Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)最新文献
Prior results show that most bounded query hierarchies cannot contain finite gaps. For example, it is known that P/sub (m+1)-tt//sup SAT/=P/sub m-tt//sup SAT//spl rArr/P/sub btt//sup SAT/=P/sub m-tt//sup SAT/ and for all sets A/spl middot/FP/sub (m=1)-tt//sup A/=FP/sub m-tt//sup A//spl rArr/FP/sub btt//sup A/=FP/sub m-tt//sup A//spl middot/P/sub (m+1)-T//sup A/=P/sub m-T//sup A/=P/sub bT//sup A//spl middot/FP/sub (m+1)-T//sup A/=FP/sub m-T//sup A//spl rArr/FP/sub bT//sup A/=FP/sub m-T//sup A/ where P/sub m-tt//sup A/ is the set of languages computable by polynomial-time Turing machines that make m nonadaptive queries to A; P/sub btt//sup A/=/spl cup//sub m/P/sub m-tt//sup A/, P/sub m-t//sup A/ and P/sub bT//sup A/ are the analogous adaptive queries classes; and FP/sub m-tt//sup A/, FP/sub btt//sup A/, FP/sub m-T//sup A/, and FP/sub bT//sup A/ in turn are the analogous function classes. It was widely expected that these general results would extend to the remaining case-languages computed with nonadaptive queries-yet results remained elusive. The best known was that P/sub 2m-tt//sup A/=P/sub m-tt//sup A//spl rArr/P/sub btt//sup A/=P/sub m-tt//sup A/. We disprove the conjecture, in fact, P/sub [4/3m]-tt//sup A/=P/sub m-tt//sup A/not/spl rArr/P/sub ([4/3m]+1)-tt/=P/sub [4/3m]-tt//sup A/. Thus there is a P/sub m-tt//sup A/ hierarchy that contains a finite gap. We also make progress on the 3-tt vs. 2-tt case: P/sub 3-tt//sup A/=P/sub 2-tt//sup A//spl rArr/P/sub btt//sup A//spl sube/P/sub 2-tt//sup A//poly.
{"title":"Gaps in bounded query hierarchies","authors":"R. Beigel","doi":"10.1109/CCC.1999.766271","DOIUrl":"https://doi.org/10.1109/CCC.1999.766271","url":null,"abstract":"Prior results show that most bounded query hierarchies cannot contain finite gaps. For example, it is known that P/sub (m+1)-tt//sup SAT/=P/sub m-tt//sup SAT//spl rArr/P/sub btt//sup SAT/=P/sub m-tt//sup SAT/ and for all sets A/spl middot/FP/sub (m=1)-tt//sup A/=FP/sub m-tt//sup A//spl rArr/FP/sub btt//sup A/=FP/sub m-tt//sup A//spl middot/P/sub (m+1)-T//sup A/=P/sub m-T//sup A/=P/sub bT//sup A//spl middot/FP/sub (m+1)-T//sup A/=FP/sub m-T//sup A//spl rArr/FP/sub bT//sup A/=FP/sub m-T//sup A/ where P/sub m-tt//sup A/ is the set of languages computable by polynomial-time Turing machines that make m nonadaptive queries to A; P/sub btt//sup A/=/spl cup//sub m/P/sub m-tt//sup A/, P/sub m-t//sup A/ and P/sub bT//sup A/ are the analogous adaptive queries classes; and FP/sub m-tt//sup A/, FP/sub btt//sup A/, FP/sub m-T//sup A/, and FP/sub bT//sup A/ in turn are the analogous function classes. It was widely expected that these general results would extend to the remaining case-languages computed with nonadaptive queries-yet results remained elusive. The best known was that P/sub 2m-tt//sup A/=P/sub m-tt//sup A//spl rArr/P/sub btt//sup A/=P/sub m-tt//sup A/. We disprove the conjecture, in fact, P/sub [4/3m]-tt//sup A/=P/sub m-tt//sup A/not/spl rArr/P/sub ([4/3m]+1)-tt/=P/sub [4/3m]-tt//sup A/. Thus there is a P/sub m-tt//sup A/ hierarchy that contains a finite gap. We also make progress on the 3-tt vs. 2-tt case: P/sub 3-tt//sup A/=P/sub 2-tt//sup A//spl rArr/P/sub btt//sup A//spl sube/P/sub 2-tt//sup A//poly.","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":"129518949","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, as follows. In the Ramsey theory of graphs F/spl rarr/(G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H as a subgraph. The problem ARROWING of deciding whether F/spl rarr/(G, H) lies in /spl Pi//sub 2//sup P/=coNP/sup NP/ and it was shown to be coNP-hard by S.A. Burr (1990). We prove that ARROWING is actually /spl Pi//sub 2//sup P/-complete, simultaneously settling a conjecture of Burr and providing a natural example of a problem complete for a higher level of the polynomial hierarchy. We also consider several specific variants of ARROWING, where G and H are restricted to particular families of graphs. We have a general completeness result for this case under the assumption that certain graphs are constructible in polynomial time. Furthermore we show that STRONG ARROWING, the version of ARROWING for induced subgraphs, is /spl Pi//sub 2//sup P/-complete.
{"title":"Graph Ramsey theory and the polynomial hierarchy","authors":"M. Schaefer","doi":"10.1145/301250.301411","DOIUrl":"https://doi.org/10.1145/301250.301411","url":null,"abstract":"Summary form only given, as follows. In the Ramsey theory of graphs F/spl rarr/(G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H as a subgraph. The problem ARROWING of deciding whether F/spl rarr/(G, H) lies in /spl Pi//sub 2//sup P/=coNP/sup NP/ and it was shown to be coNP-hard by S.A. Burr (1990). We prove that ARROWING is actually /spl Pi//sub 2//sup P/-complete, simultaneously settling a conjecture of Burr and providing a natural example of a problem complete for a higher level of the polynomial hierarchy. We also consider several specific variants of ARROWING, where G and H are restricted to particular families of graphs. We have a general completeness result for this case under the assumption that certain graphs are constructible in polynomial time. Furthermore we show that STRONG ARROWING, the version of ARROWING for induced subgraphs, is /spl Pi//sub 2//sup P/-complete.","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":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116747110","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}
Two important algebraic proof systems are the Nullstellensatz system and the polynomial calculus (also called the Grobner system). The Nullstellensatz system is a propositional proof system based on Hilbert's Nullstellensatz, and the polynomial calculus (PC) is a proof system which allows derivations of polynomials, over some field. The complexity of a proof in these systems is measured in terms of the degree of the polynomials used in the proof. The mod p counting principle can be formulated as a set MOD/sub p//sup n/ of constant-degree polynomials expressing the negation of the counting principle. The Tseitin mod p principles, TS/sub n/(p), are translations of the MOD/sub p//sup n/ into the Fourier basis. The present paper gives linear lower bounds on the degree of polynomial calculus refutations of MOD/sub p//sup n/ over p fields of characteristic q /spl ne/ p and over rings Z/sub q/ with q,p relatively prime. These are the first linear lower bounds for the polynomial calculus. As it is well-known to be easy to give constant degree polynomial calculus (and even Nullstellensatz) refutations of the MOD/sub p//sup n/ polynomials over F/sub p/, our results imply that the MOD/sub p//sup n/ polynomials have a linear gap between proof complexity for the polynomial calculus over F/sub p/ and over F/sub q/. We also obtain a linear gap for the polynomial calculus over rings Z/sub p/ and Z/sub q/ where p, q do not have identical prime factors.
{"title":"Linear gaps between degrees for the polynomial calculus modulo distinct primes","authors":"S. Buss, D. Grigoriev, R. Impagliazzo, T. Pitassi","doi":"10.1145/301250.301399","DOIUrl":"https://doi.org/10.1145/301250.301399","url":null,"abstract":"Two important algebraic proof systems are the Nullstellensatz system and the polynomial calculus (also called the Grobner system). The Nullstellensatz system is a propositional proof system based on Hilbert's Nullstellensatz, and the polynomial calculus (PC) is a proof system which allows derivations of polynomials, over some field. The complexity of a proof in these systems is measured in terms of the degree of the polynomials used in the proof. The mod p counting principle can be formulated as a set MOD/sub p//sup n/ of constant-degree polynomials expressing the negation of the counting principle. The Tseitin mod p principles, TS/sub n/(p), are translations of the MOD/sub p//sup n/ into the Fourier basis. The present paper gives linear lower bounds on the degree of polynomial calculus refutations of MOD/sub p//sup n/ over p fields of characteristic q /spl ne/ p and over rings Z/sub q/ with q,p relatively prime. These are the first linear lower bounds for the polynomial calculus. As it is well-known to be easy to give constant degree polynomial calculus (and even Nullstellensatz) refutations of the MOD/sub p//sup n/ polynomials over F/sub p/, our results imply that the MOD/sub p//sup n/ polynomials have a linear gap between proof complexity for the polynomial calculus over F/sub p/ and over F/sub q/. We also obtain a linear gap for the polynomial calculus over rings Z/sub p/ and Z/sub q/ where p, q do not have identical prime factors.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123081272","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 following pointer chasing problem plays a central role in the study of bounded round communication complexity. There are two players A and B. There are two sets of vertices V/sub A/ and V/sub B/ of size n each. Player A is given a function f/sub A/: VA/spl rarr/VB and player B is given a function f/sub B/: VB/spl rarr/VA. In the problem g/sub k/ the players have to determine the vertex reached by applying f/sub A/ and f/sub B/ alternately, k times starting with a fixed vertex v/sub 0//spl isin/V/sub A/. That is, in g/sub 1/, they must determine f/sub A/(v/sub 0/), in g/sub 2/ they must determine f/sub B/(f/sub A/(v/sub 0/)), in g/sub 3/ they must determine f/sub A/(f/sub B/(f/sub A/(v/sub 0/))), and so on.
{"title":"The communication complexity of pointer chasing. Applications of entropy and sampling","authors":"S. Ponzio, J. Radhakrishnan, Venkatesh Srinivasan","doi":"10.1145/301250.301413","DOIUrl":"https://doi.org/10.1145/301250.301413","url":null,"abstract":"The following pointer chasing problem plays a central role in the study of bounded round communication complexity. There are two players A and B. There are two sets of vertices V/sub A/ and V/sub B/ of size n each. Player A is given a function f/sub A/: VA/spl rarr/VB and player B is given a function f/sub B/: VB/spl rarr/VA. In the problem g/sub k/ the players have to determine the vertex reached by applying f/sub A/ and f/sub B/ alternately, k times starting with a fixed vertex v/sub 0//spl isin/V/sub A/. That is, in g/sub 1/, they must determine f/sub A/(v/sub 0/), in g/sub 2/ they must determine f/sub B/(f/sub A/(v/sub 0/)), in g/sub 3/ they must determine f/sub A/(f/sub B/(f/sub A/(v/sub 0/))), and so on.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122278094","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 combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query, complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: P/sub /spl par///sup NP[2k]//spl sube/EQP/sub /spl par///sup NP[k]/; P/sub /spl par///sup NP[2k+1-2]//spl sube/EQP/sup NP[k]/; FP/sub /spl par///sup NP[2k=1-2]//spl sube/FEQP/sup NP[2k]/; FP/sub /spl par///sup NP/spl sube/FEQP(NP[Olog n)]/. For sets A that are many-one complete for PSPACE or EXP we show that Fp/sup A//spl sube/FEQP/sup A[1]/. Sets A that are many-one complete for PP have the property that FP/sub /spl par///sup A//spl sube/FEQP/sup A[1]/. In general we prove that for any set A there is a set X such that FP/sup A//spl sube/FEQP/sup X[1]/, establishing that no set is superterse in the quantum setting.
我们将有界查询类的经典概念和技术与量子计算中开发的概念和技术相结合。我们给出了强有力的证据，证明对NP类的oracle的量子查询确实降低了决策问题的查询复杂性。在传统的复杂度假设下，我们得到了函数类的量子查询复杂度与经典查询复杂度之间的指数级加速。对于决策问题和函数类，我们得到了以下结果:P/sub /spl par///sup NP[2k]//spl sub /EQP/sub /spl par///sup NP[k]/;P/sub /spl par///sup NP[2k+1-2]//spl sub /EQP/sup NP[k]/;FP/sub /spl par///sup NP[2k=1-2]//spl sub /FEQP/sup NP[2k]/;FP/sub /spl par///sup NP/spl sub /FEQP(NP[Olog n)]/。对于PSPACE或EXP的多一完全集合A，我们证明了Fp/sup A//spl sub /FEQP/sup A[1]/。对于PP来说，多一完全集合A具有FP/sub /spl par///sup A//spl sub /FEQP/sup A[1]/的属性。一般地，我们证明了对于任意集合A，存在一个集合X使得FP/sup A//spl subbe /FEQP/sup X[1]/，建立了在量子集合中没有超简洁的集合。
{"title":"Quantum bounded query complexity","authors":"H. Buhrman, W. V. Dam","doi":"10.1109/CCC.1999.766273","DOIUrl":"https://doi.org/10.1109/CCC.1999.766273","url":null,"abstract":"We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query, complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: P/sub /spl par///sup NP[2k]//spl sube/EQP/sub /spl par///sup NP[k]/; P/sub /spl par///sup NP[2k+1-2]//spl sube/EQP/sup NP[k]/; FP/sub /spl par///sup NP[2k=1-2]//spl sube/FEQP/sup NP[2k]/; FP/sub /spl par///sup NP/spl sube/FEQP(NP[Olog n)]/. For sets A that are many-one complete for PSPACE or EXP we show that Fp/sup A//spl sube/FEQP/sup A[1]/. Sets A that are many-one complete for PP have the property that FP/sub /spl par///sup A//spl sube/FEQP/sup A[1]/. In general we prove that for any set A there is a set X such that FP/sup A//spl sube/FEQP/sup X[1]/, establishing that no set is superterse in the quantum setting.","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":"1999-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122548866","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}
Heilbronn's triangle problem asks for the least /spl Delta/ such that n points lying in the unit disc necessarily contain a triangle of area at most /spl Delta/. Heilbronn initially conjectured /spl Delta/=O(1/n/sup 2/). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n/sup 2//spl les//spl Delta//spl les/C/n/sup 8/7-/spl epsiv// for every constant /spl epsiv/>0. We resolve Heilbronn's problem in the expected case: If we uniformly at random put n points in the unit disc then (i) the area of the smallest triangle has expectation /spl Theta/(1/n/sup 3/); and (ii) the smallest triangle has area /spl Theta/(1/n/sup 3/) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.
{"title":"The expected size of Heilbronn's triangles","authors":"Tao Jiang, Ming Li, P. Vitányi","doi":"10.1109/CCC.1999.766269","DOIUrl":"https://doi.org/10.1109/CCC.1999.766269","url":null,"abstract":"Heilbronn's triangle problem asks for the least /spl Delta/ such that n points lying in the unit disc necessarily contain a triangle of area at most /spl Delta/. Heilbronn initially conjectured /spl Delta/=O(1/n/sup 2/). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n/sup 2//spl les//spl Delta//spl les/C/n/sup 8/7-/spl epsiv// for every constant /spl epsiv/>0. We resolve Heilbronn's problem in the expected case: If we uniformly at random put n points in the unit disc then (i) the area of the smallest triangle has expectation /spl Theta/(1/n/sup 3/); and (ii) the smallest triangle has area /spl Theta/(1/n/sup 3/) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"226 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120866776","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}
There are a number of questions in quantum complexity that have been resolved in the time-bounded setting, but remain open in the space-bounded setting. For example, it is not currently known if space-bounded probabilistic computations can be simulated by space-bounded quantum machines without allowing measurements during the computation, while it is known that an analogous statement holds in the time-bounded case. A more general question asks if measurements during a quantum computation can allow for more space-efficient solutions to certain problems. In this paper we show that space-bounded quantum Turing machines can efficiently simulate a limited class of random processes-random walks on undirected graphs-without relying on measurements during the computation. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in the quantum analogue of randomized logspace, i.e., SL/spl sube/QR/sub H/L.
{"title":"Quantum simulations of classical random walks and undirected graph connectivity","authors":"John Watrous","doi":"10.1109/CCC.1999.766275","DOIUrl":"https://doi.org/10.1109/CCC.1999.766275","url":null,"abstract":"There are a number of questions in quantum complexity that have been resolved in the time-bounded setting, but remain open in the space-bounded setting. For example, it is not currently known if space-bounded probabilistic computations can be simulated by space-bounded quantum machines without allowing measurements during the computation, while it is known that an analogous statement holds in the time-bounded case. A more general question asks if measurements during a quantum computation can allow for more space-efficient solutions to certain problems. In this paper we show that space-bounded quantum Turing machines can efficiently simulate a limited class of random processes-random walks on undirected graphs-without relying on measurements during the computation. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in the quantum analogue of randomized logspace, i.e., SL/spl sube/QR/sub H/L.","PeriodicalId":432015,"journal":{"name":"Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121768629","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. R. Impagliazzo and A. Wigderson (1997) have recently shown that if there exists a decision problem solvable in time 2/sup O(n)/ and having circuit complexity 2/sup /spl Omega/(n)/ (for all but finitely many n) then P=BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of "hardness amplification" (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with the Nisan-Wigderson (1994) generator. In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardness-randomness trade-offs.
{"title":"Pseudorandom generators without the XOR lemma","authors":"M. Sudan, L. Trevisan, S. Vadhan","doi":"10.1109/CCC.1999.766253","DOIUrl":"https://doi.org/10.1109/CCC.1999.766253","url":null,"abstract":"Summary form only given. R. Impagliazzo and A. Wigderson (1997) have recently shown that if there exists a decision problem solvable in time 2/sup O(n)/ and having circuit complexity 2/sup /spl Omega/(n)/ (for all but finitely many n) then P=BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of \"hardness amplification\" (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with the Nisan-Wigderson (1994) generator. In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardness-randomness trade-offs.","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":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122543795","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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