Pub Date : 2023-05-09DOI: 10.1007/s00224-023-10122-2
Markus Lohrey
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 $$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ p1z1p2z2⋯pkzk . Here the $$p_i$$ pi are explicit words over the generating set of the group and all $$z_i$$ zi are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $$textsf{GL}(2,mathbb {Z})$$ GL(2,Z) can be decided in polynomial time when elements of $$textsf{GL}(2,mathbb {Z})$$ GL(2,Z) 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 $$textsf{GL}(2,mathbb {Z})$$ GL(2,Z) .
摘要证明了一个几乎自由群的子群隶属问题可以在多项式时间内决定，当所有群元素都由所谓的幂词表示时，即$$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ p 1 z 1 p 2 z 2⋯p k z k。这里的$$p_i$$ pi是组的生成集上的显式单词，所有的$$z_i$$ zi都是二进制编码的整数。作为推论，当$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的元素用二进制编码的整数矩阵表示时，矩阵群$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的子群隶属性问题可以在多项式时间内确定。对于相同的输入表示，它还表明可以在多项式时间内计算给定的有限生成的子群$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的索引。
{"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}
Pub Date : 2023-04-24DOI: 10.1007/s00224-023-10121-3
Stefan Göller, Mathieu Hilaire
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.
参数时间自动机(PTA)由Alur, Henzinger和Vardi作为时间自动机的扩展引入，其中时钟可以与参数进行比较。可达性问题要求参数赋值给非负整数的存在性，使得可达性在底层时间自动机中保持。PTA的可达性问题早已被认为是无法确定的，已经超过了三个参数时钟。几年前，Bundala和Ouaknine证明了对于具有两个参数时钟和一个参数的PTA，可达性问题是可决定的，并给出了复杂度类P S P A C E N E X P的下界。我们的主要研究结果是，单参数双参数时间自动机的可达性问题是E X P S P A C E完备的。我们的贡献是双重的。对于E X P S P A C E下界，受到[13,14]的启发，我们利用了复杂性理论的深层结果，即E X P S P A C E的可串行性表征(反过来基于Barrington定理)，以及Chiu, Davida和Litow将中文剩余表示中的数字转换为二进制表示。结果表明，在两个参数时钟和一个参数时钟上使用较小的PTA可以模拟串行性计算。对于E X P S P A C E上界，我们首先给出了一个仔细的指数时间缩减，从两个参数时钟和一个参数的PTA到一个参数单计数器自动机的(轻微子类)，基于对Bundala和Ouaknine构造的轻微调整。为了解决具有一个参数的参数单计数器自动机的可达性问题，我们提供了一系列技术，将虚拟运行划分为几个精心选择的子组，使我们能够证明仅考虑指数量级的参数值是足够的。这允许我们在两个参数时钟和一个参数上显示PTA的唯一参数值的双指数上界。我们希望我们的技术的扩展导致最终建立具有两个参数时钟(和任意多个参数)的参数时间自动机中可达性的长期开放问题的可判定性，并且，如果可判定性成立，确定其精确的计算复杂性。
{"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}
Pub Date : 2023-02-11DOI: 10.1007/s00224-023-10120-4
I. Olkhovsky, A. Okhotin
{"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}
Pub Date : 2023-02-11DOI: 10.1007/s00224-023-10117-z
Kaoru Fujioka
{"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}
Pub Date : 2023-01-17DOI: 10.1007/s00224-023-10118-y
Joshua A. Grochow
{"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}
Pub Date : 2023-01-11DOI: 10.1007/s00224-022-10115-7
H. Fernau, M. Volkov
{"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}
Pub Date : 2023-01-07DOI: 10.1007/s00224-022-10116-6
Christopher Paul, M. Bläser
{"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}
Pub Date : 2022-12-27DOI: 10.1007/s00224-022-10113-9
Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida
For a size parameter (s:mathbb {N}to mathbb {N}), the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}n →{0,1} (represented by a string of length N := 2n) is at most a threshold s(n). 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 μ1 > 0, if (text {MCSP}[2^{mu _{1}cdot n}]) cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N1.01, 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 (text {MCSP}[2^{mu _{2}cdot n}]) in time N1.99, for some constant μ2 > μ1. (2) A non-deterministic (or parity) branching program of size (o(N^{1.5}/log N)) 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 (N^{1.5-oleft (1right )}). 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 (widetilde {O}(sqrt {N})) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs. (2) Any read-once co-non-deterministic branching program computing MCSP must have size at least (2^{widetilde {Omega }(N)}).
{"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}<sup><i>n</i></sup> →{0,1} (represented by a string of length <i>N</i> := 2<sup><i>n</i></sup>) 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><sub>1</sub> > 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><sup>1.01</sup>, 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><sup>1.99</sup>, for some constant <i>μ</i><sub>2</sub> > <i>μ</i><sub>1</sub>. (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}
Pub Date : 2022-12-17DOI: 10.1007/s00224-022-10109-5
P. Bille, I. L. Gørtz
{"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}
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