Pub Date : 2022-11-30DOI: 10.48550/arXiv.2211.17211
P. Beame, Sajin Koroth
Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. Several important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, from its current near-linear size down to polylogarithmic in the number of inputs $N$ of the original function or, ideally, constant. The near-linear size bound was shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with the Index function of near-linear size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; 1) The conjecture of Lovett et al. is false when the size of the Index gadget is $log N-omega(1)$. 2) Also, the Inner-Product function, which satisfies the disperser property at size $O(log N)$, does not have this property when its size is $log N-omega(1)$. 3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting theorem for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. 4) Using the ideas from this lifting theorem, we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like $Res(oplus)$ refutation size, which yields many new exponential lower bounds on such proofs.
{"title":"On Disperser/Lifting Properties of the Index and Inner-Product Functions","authors":"P. Beame, Sajin Koroth","doi":"10.48550/arXiv.2211.17211","DOIUrl":"https://doi.org/10.48550/arXiv.2211.17211","url":null,"abstract":"Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. Several important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, from its current near-linear size down to polylogarithmic in the number of inputs $N$ of the original function or, ideally, constant. The near-linear size bound was shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with the Index function of near-linear size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; 1) The conjecture of Lovett et al. is false when the size of the Index gadget is $log N-omega(1)$. 2) Also, the Inner-Product function, which satisfies the disperser property at size $O(log N)$, does not have this property when its size is $log N-omega(1)$. 3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting theorem for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. 4) Using the ideas from this lifting theorem, we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like $Res(oplus)$ refutation size, which yields many new exponential lower bounds on such proofs.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72773772","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}
Pub Date : 2022-11-30DOI: 10.48550/arXiv.2211.17214
A. Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, Suhail Sherif
We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([G"{o}"{o}s, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of $f$, which could be exponentially smaller than its deterministic counterpart when either $f$ is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to $f$. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., $textit{Res}$($oplus$), of the unsatisfiability of closely related constant-width CNF formulas.
{"title":"Lifting to Parity Decision Trees Via Stifling","authors":"A. Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, Suhail Sherif","doi":"10.48550/arXiv.2211.17214","DOIUrl":"https://doi.org/10.48550/arXiv.2211.17214","url":null,"abstract":"We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([G\"{o}\"{o}s, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of $f$, which could be exponentially smaller than its deterministic counterpart when either $f$ is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to $f$. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., $textit{Res}$($oplus$), of the unsatisfiability of closely related constant-width CNF formulas.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81641947","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}
Pub Date : 2022-11-26DOI: 10.48550/arXiv.2211.14640
Samuel Epstein
We provide another proof to the EL Theorem. We show the tradeoff between compressibility of codebooks and their communication capacity. A resource bounded version of the EL Theorem is proven. This is used to prove three instances of resource bounded derandomization. This paper is in support of the general claim that if the existence of an object can be proven with the probabilistic method, then bounds on its Kolmogorov complexity can be proven as well.
{"title":"Derandomization under Different Resource Constraints","authors":"Samuel Epstein","doi":"10.48550/arXiv.2211.14640","DOIUrl":"https://doi.org/10.48550/arXiv.2211.14640","url":null,"abstract":"We provide another proof to the EL Theorem. We show the tradeoff between compressibility of codebooks and their communication capacity. A resource bounded version of the EL Theorem is proven. This is used to prove three instances of resource bounded derandomization. This paper is in support of the general claim that if the existence of an object can be proven with the probabilistic method, then bounds on its Kolmogorov complexity can be proven as well.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"508 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76709022","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}
Pub Date : 2022-11-24DOI: 10.48550/arXiv.2211.13741
M. Braverman, Subhash Khot, Dor Minzer
We show that the value of the $n$-fold repeated GHZ game is at most $2^{-Omega(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.
{"title":"Parallel Repetition for the GHZ Game: Exponential Decay","authors":"M. Braverman, Subhash Khot, Dor Minzer","doi":"10.48550/arXiv.2211.13741","DOIUrl":"https://doi.org/10.48550/arXiv.2211.13741","url":null,"abstract":"We show that the value of the $n$-fold repeated GHZ game is at most $2^{-Omega(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"471 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78200062","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}
Pub Date : 2022-11-22DOI: 10.48550/arXiv.2211.12441
Ivan Hu, D. Melkebeek, Andrew Morgan
We consider the following computational problem: Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical psychology, with the evaluation of the Mann--Whitney statistic for detecting differences between distributions as a special case. We study the complexity of the problem in the comparison-query model, used for problems like sorting and selection. For many types of trees with $n$ leaves, we establish lower bounds close to the strongest known in the model, namely the lower bound of $log_2(n!)$ for sorting $n$ items. We show: (a) $log_2((alpha(1-alpha)n)!) - O(log n)$ queries are needed whenever the tree has a subtree that contains a fraction $alpha$ of the leaves. This implies a lower bound of $log_2((frac{k}{(k+1)^2}n)!) - O(log n)$ for trees of degree $k$. (b) $log_2(n!) - O(log n)$ queries are needed in case the tree is binary. (c) $log_2(n!) - O(k log k)$ queries are needed for certain classes of trees of degree $k$, including perfect trees with even $k$. The lower bounds are obtained by developing two novel techniques for a generic problem $Pi$ in the comparison-query model and applying them to inversion minimization on trees. Both techniques can be described in terms of the Cayley graph of the symmetric group with adjacent-rank transpositions as the generating set. Consider the subgraph consisting of the edges between vertices with the same value under $Pi$. We show that the size of any decision tree for $Pi$ must be at least: (i) the number of connected components of this subgraph, and (ii) the factorial of the average degree of the complementary subgraph, divided by $n$. Lower bounds on query complexity then follow by taking the base-2 logarithm.
{"title":"Query Complexity of Inversion Minimization on Trees","authors":"Ivan Hu, D. Melkebeek, Andrew Morgan","doi":"10.48550/arXiv.2211.12441","DOIUrl":"https://doi.org/10.48550/arXiv.2211.12441","url":null,"abstract":"We consider the following computational problem: Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical psychology, with the evaluation of the Mann--Whitney statistic for detecting differences between distributions as a special case. We study the complexity of the problem in the comparison-query model, used for problems like sorting and selection. For many types of trees with $n$ leaves, we establish lower bounds close to the strongest known in the model, namely the lower bound of $log_2(n!)$ for sorting $n$ items. We show: (a) $log_2((alpha(1-alpha)n)!) - O(log n)$ queries are needed whenever the tree has a subtree that contains a fraction $alpha$ of the leaves. This implies a lower bound of $log_2((frac{k}{(k+1)^2}n)!) - O(log n)$ for trees of degree $k$. (b) $log_2(n!) - O(log n)$ queries are needed in case the tree is binary. (c) $log_2(n!) - O(k log k)$ queries are needed for certain classes of trees of degree $k$, including perfect trees with even $k$. The lower bounds are obtained by developing two novel techniques for a generic problem $Pi$ in the comparison-query model and applying them to inversion minimization on trees. Both techniques can be described in terms of the Cayley graph of the symmetric group with adjacent-rank transpositions as the generating set. Consider the subgraph consisting of the edges between vertices with the same value under $Pi$. We show that the size of any decision tree for $Pi$ must be at least: (i) the number of connected components of this subgraph, and (ii) the factorial of the average degree of the complementary subgraph, divided by $n$. Lower bounds on query complexity then follow by taking the base-2 logarithm.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"23 1","pages":"2836-2866"},"PeriodicalIF":0.0,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78976391","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}
Pub Date : 2022-11-14DOI: 10.48550/arXiv.2211.07691
Prashanth Amireddy, A. Garg, N. Kayal, Chandan Saha, Bhargav Thankey
We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restriction. We are able to upper bound the measures for homogeneous formulas directly, without going via set-multilinearity. Our lower bounds hold for the iterated matrix multiplication as well as the Nisan-Wigderson design polynomials. We also define a subclass of homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This generalizes the superpolynomial lower bounds for regular formulas in [Kayal-Saha-Saptharishi, STOC 2014], [Fournier-Limaye-Malod-Srinivasan, STOC 2014].
{"title":"Low-depth arithmetic circuit lower bounds via shifted partials","authors":"Prashanth Amireddy, A. Garg, N. Kayal, Chandan Saha, Bhargav Thankey","doi":"10.48550/arXiv.2211.07691","DOIUrl":"https://doi.org/10.48550/arXiv.2211.07691","url":null,"abstract":"We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restriction. We are able to upper bound the measures for homogeneous formulas directly, without going via set-multilinearity. Our lower bounds hold for the iterated matrix multiplication as well as the Nisan-Wigderson design polynomials. We also define a subclass of homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This generalizes the superpolynomial lower bounds for regular formulas in [Kayal-Saha-Saptharishi, STOC 2014], [Fournier-Limaye-Malod-Srinivasan, STOC 2014].","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"147 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89229873","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}
Pub Date : 2022-11-04DOI: 10.48550/arXiv.2211.02694
Jason Vander Woude, P. Dixon, A. Pavan, Jamie Radcliffe, N. V. Vinodchandran
Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {em secluded hypercube partition problem}: Given $kin mathbb{N}$ (ideally small) and $epsilon>0$ (ideally large), is there a partition of $mathbb{R}^d$ with unit hypercubes such that for every point $p in mathbb{R}^d$, its closed $epsilon$-neighborhood (in the $ell_{infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $din mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $mathbb{R}^d$ with $k = d+1$ and $epsilon = frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $epsilon$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $epsilon$ and prove that any partition in this broad class that has $k=d+1$, must have $epsilonleqfrac{1}{2sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space.
四舍五入已被证明是理论计算机科学的一个基本工具。通过观察$mathbb{R}^d$的舍入和划分是等价的，我们引入了以下自然划分问题，我们称之为{em隐蔽超立方体划分问题}:给定$kin mathbb{N}$(理想的小)和$epsilon>0$(理想的大)，是否存在一个具有单位超立方体的$mathbb{R}^d$分区，使得对于每个点$p in mathbb{R}^d$，其封闭的$epsilon$邻域(在$ell_{infty}$范数中)最多与$k$超立方体相交?我们对这个分区问题进行了全面的研究。我们证明了对于每一个$din mathbb{N}$，存在一个显式的(且有效可计算的)包含$k = d+1$和$epsilon = frac{1}{2d}$的$mathbb{R}^d$超立方分区。我们通过证明对于包括超立方体分区在内的广泛的“合理”分区，$k=d+1$的值是可能的最佳值(对于任何$epsilon$)来补充这种构造。我们还研究了参数$epsilon$的最优性，并证明了在这个广义类中任何分区都有$k=d+1$，必须有$epsilonleqfrac{1}{2sqrt{d}}$。这些边界暗示了文献中存在的某些确定性舍入格式的局限性。此外，这个一般边界是基于目前已知的立方体解剖数的下界，对这个边界的改进将产生对我们边界的改进。虽然我们的工作是出于理解舍入算法的愿望，但我们的主要概念贡献之一是引入了{em隐蔽的超立方体划分问题}，这与数学家对欧几里得空间的各种超立方体划分/平铺的长期研究历史非常吻合。
{"title":"Geometry of Rounding","authors":"Jason Vander Woude, P. Dixon, A. Pavan, Jamie Radcliffe, N. V. Vinodchandran","doi":"10.48550/arXiv.2211.02694","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02694","url":null,"abstract":"Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {em secluded hypercube partition problem}: Given $kin mathbb{N}$ (ideally small) and $epsilon>0$ (ideally large), is there a partition of $mathbb{R}^d$ with unit hypercubes such that for every point $p in mathbb{R}^d$, its closed $epsilon$-neighborhood (in the $ell_{infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $din mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $mathbb{R}^d$ with $k = d+1$ and $epsilon = frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $epsilon$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $epsilon$ and prove that any partition in this broad class that has $k=d+1$, must have $epsilonleqfrac{1}{2sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86094891","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}
Pub Date : 2022-10-31DOI: 10.48550/arXiv.2210.17429
Yaroslav Alekseev, E. Hirsch
The (extended) Binary Value Principle (eBVP: $sum_{i=1}^n x_i2^{i-1} = -k$ for $k>0$ and $x^2_i=x_i$) has received a lot of attention recently, several lower bounds have been proved for it (Alekseev et al 2020, Alekseev 2021, Part and Tzameret 2021). Also it has been shown (Alekseev et al 2020) that the probabilistically verifiable Ideal Proof System (IPS) (Grochow and Pitassi 2018) together with eBVP polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with the algebraic version of Tseitin's extension rule (Ext-PC). Contrary to IPS, this is a Cook--Reckhow proof system. We show that in this context eBVP still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule (Grigoriev and Hirsch 2003), which is in sharp contrast with the result of (Alekseev 2021) that shows an exponential lower bound on the size of Ext-PC derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that eBVP probably does not help in proving exponential lower bounds for Boolean formulas: we show that an Ext-PC (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from eBVP must be of exponential size.
(扩展的)二进制值原理(eBVP: $sum_{i=1}^n x_i2^{i-1} = -k$ for $k>0$和$x^2_i=x_i$)最近受到了很多关注，已经证明了它的几个下界(Alekseev et al 2020, Alekseev 2021, Part and Tzameret 2021)。此外，已经证明(Alekseev等人2020)，概率可验证的理想证明系统(IPS) (Grochow和Pitassi 2018)与eBVP一起多项式地模拟了类似的半代数证明系统。本文考虑了多项式微积分的代数版本的tseittin的扩展规则(Ext-PC)。与IPS不同的是，这是库克-雷克豪的证明系统。我们表明，在这种情况下，eBVP仍然允许模拟类似的半代数系统。我们还证明了它允许模拟平方根规则(Grigoriev和Hirsch 2003)，这与(Alekseev 2021)的结果形成鲜明对比，后者显示了二进制值原理的Ext-PC衍生的大小的指数下界。另一方面，我们证明了eBVP可能对证明布尔公式的指数下界没有帮助:我们证明了从eBVP推导出CNF中任何不满足的布尔公式的Ext-PC(即使使用平方根规则)必须具有指数大小。
{"title":"The power of the Binary Value Principle","authors":"Yaroslav Alekseev, E. Hirsch","doi":"10.48550/arXiv.2210.17429","DOIUrl":"https://doi.org/10.48550/arXiv.2210.17429","url":null,"abstract":"The (extended) Binary Value Principle (eBVP: $sum_{i=1}^n x_i2^{i-1} = -k$ for $k>0$ and $x^2_i=x_i$) has received a lot of attention recently, several lower bounds have been proved for it (Alekseev et al 2020, Alekseev 2021, Part and Tzameret 2021). Also it has been shown (Alekseev et al 2020) that the probabilistically verifiable Ideal Proof System (IPS) (Grochow and Pitassi 2018) together with eBVP polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with the algebraic version of Tseitin's extension rule (Ext-PC). Contrary to IPS, this is a Cook--Reckhow proof system. We show that in this context eBVP still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule (Grigoriev and Hirsch 2003), which is in sharp contrast with the result of (Alekseev 2021) that shows an exponential lower bound on the size of Ext-PC derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that eBVP probably does not help in proving exponential lower bounds for Boolean formulas: we show that an Ext-PC (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from eBVP must be of exponential size.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75657885","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}
Pub Date : 2022-09-21DOI: 10.48550/arXiv.2209.10509
P. Harsha, Daniel Mitropolsky, Alon Rosen
A problem is emph{downward self-reducible} if it can be solved efficiently given an oracle that returns solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is well studied and it is known that all downward self-reducible problems are in textsc{PSPACE}. In this paper, we initiate the study of downward self-reducible search problems which are guaranteed to have a solution -- that is, the downward self-reducible problems in textsc{TFNP}. We show that most natural $PLS$-complete problems are downward self-reducible and any downward self-reducible problem in textsc{TFNP} is contained in textsc{PLS}. Furthermore, if the downward self-reducible problem is in textsc{TFUP} (i.e. it has a unique solution), then it is actually contained in textsc{UEOPL}, a subclass of textsc{CLS}. This implies that if integer factoring is emph{downward self-reducible} then it is in fact in textsc{UEOPL}, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.
{"title":"Downward Self-Reducibility in TFNP","authors":"P. Harsha, Daniel Mitropolsky, Alon Rosen","doi":"10.48550/arXiv.2209.10509","DOIUrl":"https://doi.org/10.48550/arXiv.2209.10509","url":null,"abstract":"A problem is emph{downward self-reducible} if it can be solved efficiently given an oracle that returns solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is well studied and it is known that all downward self-reducible problems are in textsc{PSPACE}. In this paper, we initiate the study of downward self-reducible search problems which are guaranteed to have a solution -- that is, the downward self-reducible problems in textsc{TFNP}. We show that most natural $PLS$-complete problems are downward self-reducible and any downward self-reducible problem in textsc{TFNP} is contained in textsc{PLS}. Furthermore, if the downward self-reducible problem is in textsc{TFUP} (i.e. it has a unique solution), then it is actually contained in textsc{UEOPL}, a subclass of textsc{CLS}. This implies that if integer factoring is emph{downward self-reducible} then it is in fact in textsc{UEOPL}, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"29 1","pages":"67:1-67:17"},"PeriodicalIF":0.0,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72826673","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}
Pub Date : 2022-09-16DOI: 10.48550/arXiv.2209.08042
Rahul Chugh, Supartha Podder, Swagato Sanyal
Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, both of the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0^n to 1^n has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree. Finally, we show using a lifting theorem of communication complexity by G{"{o}}{"{o}}s, Pitassi and Watson that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols.
{"title":"Decision Tree Complexity versus Block Sensitivity and Degree","authors":"Rahul Chugh, Supartha Podder, Swagato Sanyal","doi":"10.48550/arXiv.2209.08042","DOIUrl":"https://doi.org/10.48550/arXiv.2209.08042","url":null,"abstract":"Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, both of the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0^n to 1^n has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree. Finally, we show using a lifting theorem of communication complexity by G{\"{o}}{\"{o}}s, Pitassi and Watson that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72780024","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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