exponential time reduction from PTA over two parametric clocks and one parameter to a (slight subclass of) parametric one-counter automata over one parameter based on a minor adjustment of a construction due to Bundala and Ouaknine. For solving the reachability problem for parametric one-counter automata with one parameter, we provide a series of techniques to partition a fictitious run into several carefully chosen subruns that allow us to prove that it is sufficient to consider a parameter value of exponential magnitude only. This allows us to show a doubly-exponential upper bound on the value of the only parameter of a PTA over two parametric clocks and one parameter. We hope that extensions of our techniques lead to finally establishing decidability of the long-standing open problem of reachability in parametric timed automata with two parametric clocks (and arbitrarily many parameters) and, if decidability holds, determinining its precise computational complexity.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135223285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Correction to: On the Transformation of LL(k)-linear to LL(1)-linear Grammars","authors":"I. Olkhovsky, A. Okhotin","doi":"10.1007/s00224-023-10120-4","DOIUrl":"https://doi.org/10.1007/s00224-023-10120-4","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"263"},"PeriodicalIF":0.5,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47920146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Hierarchy of Swarm-automaton for the Number of Agents","authors":"Kaoru Fujioka","doi":"10.1007/s00224-023-10117-z","DOIUrl":"https://doi.org/10.1007/s00224-023-10117-z","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"1 1","pages":"1-18"},"PeriodicalIF":0.5,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47310883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial-Time Axioms of Choice and Polynomial-Time Cardinality","authors":"Joshua A. Grochow","doi":"10.1007/s00224-023-10118-y","DOIUrl":"https://doi.org/10.1007/s00224-023-10118-y","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"627-669"},"PeriodicalIF":0.5,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44649146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Forced Periodicity of Perfect Colorings","authors":"Pyry Herva, J. Kari","doi":"10.1007/s00224-023-10127-x","DOIUrl":"https://doi.org/10.1007/s00224-023-10127-x","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"732 - 759"},"PeriodicalIF":0.5,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46249853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Preface of the Special Issue Dedicated to Selected Papers from CSR 2020","authors":"H. Fernau, M. Volkov","doi":"10.1007/s00224-022-10115-7","DOIUrl":"https://doi.org/10.1007/s00224-022-10115-7","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"219-220"},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47039049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Preface of STACS 2020 Special Issue","authors":"Christopher Paul, M. Bläser","doi":"10.1007/s00224-022-10116-6","DOIUrl":"https://doi.org/10.1007/s00224-022-10116-6","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"1-3"},"PeriodicalIF":0.5,"publicationDate":"2023-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43225280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"One-Tape Turing Machine and Branching Program Lower Bounds for MCSP","authors":"Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida","doi":"10.1007/s00224-022-10113-9","DOIUrl":"https://doi.org/10.1007/s00224-022-10113-9","url":null,"abstract":"<p>For a size parameter <span>(s:mathbb {N}to mathbb {N})</span>, the Minimum Circuit Size Problem (denoted by MCSP[<i>s</i>(<i>n</i>)]) is the problem of deciding whether the minimum circuit size of a given function <i>f</i> : {0,1}^<i>n</i> →{0,1} (represented by a string of length <i>N</i> := 2^<i>n</i>) is at most a threshold <i>s</i>(<i>n</i>). A recent line of work exhibited “hardness magnification” phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant <i>μ</i>₁ &gt; 0, if <span>(text {MCSP}[2^{mu _{1}cdot n}])</span> cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time <i>N</i>^1.01, then P≠NP. In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: (1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute <span>(text {MCSP}[2^{mu _{2}cdot n}])</span> in time <i>N</i>^1.99, for some constant <i>μ</i>₂ &gt; <i>μ</i>₁. (2) A non-deterministic (or parity) branching program of size <span>(o(N^{1.5}/log N))</span> cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nečiporuk method to MKTP, which previously appeared to be difficult. (3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least <span>(N^{1.5-oleft (1right )})</span>. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models. The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola, Electron. Colloq. Comput. Complexity (ECCC) 26, 51, 2019). En route, we obtain several related results: (1) There exists a (local) hitting set generator with seed length <span>(widetilde {O}(sqrt {N}))</span> secure against read-once polynomial-size non-deterministic branching programs on <i>N</i>-bit inputs. (2) Any read-once co-non-deterministic branching program computing MCSP must have size at least <span>(2^{widetilde {Omega }(N)})</span>.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"223 ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random Access in Persistent Strings and Segment Selection","authors":"P. Bille, I. L. Gørtz","doi":"10.1007/s00224-022-10109-5","DOIUrl":"https://doi.org/10.1007/s00224-022-10109-5","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"694 - 713"},"PeriodicalIF":0.5,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43025160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}