Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)最新文献
We show how to construct proof systems for NP languages where a deterministic polynomial-time verifier can check membership, given any N/sup (2/3)+/spl epsi// bits of an N-bit witness of membership. We also provide a slightly superpolynomial time proof system where the verifier can check membership, given only N/sup (1/2)+/spl epsi// bits of an N-bit witness. These pursuits are motivated by the work of Gal et. al. (1997). In addition, we construct proof systems where a deterministic polynomial-time verifier can check membership, given an N-bit string that agrees with a legitimate witness on just (N/2)+N/sup (4/5)+/spl epsi// bits. Our results and framework have applications for two related areas of research in complexity theory: proof systems for NP, and the relative power of Cook reductions and Karp-Levin type reductions. Our proof techniques are based on algebraic coding theory and small sample space constructions.
{"title":"Proofs, codes, and polynomial-time reducibilities","authors":"Ravi Kumar, D. Sivakumar","doi":"10.1109/CCC.1999.766261","DOIUrl":"https://doi.org/10.1109/CCC.1999.766261","url":null,"abstract":"We show how to construct proof systems for NP languages where a deterministic polynomial-time verifier can check membership, given any N/sup (2/3)+/spl epsi// bits of an N-bit witness of membership. We also provide a slightly superpolynomial time proof system where the verifier can check membership, given only N/sup (1/2)+/spl epsi// bits of an N-bit witness. These pursuits are motivated by the work of Gal et. al. (1997). In addition, we construct proof systems where a deterministic polynomial-time verifier can check membership, given an N-bit string that agrees with a legitimate witness on just (N/2)+N/sup (4/5)+/spl epsi// bits. Our results and framework have applications for two related areas of research in complexity theory: proof systems for NP, and the relative power of Cook reductions and Karp-Levin type reductions. Our proof techniques are based on algebraic coding theory and small sample space constructions.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117353679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Muchnik, Andrei E. Romashchenko, A. Shen, N. Vereshchagin
In this paper we construct a structure R that is a "finite version" of the semilattice of Turing degrees. Its elements are strings (technically, sequences of strings) and x/spl les/y means that K(x|)=(conditional Kolmogorov complexity of x relative to y) is small. We construct two elements in R that do not have greatest lower bound. We give a series of examples that show how natural algebraic constructions give two elements that have lower bound O (minimal element) but significant mutual information. (A first example of that kind was constructed by Gacs-Korner (1973) using completely different technique.) We define a notion of "complexity profile" of the pair of elements of R and give (exact) upper and lower bounds for it in a particular case.
{"title":"Upper semilattice of binary strings with the relation \"x is simple conditional to y\"","authors":"A. Muchnik, Andrei E. Romashchenko, A. Shen, N. Vereshchagin","doi":"10.1109/CCC.1999.766270","DOIUrl":"https://doi.org/10.1109/CCC.1999.766270","url":null,"abstract":"In this paper we construct a structure R that is a \"finite version\" of the semilattice of Turing degrees. Its elements are strings (technically, sequences of strings) and x/spl les/y means that K(x|)=(conditional Kolmogorov complexity of x relative to y) is small. We construct two elements in R that do not have greatest lower bound. We give a series of examples that show how natural algebraic constructions give two elements that have lower bound O (minimal element) but significant mutual information. (A first example of that kind was constructed by Gacs-Korner (1973) using completely different technique.) We define a notion of \"complexity profile\" of the pair of elements of R and give (exact) upper and lower bounds for it in a particular case.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128517107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper initiates the study of deterministic amplification of space-bounded probabilistic algorithms. The straightforward implementations of known amplification methods cannot be used for such algorithms, since they consume too much space. We present a new implementation of the Ajtai-Komlos-Szemeredi method, that enables to amplify an S-space algorithm that uses r random bits and errs with probability /spl epsiv/ to an O(kS)-space algorithm that uses r+O(k) random bits and errs with probability /spl epsiv//sup /spl Omega/(k)/. This method can be used to reduce the error probability of BPL algorithms below any constant, with only a constant addition of new random bits. This is weaker than the exponential reduction that can be achieved for BPP algorithms by methods that use only O(r) random bits. However we prove that any black-box amplification method that uses O(r) random bits and makes at most p parallel simulations reduces the error to at most /spl epsiv//sup O(p)/. Hence, in BPL, where p should be a constant, the error cannot be reduced to less than a constant. This means that our method is optimal with respect to black-box amplification methods, that use O(r) random bits. The new implementation of the AKS method is based on explicit constructions of constant-space online extractors and online expanders. These are extractors and expanders, for which neighborhoods can be computed in a constant space by a Turing machine with a one-way input tape.
{"title":"Deterministic amplification of space-bounded probabilistic algorithms","authors":"Ziv Bar-Yossef, Oded Goldreich, A. Wigderson","doi":"10.1109/CCC.1999.766276","DOIUrl":"https://doi.org/10.1109/CCC.1999.766276","url":null,"abstract":"This paper initiates the study of deterministic amplification of space-bounded probabilistic algorithms. The straightforward implementations of known amplification methods cannot be used for such algorithms, since they consume too much space. We present a new implementation of the Ajtai-Komlos-Szemeredi method, that enables to amplify an S-space algorithm that uses r random bits and errs with probability /spl epsiv/ to an O(kS)-space algorithm that uses r+O(k) random bits and errs with probability /spl epsiv//sup /spl Omega/(k)/. This method can be used to reduce the error probability of BPL algorithms below any constant, with only a constant addition of new random bits. This is weaker than the exponential reduction that can be achieved for BPP algorithms by methods that use only O(r) random bits. However we prove that any black-box amplification method that uses O(r) random bits and makes at most p parallel simulations reduces the error to at most /spl epsiv//sup O(p)/. Hence, in BPL, where p should be a constant, the error cannot be reduced to less than a constant. This means that our method is optimal with respect to black-box amplification methods, that use O(r) random bits. The new implementation of the AKS method is based on explicit constructions of constant-space online extractors and online expanders. These are extractors and expanders, for which neighborhoods can be computed in a constant space by a Turing machine with a one-way input tape.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127764122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A function f is self-reducible if it can be computed given an oracle for f. In a random-self-reduction the queries must be made in such a way that the distribution of the ith query is independent of the input that gave rise to it. Random-self-reductions have many applications, including countless cryptographic protocols, probabilistically checkable proofs, average-case complexity, and program checking. A simpler model of randomized self-reducibility is coherence, in which the only condition on the queries is that the input itself may not be among the queries. We show that there is a function which is random-self-reducible with 2 rounds of queries, but which is not even coherent, even if polynomial advice is allowed, when the queries must be made in a single round.
{"title":"Stronger separations for random-self-reducibility, rounds, and advice","authors":"L. Babai, Sophie Laplante","doi":"10.1109/CCC.1999.766268","DOIUrl":"https://doi.org/10.1109/CCC.1999.766268","url":null,"abstract":"A function f is self-reducible if it can be computed given an oracle for f. In a random-self-reduction the queries must be made in such a way that the distribution of the ith query is independent of the input that gave rise to it. Random-self-reductions have many applications, including countless cryptographic protocols, probabilistically checkable proofs, average-case complexity, and program checking. A simpler model of randomized self-reducibility is coherence, in which the only condition on the queries is that the input itself may not be among the queries. We show that there is a function which is random-self-reducible with 2 rounds of queries, but which is not even coherent, even if polynomial advice is allowed, when the queries must be made in a single round.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129190483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. A. Barrington, Chi-Jen Lu, Peter Bro Miltersen, Sven Skyum
In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC/sup 0/. This provides a striking contrast to the non-planar case, where exactly NC/sup 1/ is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outerface only, can be solved in LOGDCFL/spl sube/SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outerface only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outerface can be computed by a monotone planar circuit of width O(w) and depth w/sup O(1)/. Finally, we show that monotone planar read-once circuits, with inputs on the outerface only, can be efficiently learned using membership queries.
本文给出了单调平面电路的几个结果。我们证明了有界宽度的单调平面电路，可以访问负的输入变量，精确地计算非均匀AC/sup 0/下的函数。这与非平面情况形成鲜明对比，在非平面情况下，精确地计算NC/sup 1/。我们证明了只有外表面输入的单调平面电路的电路值问题，可以在LOGDCFL/spl sub /SC中解决，改进了Dymond和Cook提出的LOGCFL上界。我们表明，对于单调平面电路，输入仅在外表面，与宽度相比，过多的深度是无用的;用宽度为w、输入在外表面的单调平面电路计算的任何函数都可以用宽度为O(w)、深度为w/sup为O(1)/的单调平面电路计算。最后，我们证明了只在外表面输入的单调平面读一次电路可以使用隶属度查询有效地学习。
{"title":"On monotone planar circuits","authors":"D. A. Barrington, Chi-Jen Lu, Peter Bro Miltersen, Sven Skyum","doi":"10.1109/CCC.1999.766259","DOIUrl":"https://doi.org/10.1109/CCC.1999.766259","url":null,"abstract":"In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC/sup 0/. This provides a striking contrast to the non-planar case, where exactly NC/sup 1/ is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outerface only, can be solved in LOGDCFL/spl sube/SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outerface only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outerface can be computed by a monotone planar circuit of width O(w) and depth w/sup O(1)/. Finally, we show that monotone planar read-once circuits, with inputs on the outerface only, can be efficiently learned using membership queries.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"06 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128729078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey some recent developments in the study of the complexity of lattice problems. After a discussion of some problems on lattices which can be algorithmically solved efficiently, our main focus is the recent progress on complexity results of intractability. We discuss Ajtai's worst-case/average-case connections, NP-hardness and non-NP-hardness, transference theorems between primal and dual lattices, and the Ajtai-Dwork cryptosystem.
{"title":"Some recent progress on the complexity of lattice problems","authors":"Jin-Yi Cai","doi":"10.1109/CCC.1999.766274","DOIUrl":"https://doi.org/10.1109/CCC.1999.766274","url":null,"abstract":"We survey some recent developments in the study of the complexity of lattice problems. After a discussion of some problems on lattices which can be algorithmically solved efficiently, our main focus is the recent progress on complexity results of intractability. We discuss Ajtai's worst-case/average-case connections, NP-hardness and non-NP-hardness, transference theorems between primal and dual lattices, and the Ajtai-Dwork cryptosystem.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133698679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We apply a new transference theorem from the geometry of numbers to Ajtai's connection of average-case to worst-case complexity of lattice problems. We also derive stronger bounds for the special class of lattices which possess n/sup /spl epsiv//-unique shortest lattice vectors. This class of lattices plays a significant role in Ajtai's connection of average-case to worst-case complexity of the shortest lattice vector problem, and in the Ajtai-Dwork public-key cryptosystem. Our proofs are non-constructive, based on methods from harmonic analysis. They yield currently the best Ajtai connection factors.
{"title":"Applications of a new transference theorem to Ajtai's connection factor","authors":"Jin-Yi Cai","doi":"10.1109/CCC.1999.766278","DOIUrl":"https://doi.org/10.1109/CCC.1999.766278","url":null,"abstract":"We apply a new transference theorem from the geometry of numbers to Ajtai's connection of average-case to worst-case complexity of lattice problems. We also derive stronger bounds for the special class of lattices which possess n/sup /spl epsiv//-unique shortest lattice vectors. This class of lattices plays a significant role in Ajtai's connection of average-case to worst-case complexity of the shortest lattice vector problem, and in the Ajtai-Dwork public-key cryptosystem. Our proofs are non-constructive, based on methods from harmonic analysis. They yield currently the best Ajtai connection factors.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"3 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123873428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The effect of query order on NP-completeness is investigated. A sequence D/spl I.oarr/=(D/sub 1/,...,D/sub k/) of decision problems is defined to be sequentially complete for NP if each D/sub i//spl isin/NP and every problem in NP can be decided in polynomial time with one query to each of D/sub 1/,...,D/sub k/ in this order. It is shown that, if NP contains a language that is p-generic in the sense of Ambos-Spies, Fleischhack, and Huwig (1987), then for every integer k/spl ges/2, there is a sequence D/spl I.oarr/=(d/sub 1/,...,D/sub k/) such that D is sequentially complete for NP, but no nontrivial permutation (D(i/sub 1/),...,D(i/sub k/)) of D/spl I.oarr/ is sequentially complete for NP. It follows that such a sequence D/spl I.oarr/ exists if there is any strongly positive, p-computable probability measure /spl nu/ such that "/sub p/(NP)/spl ne/0.
研究了查询顺序对np完备性的影响。A序列D/spl I.oarr/=(D/sub 1/，…如果每个D/下标i//spl都是/NP，并且NP中的每个问题都可以在多项式时间内通过对D/下标1/，…的查询来决定，则定义决策问题的D/下标k/)对于NP是顺序完备的。D/下标k/按这个顺序。结果表明，如果NP包含Ambos-Spies, Fleischhack, and Huwig(1987)意义上的p-泛型语言，则对于每一个整数k/spl ges/2，存在一个序列D/spl I.oarr/=(D/ sub 1/，…)，D/下标k/)使得D对于NP是顺序完全的，但不存在非平凡排列(D(i/下标1/)，…，D /spl i .oarr/的D(i/sub k/))对于NP是顺序完全的。因此，如果存在任何强正的、p可计算的概率测度/spl nu/使得“/sub p/(NP)/spl ne/0”，则存在这样的序列D/spl I.oarr/。
{"title":"Query order and NP-completeness","authors":"J. J. Dai, J. H. Lutz","doi":"10.1109/CCC.1999.766272","DOIUrl":"https://doi.org/10.1109/CCC.1999.766272","url":null,"abstract":"The effect of query order on NP-completeness is investigated. A sequence D/spl I.oarr/=(D/sub 1/,...,D/sub k/) of decision problems is defined to be sequentially complete for NP if each D/sub i//spl isin/NP and every problem in NP can be decided in polynomial time with one query to each of D/sub 1/,...,D/sub k/ in this order. It is shown that, if NP contains a language that is p-generic in the sense of Ambos-Spies, Fleischhack, and Huwig (1987), then for every integer k/spl ges/2, there is a sequence D/spl I.oarr/=(d/sub 1/,...,D/sub k/) such that D is sequentially complete for NP, but no nontrivial permutation (D(i/sub 1/),...,D(i/sub k/)) of D/spl I.oarr/ is sequentially complete for NP. It follows that such a sequence D/spl I.oarr/ exists if there is any strongly positive, p-computable probability measure /spl nu/ such that \"/sub p/(NP)/spl ne/0.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128830356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze upper and lower bounds on size of Boolean conjunctions necessary and sufficient to approximate a given DNF formula by accuracy slightly better than 1/2 (here we define the size of a Boolean conjunction as the number of distinct variables on which it depends). Such an analysis determines the performance of a naive search algorithm that exhausts Boolean conjunctions in the order of their sizes. In fact, our analysis does not depend on kinds of symmetric functions to be exhausted: instead of conjunctions, counting either disjunctions, parity functions, majority functions, or even general symmetric functions, derives the same learning results from similar analyses.
{"title":"Learning DNF by approximating inclusion-exclusion formulae","authors":"J. Tarui, Tatsuie Tsukiji","doi":"10.1109/CCC.1999.766279","DOIUrl":"https://doi.org/10.1109/CCC.1999.766279","url":null,"abstract":"We analyze upper and lower bounds on size of Boolean conjunctions necessary and sufficient to approximate a given DNF formula by accuracy slightly better than 1/2 (here we define the size of a Boolean conjunction as the number of distinct variables on which it depends). Such an analysis determines the performance of a naive search algorithm that exhausts Boolean conjunctions in the order of their sizes. In fact, our analysis does not depend on kinds of symmetric functions to be exhausted: instead of conjunctions, counting either disjunctions, parity functions, majority functions, or even general symmetric functions, derives the same learning results from similar analyses.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129111972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the square-free numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC/sup 0/ [p] for any prime p. Similar lower bounds are presented for the set of square-free numbers, and for the problem of computing the greatest common divisor of two numbers.
{"title":"A lower bound for primality","authors":"E. Allender, M. Saks, I. Shparlinski","doi":"10.1109/CCC.1999.766257","DOIUrl":"https://doi.org/10.1109/CCC.1999.766257","url":null,"abstract":"Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the square-free numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC/sup 0/ [p] for any prime p. Similar lower bounds are presented for the set of square-free numbers, and for the problem of computing the greatest common divisor of two numbers.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114667840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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