Noel Arteche, Erfan Khaniki, Ján Pich, Rahul Santhanam
Folklore in complexity theory suspects that circuit lower bounds against $mathbf{NC}^1$ or $mathbf{P}/operatorname{poly}$, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation $mathbf{NEXP} notsubseteq mathbf{P}/operatorname{poly}$, as recently observed by Pich and Santhanam (2023). We show such a connection conditionally for the Implicit Extended Frege proof system ($mathsf{iEF}$) introduced by Kraj'iv{c}ek (The Journal of Symbolic Logic, 2004), capable of formalizing most of contemporary complexity theory. In particular, we show that if $mathsf{iEF}$ proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of $mathsf{iEF}$ proofs implies $#mathbf{P} notsubseteq mathbf{FP}/operatorname{poly}$ (which would in turn imply, for example, $mathbf{PSPACE} notsubseteq mathbf{P}/operatorname{poly}$). Our proof exploits the formalization inside $mathsf{iEF}$ of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan (Journal of the ACM, 1992). This has consequences for the self-provability of circuit upper bounds in $mathsf{iEF}$. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.
复杂性理论的民间传说认为，针对$mathbf{NC}^1$或$mathbf{P}/operatorname{poly}$的电路下界，目前还遥不可及，是为弗雷格或扩展弗雷格等系统证明强证明复杂性下界的必要步骤。然而，从形式上建立这样的联系已经令人生畏，因为这将意味着突破性的分离$mathbf{NEXP}/not/subsete$。notsubseteq mathbf{P}/operatorname{poly}$, 正如 Pich 和 Santhanam (2023) 最近所观察到的。我们为克拉伊夫切克（《符号逻辑杂志》，2004 年）引入的隐式扩展弗雷格证明系统（$mathsf{iEF}$）展示了这种有条件的联系，它能够形式化大部分当代复杂性理论。特别是，我们证明了如果 $mathsf{iEF}$ 能够有效证明标准随机化假设，即对于亚指数大小的电路来说，一个具体的布尔函数平均很难，那么 $mathsf{iEF}$ 证明长度的任何超多项式下限都意味着 $#mathbf{P}不是/subseteq/mathbf{FP}/operatorname{poly}$（这反过来又意味着，例如，$mathbf{PSPACE}/operatorname{poly}$）。不是/subseteq （mathbf{P}//operatorname{poly}$）。我们的证明利用了 $mathsf{iEF}$ 内部对 Lund、Fortnow、Karloff 和 Nisan 的求和校验协议（Journal of the ACM,1992）合理性的形式化。这对 $mathsf{iEF}$ 中电路上界的自证明性产生了影响。有趣的是，要进一步改进我们的结果，似乎需要在构建具有更高效证明器的交互式证明系统方面取得进展。
{"title":"From Proof Complexity to Circuit Complexity via Interactive Protocols","authors":"Noel Arteche, Erfan Khaniki, Ján Pich, Rahul Santhanam","doi":"arxiv-2405.02232","DOIUrl":"https://doi.org/arxiv-2405.02232","url":null,"abstract":"Folklore in complexity theory suspects that circuit lower bounds against\u0000$mathbf{NC}^1$ or $mathbf{P}/operatorname{poly}$, currently out of reach,\u0000are a necessary step towards proving strong proof complexity lower bounds for\u0000systems like Frege or Extended Frege. Establishing such a connection formally,\u0000however, is already daunting, as it would imply the breakthrough separation\u0000$mathbf{NEXP} notsubseteq mathbf{P}/operatorname{poly}$, as recently\u0000observed by Pich and Santhanam (2023). We show such a connection conditionally for the Implicit Extended Frege proof\u0000system ($mathsf{iEF}$) introduced by Kraj'iv{c}ek (The Journal of Symbolic\u0000Logic, 2004), capable of formalizing most of contemporary complexity theory. In\u0000particular, we show that if $mathsf{iEF}$ proves efficiently the standard\u0000derandomization assumption that a concrete Boolean function is hard on average\u0000for subexponential-size circuits, then any superpolynomial lower bound on the\u0000length of $mathsf{iEF}$ proofs implies $#mathbf{P} notsubseteq\u0000mathbf{FP}/operatorname{poly}$ (which would in turn imply, for example,\u0000$mathbf{PSPACE} notsubseteq mathbf{P}/operatorname{poly}$). Our proof\u0000exploits the formalization inside $mathsf{iEF}$ of the soundness of the\u0000sum-check protocol of Lund, Fortnow, Karloff, and Nisan (Journal of the ACM,\u00001992). This has consequences for the self-provability of circuit upper bounds\u0000in $mathsf{iEF}$. Interestingly, further improving our result seems to require\u0000progress in constructing interactive proof systems with more efficient provers.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883818","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}
Leo van Iersel, Mark Jones, Jannik Schestag, Celine Scornavacca, Mathias Weller
Network Phylogenetic Diversity (Network-PD) is a measure for the diversity of a set of species based on a rooted phylogenetic network (with branch lengths and inheritance probabilities on the reticulation edges) describing the evolution of those species. We consider the textsc{Max-Network-PD} problem: given such a network, find~$k$ species with maximum Network-PD score. We show that this problem is fixed-parameter tractable (FPT) for binary networks, by describing an optimal algorithm running in $mathcal{O}(2^r log (k)(n+r))$~time, with~$n$ the total number of species in the network and~$r$ its reticulation number. Furthermore, we show that textsc{Max-Network-PD} is NP-hard for level-1 networks, proving that, unless P$=$NP, the FPT approach cannot be extended by using the level as parameter instead of the reticulation number.
{"title":"Maximizing Network Phylogenetic Diversity","authors":"Leo van Iersel, Mark Jones, Jannik Schestag, Celine Scornavacca, Mathias Weller","doi":"arxiv-2405.01091","DOIUrl":"https://doi.org/arxiv-2405.01091","url":null,"abstract":"Network Phylogenetic Diversity (Network-PD) is a measure for the diversity of\u0000a set of species based on a rooted phylogenetic network (with branch lengths\u0000and inheritance probabilities on the reticulation edges) describing the\u0000evolution of those species. We consider the textsc{Max-Network-PD} problem:\u0000given such a network, find~$k$ species with maximum Network-PD score. We show\u0000that this problem is fixed-parameter tractable (FPT) for binary networks, by\u0000describing an optimal algorithm running in $mathcal{O}(2^r log\u0000(k)(n+r))$~time, with~$n$ the total number of species in the network and~$r$\u0000its reticulation number. Furthermore, we show that textsc{Max-Network-PD} is\u0000NP-hard for level-1 networks, proving that, unless P$=$NP, the FPT approach\u0000cannot be extended by using the level as parameter instead of the reticulation\u0000number.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is motivated by a question whether it is possible to calculate a chaotic sequence efficiently, e.g., is it possible to get the $n$-th bit of a bit sequence generated by a chaotic map, such as $beta$-expansion, tent map and logistic map in $mathrm{o}(n)$ time/space? This paper gives an affirmative answer to the question about the space complexity of a tent map. We show that the decision problem of whether a given bit sequence is a valid tent code is solved in $mathrm{O}(log^{2} n)$ space in a sense of the smoothed complexity.
{"title":"A Smoothed Analysis of the Space Complexity of Computing a Chaotic Sequence","authors":"Naoaki Okada, Shuji Kijima","doi":"arxiv-2405.00327","DOIUrl":"https://doi.org/arxiv-2405.00327","url":null,"abstract":"This work is motivated by a question whether it is possible to calculate a\u0000chaotic sequence efficiently, e.g., is it possible to get the $n$-th bit of a\u0000bit sequence generated by a chaotic map, such as $beta$-expansion, tent map\u0000and logistic map in $mathrm{o}(n)$ time/space? This paper gives an affirmative\u0000answer to the question about the space complexity of a tent map. We show that\u0000the decision problem of whether a given bit sequence is a valid tent code is\u0000solved in $mathrm{O}(log^{2} n)$ space in a sense of the smoothed complexity.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827008","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 construct an oracle relative to which $mathrm{NP} = mathrm{PSPACE}$, but $mathrm{UP}$ has no many-one complete sets. This combines the properties of an oracle by Hartmanis and Hemachandra [HH88] and one by Ogiwara and Hemachandra [OH93]. The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. This answers several questions by Pudl'ak [Pud17], e.g., the implications $mathsf{UP} Longrightarrow mathsf{CON}^{mathsf{N}}$ and $mathsf{SAT} Longrightarrow mathsf{TFNP}$ are false relative to our oracle. Moreover, the oracle demonstrates that, in principle, it is possible that $mathrm{TFNP}$-complete problems exist, while at the same time $mathrm{SAT}$ has no p-optimal proof systems.
{"title":"An Oracle with no $mathrm{UP}$-Complete Sets, but $mathrm{NP}=mathrm{PSPACE}$","authors":"David Dingel, Fabian Egidy, Christian Glaßer","doi":"arxiv-2404.19104","DOIUrl":"https://doi.org/arxiv-2404.19104","url":null,"abstract":"We construct an oracle relative to which $mathrm{NP} = mathrm{PSPACE}$, but\u0000$mathrm{UP}$ has no many-one complete sets. This combines the properties of an\u0000oracle by Hartmanis and Hemachandra [HH88] and one by Ogiwara and Hemachandra\u0000[OH93]. The oracle provides new separations of classical conjectures on optimal proof\u0000systems and complete sets in promise classes. This answers several questions by\u0000Pudl'ak [Pud17], e.g., the implications $mathsf{UP} Longrightarrow\u0000mathsf{CON}^{mathsf{N}}$ and $mathsf{SAT} Longrightarrow mathsf{TFNP}$ are\u0000false relative to our oracle. Moreover, the oracle demonstrates that, in principle, it is possible that\u0000$mathrm{TFNP}$-complete problems exist, while at the same time $mathrm{SAT}$\u0000has no p-optimal proof systems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826973","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 random $k$-XORSAT problem is a random constraint satisfaction problem of $n$ Boolean variables and $m=rn$ clauses, which a random instance can be expressed as a $Gmathbb{F}(2)$ linear system of the form $Ax=b$, where $A$ is a random $m times n$ matrix with $k$ ones per row, and $b$ is a random vector. It is known that there exist two distinct thresholds $r_{core}(k) < r_{sat}(k)$ such that as $n rightarrow infty$ for $r < r_{sat}(k)$ the random instance has solutions with high probability, while for $r_{core} < r < r_{sat}(k)$ the solution space shatters into an exponential number of clusters. Sequential local algorithms are a natural class of algorithms which assign values to variables one by one iteratively. In each iteration, the algorithm runs some heuristics, called local rules, to decide the value assigned, based on the local neighborhood of the selected variables under the factor graph representation of the instance. We prove that for any $r > r_{core}(k)$ the sequential local algorithms with certain local rules fail to solve the random $k$-XORSAT with high probability. They include (1) the algorithm using the Unit Clause Propagation as local rule for $k ge 9$, and (2) the algorithms using any local rule that can calculate the exact marginal probabilities of variables in instances with factor graphs that are trees, for $kge 13$. The well-known Belief Propagation and Survey Propagation are included in (2). Meanwhile, the best known linear-time algorithm succeeds with high probability for $r < r_{core}(k)$. Our results support the intuition that $r_{core}(k)$ is the sharp threshold for the existence of a linear-time algorithm for random $k$-XORSAT.
{"title":"Limits of Sequential Local Algorithms on the Random $k$-XORSAT Problem","authors":"Kingsley Yung","doi":"arxiv-2404.17775","DOIUrl":"https://doi.org/arxiv-2404.17775","url":null,"abstract":"The random $k$-XORSAT problem is a random constraint satisfaction problem of\u0000$n$ Boolean variables and $m=rn$ clauses, which a random instance can be\u0000expressed as a $Gmathbb{F}(2)$ linear system of the form $Ax=b$, where $A$ is\u0000a random $m times n$ matrix with $k$ ones per row, and $b$ is a random vector.\u0000It is known that there exist two distinct thresholds $r_{core}(k) < r_{sat}(k)$\u0000such that as $n rightarrow infty$ for $r < r_{sat}(k)$ the random instance\u0000has solutions with high probability, while for $r_{core} < r < r_{sat}(k)$ the\u0000solution space shatters into an exponential number of clusters. Sequential\u0000local algorithms are a natural class of algorithms which assign values to\u0000variables one by one iteratively. In each iteration, the algorithm runs some\u0000heuristics, called local rules, to decide the value assigned, based on the\u0000local neighborhood of the selected variables under the factor graph\u0000representation of the instance. We prove that for any $r > r_{core}(k)$ the sequential local algorithms with\u0000certain local rules fail to solve the random $k$-XORSAT with high probability.\u0000They include (1) the algorithm using the Unit Clause Propagation as local rule\u0000for $k ge 9$, and (2) the algorithms using any local rule that can calculate\u0000the exact marginal probabilities of variables in instances with factor graphs\u0000that are trees, for $kge 13$. The well-known Belief Propagation and Survey\u0000Propagation are included in (2). Meanwhile, the best known linear-time\u0000algorithm succeeds with high probability for $r < r_{core}(k)$. Our results\u0000support the intuition that $r_{core}(k)$ is the sharp threshold for the\u0000existence of a linear-time algorithm for random $k$-XORSAT.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826977","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 address a specific case of the matroid intersection problem: given a set of graphs sharing the same set of vertices, select a minimum cycle basis for each graph to maximize the size of their intersection. We provide a comprehensive complexity analysis of this problem, which finds applications in chemoinformatics. We establish a complete partition of subcases based on intrinsic parameters: the number of graphs, the maximum degree of the graphs, and the size of the longest cycle in the minimum cycle bases. Additionally, we present results concerning the approximability and parameterized complexity of the problem.
{"title":"Maximizing Minimum Cycle Bases Intersection","authors":"Dimitri WatelSAMOVAR, ENSIIE, Marc-Antoine WeisserGALaC, Dominique BarthUVSQ, DAVID, Ylène AboulfathUVSQ, DAVID, Thierry MautorUVSQ, DAVID","doi":"arxiv-2404.17223","DOIUrl":"https://doi.org/arxiv-2404.17223","url":null,"abstract":"We address a specific case of the matroid intersection problem: given a set\u0000of graphs sharing the same set of vertices, select a minimum cycle basis for\u0000each graph to maximize the size of their intersection. We provide a\u0000comprehensive complexity analysis of this problem, which finds applications in\u0000chemoinformatics. We establish a complete partition of subcases based on\u0000intrinsic parameters: the number of graphs, the maximum degree of the graphs,\u0000and the size of the longest cycle in the minimum cycle bases. Additionally, we\u0000present results concerning the approximability and parameterized complexity of\u0000the problem.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140809364","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 noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in $n$ noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational formula in $n$ noncommuting variables is zero on its domain of definition. Motivated by the question whether these problems have deterministic NC algorithms, we revisit their interrelationship from a parallel complexity point of view. We show the following results: 1. Based on Cohn's embedding theorem cite{Co90,Cohnfir} we show deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and from multivariate RIT to bivariate RIT. 2. We obtain a deterministic NC-Turing reduction from bivariate $RIT$ to bivariate ncRANK, thereby proving that a deterministic NC algorithm for bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK are in deterministic NC.
{"title":"A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results","authors":"Vikraman Arvind, Pushkar S Joglekar","doi":"arxiv-2404.16382","DOIUrl":"https://doi.org/arxiv-2404.16382","url":null,"abstract":"We study the noncommutative rank problem, ncRANK, of computing the rank of\u0000matrices with linear entries in $n$ noncommuting variables and the problem of\u0000noncommutative Rational Identity Testing, RIT, which is to decide if a given\u0000rational formula in $n$ noncommuting variables is zero on its domain of\u0000definition. Motivated by the question whether these problems have deterministic\u0000NC algorithms, we revisit their interrelationship from a parallel complexity\u0000point of view. We show the following results: 1. Based on Cohn's embedding theorem cite{Co90,Cohnfir} we show\u0000deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and\u0000from multivariate RIT to bivariate RIT. 2. We obtain a deterministic NC-Turing reduction from bivariate $RIT$ to\u0000bivariate ncRANK, thereby proving that a deterministic NC algorithm for\u0000bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK\u0000are in deterministic NC.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This short note outlines some of the issues in Czerwinski's paper [Cze23] claiming that NP-hard problems are not in BQP. We outline one major issue and two minor issues, and conclude that their paper does not establish what they claim it does.
{"title":"A Brief Note on a Recent Claim About NP-Hard Problems and BQP","authors":"Michael C. Chavrimootoo","doi":"arxiv-2406.08495","DOIUrl":"https://doi.org/arxiv-2406.08495","url":null,"abstract":"This short note outlines some of the issues in Czerwinski's paper [Cze23]\u0000claiming that NP-hard problems are not in BQP. We outline one major issue and\u0000two minor issues, and conclude that their paper does not establish what they\u0000claim it does.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141518335","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}
Susanna F. de Rezende, Aaron Potechin, Kilian Risse
We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{Omega(d)}$ to rule out the existence of an $n^{Theta(1)}$-clique in ErdH{o}s-R'{e}nyi random graphs whose maximum clique is of size $dleq 2log n$. This lower bound is tight up to the multiplicative constant in the exponent. We obtain this result by introducing a technique inspired by pseudo-calibration which may be of independent interest. The technique involves defining a measure on monomials that precisely captures the contribution of a monomial to a refutation. This measure intuitively captures progress and should have further applications in proof complexity.
{"title":"Clique Is Hard on Average for Sherali-Adams with Bounded Coefficients","authors":"Susanna F. de Rezende, Aaron Potechin, Kilian Risse","doi":"arxiv-2404.16722","DOIUrl":"https://doi.org/arxiv-2404.16722","url":null,"abstract":"We prove that Sherali-Adams with polynomially bounded coefficients requires\u0000proofs of size $n^{Omega(d)}$ to rule out the existence of an\u0000$n^{Theta(1)}$-clique in ErdH{o}s-R'{e}nyi random graphs whose maximum\u0000clique is of size $dleq 2log n$. This lower bound is tight up to the\u0000multiplicative constant in the exponent. We obtain this result by introducing a\u0000technique inspired by pseudo-calibration which may be of independent interest.\u0000The technique involves defining a measure on monomials that precisely captures\u0000the contribution of a monomial to a refutation. This measure intuitively\u0000captures progress and should have further applications in proof complexity.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801528","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}
One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $mathbf{P}$ versus $mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))cdot log n$, and it is widely open how to prove a super-$3log n$ depth lower bound. Recently Mihajlin and Sofronova (CCC'22) show if considering formulas with restriction on top, we can break the $3log n$ barrier. Formally, they prove there exist two functions $f:{0,1}^n rightarrow {0,1},g:{0,1}^n rightarrow {0,1}^n$, such that for any constant $00$. They ask whether the parameter $alpha$ can be push up to nearly $1$ thus implying a nearly-$3.5log n$ depth lower bound. In this paper, we provide a stronger answer to their question. We show there exist two functions $f:{0,1}^n rightarrow {0,1},g:{0,1}^n rightarrow {0,1}^n$, such that for any constant $0
{"title":"A nearly-$4log n$ depth lower bound for formulas with restriction on top","authors":"Hao Wu","doi":"arxiv-2404.15613","DOIUrl":"https://doi.org/arxiv-2404.15613","url":null,"abstract":"One of the major open problems in complexity theory is to demonstrate an\u0000explicit function which requires super logarithmic depth, a.k.a, the\u0000$mathbf{P}$ versus $mathbf{NC^1}$ problem. The current best depth lower bound\u0000is $(3-o(1))cdot log n$, and it is widely open how to prove a super-$3log n$\u0000depth lower bound. Recently Mihajlin and Sofronova (CCC'22) show if considering\u0000formulas with restriction on top, we can break the $3log n$ barrier. Formally,\u0000they prove there exist two functions $f:{0,1}^n rightarrow\u0000{0,1},g:{0,1}^n rightarrow {0,1}^n$, such that for any constant\u0000$0<alpha<0.4$ and constant $0<epsilon<alpha/2$, their XOR composition\u0000$f(g(x)oplus y)$ is not computable by an AND of $2^{(alpha-epsilon)n}$\u0000formulas of size at most $2^{(1-alpha/2-epsilon)n}$. This implies a modified\u0000version of Andreev function is not computable by any circuit of depth\u0000$(3.2-epsilon)log n$ with the restriction that top $0.4-epsilon$ layers only\u0000consist of AND gates for any small constant $epsilon>0$. They ask whether the\u0000parameter $alpha$ can be push up to nearly $1$ thus implying a nearly-$3.5log\u0000n$ depth lower bound. In this paper, we provide a stronger answer to their question. We show there\u0000exist two functions $f:{0,1}^n rightarrow {0,1},g:{0,1}^n rightarrow\u0000{0,1}^n$, such that for any constant $0<alpha<2-o(1)$, their XOR composition\u0000$f(g(x)oplus y)$ is not computable by an AND of $2^{alpha n}$ formulas of\u0000size at most $2^{(1-alpha/2-o(1))n}$. This implies a $(4-o(1))log n$ depth\u0000lower bound with the restriction that top $2-o(1)$ layers only consist of AND\u0000gates. We prove it by observing that one crucial component in Mihajlin and\u0000Sofronova's work, called the well-mixed set of functions, can be significantly\u0000simplified thus improved. Then with this observation and a more careful\u0000analysis, we obtain these nearly tight results.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801326","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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