For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Zm of degree O(rradicn) that is a generalized representation (commonly also called weak representation) of the MODl function. We give a detailed study of the case when m has exactly two distinct prime factors, and classify the minimum possible degree for a symmetric representing polynomial
{"title":"On modular counting with polynomials","authors":"Kristoffer Arnsfelt Hansen","doi":"10.1109/CCC.2006.29","DOIUrl":"https://doi.org/10.1109/CCC.2006.29","url":null,"abstract":"For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Zm of degree O(rradicn) that is a generalized representation (commonly also called weak representation) of the MODl function. We give a detailed study of the case when m has exactly two distinct prime factors, and classify the minimum possible degree for a symmetric representing polynomial","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115209132","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}
Lovdsz and Schrijver (1991) defined three progressively stronger procedures LS0, LS and LS+, for systematically tightening linear relaxations over many rounds. All three procedures yield the integral hull after at most n rounds. On the other hand, constant rounds of LS+ can derive the relaxations behind many famous approximation algorithms such as those for MAX-CUT, SPARSEST-CUT. So proving round lower bounds for these procedures on specific problems may give evidence about inapproximability. We prove new round lower bounds for vertex cover in the LS hierarchy. Arora et al. (2006) showed that the integrality gap for VERTEX COVER relaxations remains 2 - o(1) even after Omega(log n) rounds LS. However, their method can only prove round lower bounds as large as the girth of the input graph, which is O(log n) for interesting graphs. We break through this "girth barrier" and show that the integrality gap for VERTEX COVER remains 1.5 - epsi even after Omega(log2 n) rounds of LS. In contrast, the best PCP-based results only rule out 1.36-approximations. Moreover, we conjecture that the new technique we introduce to prove our lower bound, the "fence" method, may lead to linear or nearly linear LS round lower bounds for VERTEX COVER
{"title":"New Lower Bounds for Vertex Cover in the Lovasz-Schrijver Hierarchy","authors":"Iannis Tourlakis","doi":"10.1109/CCC.2006.28","DOIUrl":"https://doi.org/10.1109/CCC.2006.28","url":null,"abstract":"Lovdsz and Schrijver (1991) defined three progressively stronger procedures LS0, LS and LS+, for systematically tightening linear relaxations over many rounds. All three procedures yield the integral hull after at most n rounds. On the other hand, constant rounds of LS+ can derive the relaxations behind many famous approximation algorithms such as those for MAX-CUT, SPARSEST-CUT. So proving round lower bounds for these procedures on specific problems may give evidence about inapproximability. We prove new round lower bounds for vertex cover in the LS hierarchy. Arora et al. (2006) showed that the integrality gap for VERTEX COVER relaxations remains 2 - o(1) even after Omega(log n) rounds LS. However, their method can only prove round lower bounds as large as the girth of the input graph, which is O(log n) for interesting graphs. We break through this \"girth barrier\" and show that the integrality gap for VERTEX COVER remains 1.5 - epsi even after Omega(log2 n) rounds of LS. In contrast, the best PCP-based results only rule out 1.36-approximations. Moreover, we conjecture that the new technique we introduce to prove our lower bound, the \"fence\" method, may lead to linear or nearly linear LS round lower bounds for VERTEX COVER","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126254133","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}
An exposure-resilient extractor is an efficient procedure that, from a random variable with imperfect min-entropy, produces randomness that passes all statistical tests including those that have bounded access to the random variable, with adaptive queries that can depend on the string being tested. More precisely, EXT : {0, 1}n times {0, 1}d rarr {0, 1}m is a (k, epsi)-exposure resilient extractor resistant to q queries if, when the min-entropy of x is at least k and y is random, EXT(x, y) looks epsi-random to all statistical tests modeled by oracle circuits of unbounded complexity that can query q bits of x. We construct, for any delta < 1, a(k, epsi)-exposure resilient extractor with query resistance ndelta, k = n - nOmega(1), epsi = n-Omega(1), m = nOmega(1) and d = O(log n)
暴露弹性提取器是一种有效的程序，它从具有不完美最小熵的随机变量中产生通过所有统计测试的随机性，包括那些对随机变量有有限访问的随机性，并具有可依赖于被测试字符串的自适应查询。更准确地说,EXT: {0,1} n次{0,1}d rarr {0,1} m是一个(k, epsi)接触弹性器耐问查询,如果当x至少是k的最小熵和y是随机的,EXT (x, y)看起来epsi-random所有统计测试建模通过甲骨文无限复杂的电路,可以查询问x。我们构造,对于任何三角洲< 1,(k, epsi)接触弹性与查询器阻力ndelta, k = n - nOmega (1), epsi =ω(1),m = nOmega(1)和d = O (log n)
{"title":"Exposure-resilient extractors","authors":"Marius Zimand","doi":"10.1109/CCC.2006.19","DOIUrl":"https://doi.org/10.1109/CCC.2006.19","url":null,"abstract":"An exposure-resilient extractor is an efficient procedure that, from a random variable with imperfect min-entropy, produces randomness that passes all statistical tests including those that have bounded access to the random variable, with adaptive queries that can depend on the string being tested. More precisely, EXT : {0, 1}n times {0, 1}d rarr {0, 1}m is a (k, epsi)-exposure resilient extractor resistant to q queries if, when the min-entropy of x is at least k and y is random, EXT(x, y) looks epsi-random to all statistical tests modeled by oracle circuits of unbounded complexity that can query q bits of x. We construct, for any delta < 1, a(k, epsi)-exposure resilient extractor with query resistance ndelta, k = n - nOmega(1), epsi = n-Omega(1), m = nOmega(1) and d = O(log n)","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131685261","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}
E. Allender, L. Hellerstein, Paul McCabe, T. Pitassi, M. Saks
For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek (1979), which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (log N)Upsi, for some constant Upsi > 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for set cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(log N) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Omega(log N) larger than optimal. Finally, we extend known hardness results for Min-TC0 d to obtain new hardness results for Min-AC 0 d, under cryptographic assumptions
{"title":"Minimizing DNF formulas and AC/sup 0//sub d/ circuits given a truth table","authors":"E. Allender, L. Hellerstein, Paul McCabe, T. Pitassi, M. Saks","doi":"10.1109/CCC.2006.27","DOIUrl":"https://doi.org/10.1109/CCC.2006.27","url":null,"abstract":"For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek (1979), which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (log N)Upsi, for some constant Upsi > 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for set cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(log N) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Omega(log N) larger than optimal. Finally, we extend known hardness results for Min-TC0 d to obtain new hardness results for Min-AC 0 d, under cryptographic assumptions","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121141417","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}
In this paper we prove that the class of functions expressible by first order formulas with only two variables coincides with the class of functions computable by AC0 circuits with a linear number of gates. We then investigate the feasibility of using Ehrenfeucht-Fraisse games to prove lower bounds for that class of circuits, as well as for general AC0 circuits
{"title":"Circuit lower bounds via Ehrenfeucht-Fraisse games","authors":"M. Koucký, C. Lautemann, S. Poloczek, D. Thérien","doi":"10.1109/CCC.2006.12","DOIUrl":"https://doi.org/10.1109/CCC.2006.12","url":null,"abstract":"In this paper we prove that the class of functions expressible by first order formulas with only two variables coincides with the class of functions computable by AC0 circuits with a linear number of gates. We then investigate the feasibility of using Ehrenfeucht-Fraisse games to prove lower bounds for that class of circuits, as well as for general AC0 circuits","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130829488","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}
E. Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen
We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSLP; given a division-free straight-line program producing an integer N, decide whether N > 0. We show that PosSLP lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean traveling salesman problem lies in the counting hierarchy - the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE
{"title":"On the complexity of numerical analysis","authors":"E. Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen","doi":"10.1109/CCC.2006.30","DOIUrl":"https://doi.org/10.1109/CCC.2006.30","url":null,"abstract":"We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSLP; given a division-free straight-line program producing an integer N, decide whether N > 0. We show that PosSLP lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean traveling salesman problem lies in the counting hierarchy - the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127830529","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 show that for any reasonable semantic model of computation and for any positive integer a and rationals 1 < c < d, there exists a language computable in time nd with a bits of advice but not in time nc with a bits of advice. A semantic model is one for which there exists a computable enumeration that contains all machines in the model but may also contain others. We call such a model reasonable if it has an efficient universal machine that can be complemented within the model in exponential time and if it is efficiently closed under deterministic transducers. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one- or zero-sided error, unambiguous machines, symmetric alternation, Arthur-Merlin games of any signature, etc. Our argument yields considerably simpler proofs of known hierarchy theorems with one bit of advice for randomized and quantum machines with two-sided error. Our paradigm also allows us to derive stronger separation results in a unified way. For models that have an efficient universal machine that can be simulated deterministically in exponential time and that are efficiently closed under randomized reductions with two-sided error, we establish the following: For any constants a and c, there exists a language computable in polynomial time with one bit of advice but not in time nc with a log n bits of advice. The result applies to randomized and quantum machines with two-sided error. For randomized machines with one-sided error, our approach yields that for any constants a and c there exists a language computable in polynomial time with one bit of advice but not in time nc with a (log n)1c/ bits of advice
我们证明了对于任何合理的计算语义模型，以及对于任何正整数a和有理数1 < c < d，存在一种语言，它在时间nd上具有1位通知，但在时间nc上具有1位通知。语义模型是一种存在可计算枚举的模型，该枚举包含模型中的所有机器，但也可能包含其他机器。如果该模型具有一个能在指数时间内对模型进行补充的有效的通用机，并且在确定性换能器下有效地闭合，则该模型是合理的。我们的结果为零边误差随机机、单边或零边误差量子机、无二义机、对称交替、任意签名的Arthur-Merlin对策等提供了第一个层次定理。我们的论证为已知的层次定理提供了相当简单的证明，并为具有双面误差的随机机器和量子机器提供了一点建议。我们的范例还允许我们以统一的方式得出更强的分离结果。对于具有有效的通用机器的模型，它可以在指数时间内确定地模拟，并且在具有双边误差的随机约简下有效地关闭，我们建立了以下内容:对于任意常数a和c，存在一种可在多项式时间内计算的语言，具有1位建议，但不能在时间nc中计算，具有log n位建议。结果适用于随机和量子机器的双边误差。对于具有单侧误差的随机机器，我们的方法得出，对于任何常数a和c，存在一种可在多项式时间内计算的语言，具有1位建议，但不具有(log n)1c/ bits建议的时间nc
{"title":"A generic time hierarchy for semantic models with one bit of advice","authors":"D. Melkebeek, Konstantin Pervyshev","doi":"10.1109/CCC.2006.7","DOIUrl":"https://doi.org/10.1109/CCC.2006.7","url":null,"abstract":"We show that for any reasonable semantic model of computation and for any positive integer a and rationals 1 < c < d, there exists a language computable in time nd with a bits of advice but not in time nc with a bits of advice. A semantic model is one for which there exists a computable enumeration that contains all machines in the model but may also contain others. We call such a model reasonable if it has an efficient universal machine that can be complemented within the model in exponential time and if it is efficiently closed under deterministic transducers. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one- or zero-sided error, unambiguous machines, symmetric alternation, Arthur-Merlin games of any signature, etc. Our argument yields considerably simpler proofs of known hierarchy theorems with one bit of advice for randomized and quantum machines with two-sided error. Our paradigm also allows us to derive stronger separation results in a unified way. For models that have an efficient universal machine that can be simulated deterministically in exponential time and that are efficiently closed under randomized reductions with two-sided error, we establish the following: For any constants a and c, there exists a language computable in polynomial time with one bit of advice but not in time nc with a log n bits of advice. The result applies to randomized and quantum machines with two-sided error. For randomized machines with one-sided error, our approach yields that for any constants a and c there exists a language computable in polynomial time with one bit of advice but not in time nc with a (log n)1c/ bits of advice","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"397 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115214733","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}
Summary form only given. About two years ago Bourgain, Katz and Tao (2004) proved the following theorem, essentially stating that in every finite field, a set which does not grow much when we add all pairs of elements, and when we multiply all pairs of elements, must be very close to a subfield. Theorem 1: (Bourgain et al., 2004) For every epsi > 0 there exists a delta > 0 such that the following holds. Let F be any field with no subfield of size ges |F|epsi. For every set A sube F, with |F|epsi < |A| < |F|1 - epsi, either the sumset |A + A| > |A|1 + delta or the product set |A times A| > |A|1 + delta. This theorem revealed its fundamental nature quickly. Shortly afterwards it has found many diverse applications, including in number theory, group theory, combinatorial geometry, and the explicit construction of extractors and Ramsey graphs, mostly described in the references below. In my talk I plan to explain some of the applications, as well as to sketch the main ideas of the proof of the sum-product theorem
{"title":"Applications of the sum-product theorem in finite fields","authors":"A. Wigderson","doi":"10.1109/CCC.2006.9","DOIUrl":"https://doi.org/10.1109/CCC.2006.9","url":null,"abstract":"Summary form only given. About two years ago Bourgain, Katz and Tao (2004) proved the following theorem, essentially stating that in every finite field, a set which does not grow much when we add all pairs of elements, and when we multiply all pairs of elements, must be very close to a subfield. Theorem 1: (Bourgain et al., 2004) For every epsi > 0 there exists a delta > 0 such that the following holds. Let F be any field with no subfield of size ges |F|epsi. For every set A sube F, with |F|epsi < |A| < |F|1 - epsi, either the sumset |A + A| > |A|1 + delta or the product set |A times A| > |A|1 + delta. This theorem revealed its fundamental nature quickly. Shortly afterwards it has found many diverse applications, including in number theory, group theory, combinatorial geometry, and the explicit construction of extractors and Ramsey graphs, mostly described in the references below. In my talk I plan to explain some of the applications, as well as to sketch the main ideas of the proof of the sum-product theorem","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114684451","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 consider the problem of finding a monomial (or a term) that maximizes the agreement rate with a given set of examples over the Boolean hypercube. The problem is motivated by learning of monomials in the agnostic framework of Haussler (Hastad, 2001) and Kearns et al. (1994). Finding a monomial with the highest agreement rate was proved to be NP-hard by Kearns and Li (1993). Ben-David et al. gave the first inapproximability result for this problem, proving that the maximum agreement rate is NP-hard to approximate within 770/767 - epsi, for any constant epsi > 0 (Ben-David et al., 2003). The strongest known hardness of approximation result is due to Bshouty and Burroughs, who proved an inapproximability factor of 59/58 - epsi (2002). We show that the agreement rate NP-hard to approximate within 2 - epsi for any constant epsi > 0. This is optimal up to the second order terms and resolves an open question due to Blum (2002). We extend this result to epsi = 2-log1-lambda;n for any constant lambda > 0 under the assumption that NP nsube RTIME(npoly log(n)), thus also obtaining an inapproximability factor of 2log1-lambda n for the symmetric problem of minimizing disagreements. This improves on the log n hardness of approximation factor due to Kearns et al. (1994) and Hoffgen et al. (1995)
{"title":"Optimal hardness results for maximizing agreements with monomials","authors":"Vitaly Feldman","doi":"10.1109/CCC.2006.31","DOIUrl":"https://doi.org/10.1109/CCC.2006.31","url":null,"abstract":"We consider the problem of finding a monomial (or a term) that maximizes the agreement rate with a given set of examples over the Boolean hypercube. The problem is motivated by learning of monomials in the agnostic framework of Haussler (Hastad, 2001) and Kearns et al. (1994). Finding a monomial with the highest agreement rate was proved to be NP-hard by Kearns and Li (1993). Ben-David et al. gave the first inapproximability result for this problem, proving that the maximum agreement rate is NP-hard to approximate within 770/767 - epsi, for any constant epsi > 0 (Ben-David et al., 2003). The strongest known hardness of approximation result is due to Bshouty and Burroughs, who proved an inapproximability factor of 59/58 - epsi (2002). We show that the agreement rate NP-hard to approximate within 2 - epsi for any constant epsi > 0. This is optimal up to the second order terms and resolves an open question due to Blum (2002). We extend this result to epsi = 2-log1-lambda;n for any constant lambda > 0 under the assumption that NP nsube RTIME(npoly log(n)), thus also obtaining an inapproximability factor of 2log1-lambda n for the symmetric problem of minimizing disagreements. This improves on the log n hardness of approximation factor due to Kearns et al. (1994) and Hoffgen et al. (1995)","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124798524","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 may believe SAT does not have small Boolean circuits. But is it possible that some language with small circuits looks indistinguishable from SAT to every polynomial-time bounded adversary? We rule out this possibility. More precisely, assuming SAT does not have small circuits, we show that for every language A with small circuits, there exists a probabilistic polynomial-time algorithm that makes black-box queries to A, and produces, for a given input length, a Boolean formula on which A differs from SAT. A key step for obtaining this result is a new proof of the main result by Gutfreund, Shaltiel, and Ta-Shma reducing average-case hardness to worst-case hardness via uniform adversaries that know the algorithm they fool. The new adversary we construct has the feature of being black-box on the algorithm it fools, so it makes sense in the non-uniform setting as well. Our proof makes use of a refined analysis of the learning algorithm of Bshouty et al
{"title":"Distinguishing SAT from polynomial-size circuits, through black-box queries","authors":"Albert Atserias","doi":"10.1109/CCC.2006.17","DOIUrl":"https://doi.org/10.1109/CCC.2006.17","url":null,"abstract":"We may believe SAT does not have small Boolean circuits. But is it possible that some language with small circuits looks indistinguishable from SAT to every polynomial-time bounded adversary? We rule out this possibility. More precisely, assuming SAT does not have small circuits, we show that for every language A with small circuits, there exists a probabilistic polynomial-time algorithm that makes black-box queries to A, and produces, for a given input length, a Boolean formula on which A differs from SAT. A key step for obtaining this result is a new proof of the main result by Gutfreund, Shaltiel, and Ta-Shma reducing average-case hardness to worst-case hardness via uniform adversaries that know the algorithm they fool. The new adversary we construct has the feature of being black-box on the algorithm it fools, so it makes sense in the non-uniform setting as well. Our proof makes use of a refined analysis of the learning algorithm of Bshouty et al","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128059080","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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