Devising an efficient deterministic -- or even a non-deterministic sub-exponential time -- algorithm for testing polynomial identities is a fundamental problem in algebraic complexity and complexity at large. Motivated by this problem, as well as by results from proof complexity, we investigate the complexity of _proving_ polynomial identities. To this end, we study a class of equational proof systems, of varying strength, operating with polynomial identities written as arithmetic formulas over a given ring. A proof in these systems establishes that two arithmetic formulas compute the same polynomial, and consists of a sequence of equations between polynomials, written as arithmetic formulas, where each equation in the sequence is derived from previous equations by means of the polynomial-ring axioms. We establish the first non-trivial upper and lower bounds on the size of equational proofs of polynomial identities, as follows: 1. Polynomial-size upper bounds on equational proofs of identities involving symmetric polynomials and interpolation-based identities. In particular, we show that basic properties of the elementary symmetric polynomials are efficiently provable already in equational proofs operating with depth-4 formulas, over infinite fields. This also yields polynomial-size depth-4 proofs of the Newton identities, providing a positive answer to a question posed by Grigoriev and Hirsch [GH03]. 2. Exponential-size lower bounds on (full, unrestricted) equational proofs of identities over certain specific rings. 3. Exponential-size lower bounds on analytic proofs operating with depth-3 formulas, under a certain regularity condition. The ``analytic'' requirement is, roughly, a condition that forbids introducing arbitrary formulas in a proof and the regularity condition is an additional structural restriction. 4. Exponential-size lower bounds on one-way proofs (of unrestricted depth) over infinite fields. Here, one-way proofs are analytic proofs, in which one is also not allowed to introduce arbitrary constants. Furthermore, we determine basic structural characterizations of equational proofs, and consider relations with polynomial identity testing procedures. Specifically, we show that equational proofs efficiently simulate the polynomial identity testing algorithm provided by Dvir and Shpilka [DS04].
{"title":"The Proof Complexity of Polynomial Identities","authors":"P. Hrubes, Iddo Tzameret","doi":"10.1109/CCC.2009.9","DOIUrl":"https://doi.org/10.1109/CCC.2009.9","url":null,"abstract":"Devising an efficient deterministic -- or even a non-deterministic sub-exponential time -- algorithm for testing polynomial identities is a fundamental problem in algebraic complexity and complexity at large. Motivated by this problem, as well as by results from proof complexity, we investigate the complexity of _proving_ polynomial identities. To this end, we study a class of equational proof systems, of varying strength, operating with polynomial identities written as arithmetic formulas over a given ring. A proof in these systems establishes that two arithmetic formulas compute the same polynomial, and consists of a sequence of equations between polynomials, written as arithmetic formulas, where each equation in the sequence is derived from previous equations by means of the polynomial-ring axioms. We establish the first non-trivial upper and lower bounds on the size of equational proofs of polynomial identities, as follows: 1. Polynomial-size upper bounds on equational proofs of identities involving symmetric polynomials and interpolation-based identities. In particular, we show that basic properties of the elementary symmetric polynomials are efficiently provable already in equational proofs operating with depth-4 formulas, over infinite fields. This also yields polynomial-size depth-4 proofs of the Newton identities, providing a positive answer to a question posed by Grigoriev and Hirsch [GH03]. 2. Exponential-size lower bounds on (full, unrestricted) equational proofs of identities over certain specific rings. 3. Exponential-size lower bounds on analytic proofs operating with depth-3 formulas, under a certain regularity condition. The ``analytic'' requirement is, roughly, a condition that forbids introducing arbitrary formulas in a proof and the regularity condition is an additional structural restriction. 4. Exponential-size lower bounds on one-way proofs (of unrestricted depth) over infinite fields. Here, one-way proofs are analytic proofs, in which one is also not allowed to introduce arbitrary constants. Furthermore, we determine basic structural characterizations of equational proofs, and consider relations with polynomial identity testing procedures. Specifically, we show that equational proofs efficiently simulate the polynomial identity testing algorithm provided by Dvir and Shpilka [DS04].","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127278922","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 1982, Kannan showed that $Sigma^P_2$ does not have $n^k$-sized circuits for any $k$. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan's result, we still cannot prove that $P^NP$ does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence -- the ``algebrization'' barrier -- that current techniques have inherent limitations in this respect. We explore questions about fixed-polynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including begin{itemize} item The following are equivalent: begin{itemize} item $NP$ is in $SIZE(n^k)$ (has $O(n^k)$-size circuit families) for some $k$ item For each $c$, $P^{NP[n^c]}$ is in $SIZE(n^k)$ for some $k$ item $ONP/1$ is in $SIZE(n^k)$ for some $k$, where $ONP$ is the class of languages accepted {it obliviously} by $NP$ machines, with witnesses for ``yes'' instances depending only on the input length. end{itemize} item For a large number of natural classes ${cal C}$ and all $k geq 1$, ${cal C}$ is in $SIZE(n^k)$ if and only if ${cal C}/1capP/poly$ is in $SIZE(n^k)$. item If there is a $d$ such that $MATIME(n) subseteq NTIME(n^d)$, then $P^{NP}$ does not have $O(n^k)$ size circuits for any $k ≫ 0$. item One cannot show $n^2$-size circuit lower bounds for $oplus P$ without new nonrelativizing techniques. In particular, the proof that $PP notsubseteq SIZE(n^k)$ for all $k$ relies on the (relativizing) result that $P^{PP} subseteq MA Longrightarrow PP notsubseteq SIZE(n^k)$, and we give an oracle relative to which $P^{oplus P} subseteq MA$ and $oplus P subseteq SIZE(n^2)$ both hold. end{itemize}
{"title":"Fixed-Polynomial Size Circuit Bounds","authors":"L. Fortnow, R. Santhanam","doi":"10.1109/CCC.2009.21","DOIUrl":"https://doi.org/10.1109/CCC.2009.21","url":null,"abstract":"In 1982, Kannan showed that $Sigma^P_2$ does not have $n^k$-sized circuits for any $k$. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan's result, we still cannot prove that $P^NP$ does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence -- the ``algebrization'' barrier -- that current techniques have inherent limitations in this respect. We explore questions about fixed-polynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including begin{itemize} item The following are equivalent: begin{itemize} item $NP$ is in $SIZE(n^k)$ (has $O(n^k)$-size circuit families) for some $k$ item For each $c$, $P^{NP[n^c]}$ is in $SIZE(n^k)$ for some $k$ item $ONP/1$ is in $SIZE(n^k)$ for some $k$, where $ONP$ is the class of languages accepted {it obliviously} by $NP$ machines, with witnesses for ``yes'' instances depending only on the input length. end{itemize} item For a large number of natural classes ${cal C}$ and all $k geq 1$, ${cal C}$ is in $SIZE(n^k)$ if and only if ${cal C}/1capP/poly$ is in $SIZE(n^k)$. item If there is a $d$ such that $MATIME(n) subseteq NTIME(n^d)$, then $P^{NP}$ does not have $O(n^k)$ size circuits for any $k ≫ 0$. item One cannot show $n^2$-size circuit lower bounds for $oplus P$ without new nonrelativizing techniques. In particular, the proof that $PP notsubseteq SIZE(n^k)$ for all $k$ relies on the (relativizing) result that $P^{PP} subseteq MA Longrightarrow PP notsubseteq SIZE(n^k)$, and we give an oracle relative to which $P^{oplus P} subseteq MA$ and $oplus P subseteq SIZE(n^2)$ both hold. end{itemize}","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"121 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123708451","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}
Samir Datta, N. Limaye, Prajakta Nimbhorkar, T. Thierauf, Fabian Wagner
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in log-space.
{"title":"Planar Graph Isomorphism is in Log-Space","authors":"Samir Datta, N. Limaye, Prajakta Nimbhorkar, T. Thierauf, Fabian Wagner","doi":"10.1145/3543686","DOIUrl":"https://doi.org/10.1145/3543686","url":null,"abstract":"Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in log-space.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114136528","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 study the one-way number-on-the-forhead (NOF) communication complexity of the $k$-layer pointer jumping problem. Strong lower bounds for this problem would have important implications in circuit complexity. All of our results apply to myopic protocols (where players see only one layer ahead, but can still see arbitrarily far behind them.) Furthermore, our results apply to the maximum communication complexity, where a protocol is charged for the maximum communication sent by a single player rather than the total communication sent by all players. Our main result is a lower bound of $n/2$ bits for deterministic protocols, independent of the number of players. We also provide a matching upper bound, as well as an $Omega(n/klog n)$ lower bound for randomized protocols, improving on the bounds of Chakrabarti. In the non-Boolean version of the problem, we give a lower bound of $n (log^{(k-1)} n)(1-o(1))$ bits, essentially matching the upper bound from Damm et al.
{"title":"The Maximum Communication Complexity of Multi-Party Pointer Jumping","authors":"Joshua Brody","doi":"10.1109/CCC.2009.30","DOIUrl":"https://doi.org/10.1109/CCC.2009.30","url":null,"abstract":"We study the one-way number-on-the-forhead (NOF) communication complexity of the $k$-layer pointer jumping problem. Strong lower bounds for this problem would have important implications in circuit complexity. All of our results apply to myopic protocols (where players see only one layer ahead, but can still see arbitrarily far behind them.) Furthermore, our results apply to the maximum communication complexity, where a protocol is charged for the maximum communication sent by a single player rather than the total communication sent by all players. Our main result is a lower bound of $n/2$ bits for deterministic protocols, independent of the number of players. We also provide a matching upper bound, as well as an $Omega(n/klog n)$ lower bound for randomized protocols, improving on the bounds of Chakrabarti. In the non-Boolean version of the problem, we give a lower bound of $n (log^{(k-1)} n)(1-o(1))$ bits, essentially matching the upper bound from Damm et al.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121952903","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}
Learning is a central task in computer science, and there are various formalisms for capturing the notion. One important model studied in computational learning theory is the PAC model of Valiant (CACM 1984). On the other hand, in cryptography the notion of "learning nothing'' is often modelled by the simulation paradigm: in an interactive protocol, a party learns nothing if it can produce a transcript of the protocol by itself that is indistinguishable from what it gets by interacting with other parties. The most famous example of this paradigm is zero knowledge proofs, introduced by Goldwasser, Micali, and Rackoff (SICOMP 1989). Applebaum et al. (FOCS 2008) observed that a theorem of Ostrovsky and Wigderson (ISTCS 1993) combined with the transformation of one-way functions to pseudo-random functions (Hastad et al. SICOMP 1999, Goldreich et al. J. ACM 1986) implies that if there exist non-trivial languages with zero-knowledge arguments, then no efficient algorithm can PAC learn polynomial-size circuits. They also prove a weak reverse implication, that if a certain non-standard learning task is hard, then zero knowledge is non-trivial. This motivates the question we explore here: can one prove that hardness of PAC learning is equivalent to non-triviality of zero-knowledge? We show that this statement cannot be proven via the following techniques: 1. Relativizing techniques: there exists an oracle relative to which learning polynomial-size circuits is hard and yet the class of languages with zero knowledge arguments is trivial. 2. Semi-black-box techniques: if there is a black-box construction of a zero-knowledge argument for an NP-complete language (possibly with a non-black-box security reduction) based on hardness of PAC learning, then NP has statistical zero knowledge proofs, namely NP is contained in SZK. Under the standard conjecture that NP is not contained in SZK, our results imply that most standard techniques do not suffice to prove the equivalence between the non-triviality of zero knowledge and the hardness of PAC learning. Our results hold even when considering non-uniform hardness of PAC learning with membership queries. In addition, our technique relies on a new kind of separating oracle that may be of independent interest.
学习是计算机科学的核心任务，有各种各样的形式来表达这个概念。计算学习理论中研究的一个重要模型是Valiant (ccm, 1984)的PAC模型。另一方面，在密码学中，“什么也学不到”的概念通常是通过模拟范式来建模的:在交互式协议中，如果一方能够自己生成协议的副本，并且该副本与通过与其他各方交互获得的副本无法区分，那么它就什么也学不到。这种范式最著名的例子是零知识证明，由Goldwasser、Micali和Rackoff (SICOMP 1989)提出。Applebaum et al. (fos 2008)观察到Ostrovsky和Wigderson (ISTCS 1993)的一个定理与单向函数到伪随机函数的变换(Hastad et al.)相结合。SICOMP 1999, Goldreich等。J. ACM 1986)表明，如果存在具有零知识参数的非平凡语言，则没有有效的算法可以PAC学习多项式大小的电路。他们还证明了一个微弱的反向暗示，即如果某个非标准的学习任务很难，那么零知识是非平凡的。这激发了我们在这里探讨的问题:能否证明PAC学习的硬度等同于零知识的非平凡性?我们证明这个说法不能通过以下技术来证明:1。相对化技术:存在一种相对于其学习多项式大小的电路是困难的oracle，而具有零知识参数的语言类是微不足道的。2. 半黑箱技术:如果存在基于PAC学习硬度的NP完备语言(可能具有非黑箱安全性约简)的零知识论证的黑箱构造，则NP具有统计零知识证明，即NP包含在SZK中。在SZK中不包含NP的标准猜想下，我们的结果表明大多数标准技术不足以证明零知识的非平凡性与PAC学习的硬度之间的等价性。我们的结果甚至在考虑带有成员查询的PAC学习的非均匀硬度时也成立。此外，我们的技术依赖于一种新的分离oracle，它可能具有独立的兴趣。
{"title":"On Basing ZK ≠ BPP on the Hardness of PAC Learning","authors":"David Xiao","doi":"10.1109/CCC.2009.11","DOIUrl":"https://doi.org/10.1109/CCC.2009.11","url":null,"abstract":"Learning is a central task in computer science, and there are various formalisms for capturing the notion. One important model studied in computational learning theory is the PAC model of Valiant (CACM 1984). On the other hand, in cryptography the notion of \"learning nothing'' is often modelled by the simulation paradigm: in an interactive protocol, a party learns nothing if it can produce a transcript of the protocol by itself that is indistinguishable from what it gets by interacting with other parties. The most famous example of this paradigm is zero knowledge proofs, introduced by Goldwasser, Micali, and Rackoff (SICOMP 1989). Applebaum et al. (FOCS 2008) observed that a theorem of Ostrovsky and Wigderson (ISTCS 1993) combined with the transformation of one-way functions to pseudo-random functions (Hastad et al. SICOMP 1999, Goldreich et al. J. ACM 1986) implies that if there exist non-trivial languages with zero-knowledge arguments, then no efficient algorithm can PAC learn polynomial-size circuits. They also prove a weak reverse implication, that if a certain non-standard learning task is hard, then zero knowledge is non-trivial. This motivates the question we explore here: can one prove that hardness of PAC learning is equivalent to non-triviality of zero-knowledge? We show that this statement cannot be proven via the following techniques: 1. Relativizing techniques: there exists an oracle relative to which learning polynomial-size circuits is hard and yet the class of languages with zero knowledge arguments is trivial. 2. Semi-black-box techniques: if there is a black-box construction of a zero-knowledge argument for an NP-complete language (possibly with a non-black-box security reduction) based on hardness of PAC learning, then NP has statistical zero knowledge proofs, namely NP is contained in SZK. Under the standard conjecture that NP is not contained in SZK, our results imply that most standard techniques do not suffice to prove the equivalence between the non-triviality of zero knowledge and the hardness of PAC learning. Our results hold even when considering non-uniform hardness of PAC learning with membership queries. In addition, our technique relies on a new kind of separating oracle that may be of independent interest.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"140 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127543359","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 Berman-Hartmanis conjecture states that all NP-complete sets are P-isomorphic each other. On this conjecture, we first improve the previous result of Agrawal and show that all NP-complete sets are P/poly-time computable 1,li-reducible to each other based on the assumption that there exist regular one-way functions that cannot be inverted by randomized polynomial-time algorithms. Secondly, we show that, besides the above assumption, if all one-way functions have some easy part to invert, then all NP-complete sets are P/poly-isomorphic to each other.
{"title":"One-Way Functions and the Berman-Hartmanis Conjecture","authors":"Manindra Agrawal, O. Watanabe","doi":"10.1109/CCC.2009.17","DOIUrl":"https://doi.org/10.1109/CCC.2009.17","url":null,"abstract":"The Berman-Hartmanis conjecture states that all NP-complete sets are P-isomorphic each other. On this conjecture, we first improve the previous result of Agrawal and show that all NP-complete sets are P/poly-time computable 1,li-reducible to each other based on the assumption that there exist regular one-way functions that cannot be inverted by randomized polynomial-time algorithms. Secondly, we show that, besides the above assumption, if all one-way functions have some easy part to invert, then all NP-complete sets are P/poly-isomorphic to each other.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123685330","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 every bounded function g: {0,1}^n -≫ [0,1] admits an efficiently computable "simulator" function h: {0,1}^n-≫[0,1] such that every fixed polynomial size circuit has approximately the same correlation with g as with h. If g describes (up to scaling) a high min-entropy distribution D, then h can be used to efficiently sample a distribution D' of the same min-entropy that is indistinguishable from D by circuits of fixed polynomial size. We state and prove our result in a more abstract setting, in which we allow arbitrary finite domains instead of {0,1}^n, and arbitrary families of distinguishers, instead of fixed polynomial size circuits. Our result implies (a) the Weak Szemeredi Regularity Lemma of Frieze and Kannan (b) a constructive version of the Dense Model Theorem of Green, Tao and Ziegler with better quantitative parameters (polynomial rather than exponential in the distinguishing probability), and (c) the Impagliazzo Hardcore Set Lemma. It appears to be the general result underlying the known connections between "regularity" results in graph theory, "decomposition" results in additive combinatorics, and the Hardcore Lemma in complexity theory. We present two proofs of our result, one in the spirit of Nisan's proof of the Hardcore Lemma via duality of linear programming, and one similar to Impagliazzo's "boosting" proof. A third proof by iterative partitioning, which gives the complexity of the sampler to be exponential in the distinguishing probability, is also implicit in the Green-Tao-Ziegler proofs of the Dense Model Theorem.
{"title":"Regularity, Boosting, and Efficiently Simulating Every High-Entropy Distribution","authors":"L. Trevisan, Madhur Tulsiani, S. Vadhan","doi":"10.1109/CCC.2009.41","DOIUrl":"https://doi.org/10.1109/CCC.2009.41","url":null,"abstract":"We show that every bounded function g: {0,1}^n -≫ [0,1] admits an efficiently computable \"simulator\" function h: {0,1}^n-≫[0,1] such that every fixed polynomial size circuit has approximately the same correlation with g as with h. If g describes (up to scaling) a high min-entropy distribution D, then h can be used to efficiently sample a distribution D' of the same min-entropy that is indistinguishable from D by circuits of fixed polynomial size. We state and prove our result in a more abstract setting, in which we allow arbitrary finite domains instead of {0,1}^n, and arbitrary families of distinguishers, instead of fixed polynomial size circuits. Our result implies (a) the Weak Szemeredi Regularity Lemma of Frieze and Kannan (b) a constructive version of the Dense Model Theorem of Green, Tao and Ziegler with better quantitative parameters (polynomial rather than exponential in the distinguishing probability), and (c) the Impagliazzo Hardcore Set Lemma. It appears to be the general result underlying the known connections between \"regularity\" results in graph theory, \"decomposition\" results in additive combinatorics, and the Hardcore Lemma in complexity theory. We present two proofs of our result, one in the spirit of Nisan's proof of the Hardcore Lemma via duality of linear programming, and one similar to Impagliazzo's \"boosting\" proof. A third proof by iterative partitioning, which gives the complexity of the sampler to be exponential in the distinguishing probability, is also implicit in the Green-Tao-Ziegler proofs of the Dense Model Theorem.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128557646","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 prove that poly-sized AC^0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log^2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has later given a much simpler proof for Bazzi’s theorem.
{"title":"Poly-logarithmic Independence Fools AC^0 Circuits","authors":"M. Braverman","doi":"10.1145/1754399.1754401","DOIUrl":"https://doi.org/10.1145/1754399.1754401","url":null,"abstract":"We prove that poly-sized AC^0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log^2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has later given a much simpler proof for Bazzi’s theorem.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128647996","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 coding theorem is a fundamental result of algorithmic information theory. A well known theorem of Gács shows that the analog of the coding theorem fails for continuous sample spaces. This means that descriptional monotonic complexity does not coincide within an additive constant with the negative logarithm of algorithmic probability. Gács's proof provided a lower bound on the difference between these values. He showed that for infinitely many finite binary strings, this difference was greater than a version of the inverse Ackermann function applied to string length. This paper establishes that this lower bound can be substantially improved. The inverse Ackermann function can be replaced with a function O(log(log(x))). This shows that in continuous sample spaces, descriptional monotonic complexity and algorithmic probability are very different. While this proof builds on the original work by Gács, it does have a number of new features, in particular, the algorithm at the heart of the proof works on sets of strings as opposed to individual strings.
{"title":"Increasing the Gap between Descriptional Complexity and Algorithmic Probability","authors":"A. Day","doi":"10.1109/CCC.2009.13","DOIUrl":"https://doi.org/10.1109/CCC.2009.13","url":null,"abstract":"The coding theorem is a fundamental result of algorithmic information theory. A well known theorem of Gács shows that the analog of the coding theorem fails for continuous sample spaces. This means that descriptional monotonic complexity does not coincide within an additive constant with the negative logarithm of algorithmic probability. Gács's proof provided a lower bound on the difference between these values. He showed that for infinitely many finite binary strings, this difference was greater than a version of the inverse Ackermann function applied to string length. This paper establishes that this lower bound can be substantially improved. The inverse Ackermann function can be replaced with a function O(log(log(x))). This shows that in continuous sample spaces, descriptional monotonic complexity and algorithmic probability are very different. While this proof builds on the original work by Gács, it does have a number of new features, in particular, the algorithm at the heart of the proof works on sets of strings as opposed to individual strings.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116673596","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 major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2^k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma_2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma_2 and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma_2 norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma_2 is proved by a multi-dimensional version of Grothendieck's inequality.
{"title":"Lower Bounds on Quantum Multiparty Communication Complexity","authors":"Troy Lee, G. Schechtman, A. Shraibman","doi":"10.1109/CCC.2009.24","DOIUrl":"https://doi.org/10.1109/CCC.2009.24","url":null,"abstract":"A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2^k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma_2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma_2 and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma_2 norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma_2 is proved by a multi-dimensional version of Grothendieck's inequality.","PeriodicalId":158572,"journal":{"name":"2009 24th Annual IEEE Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121567883","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：