Pub Date : 2023-01-04DOI: 10.48550/arXiv.2301.01676
Prerona Chatterjee, Pavel Hrubevs
We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $Omega(d/log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $Omega(nd)$, if $dleq n$, or $Omega(nd frac{log n}{log d})$, if $dgeq n$. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.
{"title":"New Lower Bounds against Homogeneous Non-Commutative Circuits","authors":"Prerona Chatterjee, Pavel Hrubevs","doi":"10.48550/arXiv.2301.01676","DOIUrl":"https://doi.org/10.48550/arXiv.2301.01676","url":null,"abstract":"We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $Omega(d/log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $Omega(nd)$, if $dleq n$, or $Omega(nd frac{log n}{log d})$, if $dgeq n$. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"58 1","pages":"13:1-13:10"},"PeriodicalIF":0.0,"publicationDate":"2023-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76020974","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.15
D. Kush, Shubhangi Saraf
The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas. In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP ) defined over Θ( n 2 ) variables and of degree Θ( n ), such that any product-depth ∆ set-multilinear formula computing f has size at least n Ω( n 1 / ∆ / ∆) . Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n Ω(log n ) . If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022). Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM
Raz的开创性工作(J. ACM 2013)以及Limaye、Srinivasan和Tavenas最近的突破性成果(FOCS 2021, STOC 2022)已经证明了通过集合多元线性公式的足够强的下界来获得一般代数公式的下界的潜在途径。在本文中，我们通过证明低深度和无界深度集多元线性公式的近最优下界，在这个方向上取得了进展。更准确地说，我们证明了在任何特征为零的域上，存在一个多项式f，它是由一个多项式大小的集-多线性分支程序(即f在集-多线性VBP中)计算的，它定义在Θ(n 2)个变量上，并且度数为Θ(n)，使得任何计算f的积深∆集-多线性公式的大小至少为n Ω(n 1 /∆/∆)。此外，我们证明了任何无界深度集的大小至少为n Ω(log n)。如果这种强下界被证明为迭代矩阵乘法(IMM)多项式，或者更确切地说，任何由有序集多线性分支程序计算的多项式(即集多线性VBP的进一步限制)，那么这将产生戏剧性的后果，因为它将意味着一般代数公式的超多项式下界(Raz, J. ACM 2013;Tavenas, Limaye, and Srinivasan, STOC 2022)。在我们的工作之前，要么只知道IMM多项式的较弱下界(Tavenas, Limaye, and Srinivasan, STOC 2022)，要么只知道类似的强下界，但即使在集多元线性VP中也不知道硬多项式(Kush and Saraf, CCC 2022;拉兹，J. ACM
{"title":"Near-Optimal Set-Multilinear Formula Lower Bounds","authors":"D. Kush, Shubhangi Saraf","doi":"10.4230/LIPIcs.CCC.2023.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.15","url":null,"abstract":"The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas. In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP ) defined over Θ( n 2 ) variables and of degree Θ( n ), such that any product-depth ∆ set-multilinear formula computing f has size at least n Ω( n 1 / ∆ / ∆) . Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n Ω(log n ) . If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022). Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77500590","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 : 2023-01-01DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.47
Joshua Cook, R. Rothblum
The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S 2 ). This improves on the best previous bound of ˜ O ( n + S 3 ) and matches the result for deterministic space bounded algorithms, up to polylog( S ) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T , space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S log( T ) + Sd ) and the prover runs in time 2 O ( S ) . For d = O (log( T )), this matches the best known interactive proofs for deterministic algorithms, up to polylog( S ) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log( T ). We also improve the best prior verifier time for unbounded alternations by a factor of S . Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in.
著名的IP = PSPACE定理为任何有界空间算法提供了一个有效的交互式证明。在这项工作中，我们研究了非确定性有界空间计算的交互证明。虽然Savitch定理表明非确定性有界空间算法可以用确定性有界空间算法来模拟，但这种模拟具有二次元开销。我们直接给出了不确定性算法的交互协议，以获得更快的验证器。更具体地说，对于任何不确定的空间S算法，我们构造了一个交互式证明，其中验证者运行时间为~ O (n + s2)。这改进了最好的前界~ O (n + s3)，并匹配确定性空间有界算法的结果，最多可达polylog(S)因子。我们进一步推广到交替有界空间算法。对于任何语言L由时间T决定，空间S算法使用d个交替，我们构造了一个交互式证明，其中验证者运行时间为~ O (n + S log(T) + Sd)，证明者运行时间为2o (S)。对于d = O (log(T))，这与确定性算法中最著名的交互式证明相匹配，最多可达log(S)因子，并将非确定性算法的先前最佳验证器时间提高了log(T)因子。我们还将无界变更的最佳先验验证时间提高了S倍。利用已知的有界交替算法与有界深度电路的连接，我们也获得了具有无界扇入的有界深度电路的更快验证。
{"title":"Efficient Interactive Proofs for Non-Deterministic Bounded Space","authors":"Joshua Cook, R. Rothblum","doi":"10.4230/LIPIcs.APPROX/RANDOM.2023.47","DOIUrl":"https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.47","url":null,"abstract":"The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S 2 ). This improves on the best previous bound of ˜ O ( n + S 3 ) and matches the result for deterministic space bounded algorithms, up to polylog( S ) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T , space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S log( T ) + Sd ) and the prover runs in time 2 O ( S ) . For d = O (log( T )), this matches the best known interactive proofs for deterministic algorithms, up to polylog( S ) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log( T ). We also improve the best prior verifier time for unbounded alternations by a factor of S . Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88674689","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 : 2023-01-01DOI: 10.4230/LIPIcs.SAT.2023.4
Benjamin Böhm, Olaf Beyersdorff
We continue the investigation on the relations of QCDCL and QBF resolution systems. In particular, we introduce QCDCL versions that tightly characterise QU-Resolution and (a slight variant of) long-distance Q-Resolution. We show that most QCDCL variants – parameterised by different policies for decisions, unit propagations and reductions – lead to incomparable systems for almost all choices of these policies.
{"title":"QCDCL vs QBF Resolution: Further Insights","authors":"Benjamin Böhm, Olaf Beyersdorff","doi":"10.4230/LIPIcs.SAT.2023.4","DOIUrl":"https://doi.org/10.4230/LIPIcs.SAT.2023.4","url":null,"abstract":"We continue the investigation on the relations of QCDCL and QBF resolution systems. In particular, we introduce QCDCL versions that tightly characterise QU-Resolution and (a slight variant of) long-distance Q-Resolution. We show that most QCDCL variants – parameterised by different policies for decisions, unit propagations and reductions – lead to incomparable systems for almost all choices of these policies.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"17 1","pages":"4:1-4:17"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84317267","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.11
Dean Doron, R. Tell
Existing proofs that deduce BPL = L from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE [ S ] ⊆ DSPACE [ c · S ] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC 0 , that were not known before. 2012 ACM Subject Classification Theory of computation → Complexity theory and logic; Theory of computation → Pseudorandomness and derandomization; Theory of computation → Error-correcting codes
{"title":"Derandomization with Minimal Memory Footprint","authors":"Dean Doron, R. Tell","doi":"10.4230/LIPIcs.CCC.2023.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.11","url":null,"abstract":"Existing proofs that deduce BPL = L from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE [ S ] ⊆ DSPACE [ c · S ] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC 0 , that were not known before. 2012 ACM Subject Classification Theory of computation → Complexity theory and logic; Theory of computation → Pseudorandomness and derandomization; Theory of computation → Error-correcting codes","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"105 1","pages":"11:1-11:15"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73736197","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.36
B. Davis, Robert Robere
Recent work has shown that many of the standard TFNP classes – such as PLS , PPADS , PPAD , SOPL , and EOPL – have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C - PLS , C - PPADS , C - PPAD , C - SOPL , and C - EOPL . While C - PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that: Each of the classes C - PLS , C - PPADS , and C - SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C - PLS is characterized by depth-1.5 Frege (Res( polylog ( n ))). The classes C - PPAD and C - EOPL coincide exactly with the uncoloured classes PPADS and SOPL , respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively. Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C - PLS ∩ C - PPADS = C - SOPL , generalizing the intersection theorem PLS ∩ PPADS = SOPL . However, while it is known in the uncoloured world that PLS ∩ PPAD = EOPL = CLS , we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C - PLS O ∩ C - PPAD O ⊋ C - EOPL O . To prove our results, we introduce an abstract multivalued proof system – the Blockwise Calculus – which may be of independent interest.
{"title":"Colourful TFNP and Propositional Proofs","authors":"B. Davis, Robert Robere","doi":"10.4230/LIPIcs.CCC.2023.36","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.36","url":null,"abstract":"Recent work has shown that many of the standard TFNP classes – such as PLS , PPADS , PPAD , SOPL , and EOPL – have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C - PLS , C - PPADS , C - PPAD , C - SOPL , and C - EOPL . While C - PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that: Each of the classes C - PLS , C - PPADS , and C - SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C - PLS is characterized by depth-1.5 Frege (Res( polylog ( n ))). The classes C - PPAD and C - EOPL coincide exactly with the uncoloured classes PPADS and SOPL , respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively. Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C - PLS ∩ C - PPADS = C - SOPL , generalizing the intersection theorem PLS ∩ PPADS = SOPL . However, while it is known in the uncoloured world that PLS ∩ PPAD = EOPL = CLS , we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C - PLS O ∩ C - PPAD O ⊋ C - EOPL O . To prove our results, we introduce an abstract multivalued proof system – the Blockwise Calculus – which may be of independent interest.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"46 1","pages":"36:1-36:21"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86685083","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.17
Nicollas M. Sdroievski, D. Melkebeek
A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS’21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n a . We show that if every Arthur-Merlin protocol that runs in time n c for c = O (log 2 a ) can only compute f correctly on finitely many inputs, then AM is in NP . Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM , an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM , as well as an unconditional mild derandomization result for AM .
计算复杂性中的一个基本问题是，概率多项式时间算法能否以较小的时间开销确定性地模拟(BPP vs. P问题)。交互证明领域的一个相应问题是，Arthur-Merlin协议是否可以在时间开销很小的情况下进行不确定性模拟(AM与NP问题)。这两个问题都与下界有着复杂的联系。突出的是，在这两种情况下，黑箱非随机化，即通过伪随机生成器的非随机化，已被证明等同于针对电路的决策问题的下界。最近，Chen和Tell (FOCS ' 21)在白盒非随机化和针对几乎所有输入的算法的多位函数的下界之间建立了BPP设置的近似等价。其关键成分是一种将硬度以实例方式转换为目标命中集的技术，该技术基于计算给定实例上的硬函数f的均匀电路的评估的分层算法。在本文中，我们为Arthur-Merlin协议开发了相应的技术，并在AM设置中建立了类似的近等价。作为我们在非随机化方向上的结果的一个例子，考虑一个长度保持函数f，它可以通过一个运行时间为n a的不确定性算法来计算。我们表明，如果每个Arthur-Merlin协议在c = O (log 2a)的时间c内运行，在有限多个输入上只能正确计算f，那么AM在NP中。我们的主要技术贡献是基于非确定性计算的概率可检查证明构建合适的目标命中集生成器。作为我们构建的副产品，我们获得了第一个结果，表明AM的白盒非随机化可能相当于AM的目标命中集生成器的存在，这是Goldreich (LNCS, 2011)提出的一个问题。在平均情况下设置的副产品包括第一个均匀的硬度与随机AM权衡，以及无条件温和的非随机AM结果。
{"title":"Instance-Wise Hardness versus Randomness Tradeoffs for Arthur-Merlin Protocols","authors":"Nicollas M. Sdroievski, D. Melkebeek","doi":"10.4230/LIPIcs.CCC.2023.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.17","url":null,"abstract":"A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS’21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n a . We show that if every Arthur-Merlin protocol that runs in time n c for c = O (log 2 a ) can only compute f correctly on finitely many inputs, then AM is in NP . Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM , an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM , as well as an unconditional mild derandomization result for AM .","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"2010 1","pages":"17:1-17:36"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82546818","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.21
Xi Chen, Yuhao Li, M. Yannakakis
We study the problem of finding a Tarski fixed point over the k -dimensional grid [ n ] k . We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point
{"title":"Reducing Tarski to Unique Tarski (in the Black-box Model)","authors":"Xi Chen, Yuhao Li, M. Yannakakis","doi":"10.4230/LIPIcs.CCC.2023.21","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.21","url":null,"abstract":"We study the problem of finding a Tarski fixed point over the k -dimensional grid [ n ] k . We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88239665","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.6
Shuichi Hirahara, Zhenjian Lu, Hanlin Ren
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization . For a complexity class C , we say that a statement is C -relativizing if the statement holds relative to every oracle O ∈ C . It is easy to see that every result that relativizes also C -relativizes for every complexity class C . On the other hand, we observe that many non-relativizing results, such as IP = PSPACE , are in fact PSPACE -relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0,
{"title":"Bounded Relativization","authors":"Shuichi Hirahara, Zhenjian Lu, Hanlin Ren","doi":"10.4230/LIPIcs.CCC.2023.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.6","url":null,"abstract":"Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization . For a complexity class C , we say that a statement is C -relativizing if the statement holds relative to every oracle O ∈ C . It is easy to see that every result that relativizes also C -relativizes for every complexity class C . On the other hand, we observe that many non-relativizing results, such as IP = PSPACE , are in fact PSPACE -relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0,","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"12 1","pages":"6:1-6:45"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87522728","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 : 2023-01-01DOI: 10.4230/LIPIcs.CCC.2023.12
H. Goldberg, Valentine Kabanets
Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016) showed that the existence of natural properties in the sense of Razborov and Rudich (JCSS, 1997) implies PAC learning algorithms in the sense of Valiant (Comm. ACM, 1984), for boolean functions in P / poly , under the uniform distribution and with membership queries. It is still an open problem to get from natural properties learning algorithms that do not rely on membership queries but rather use randomly drawn labeled examples. Natural properties may be understood as an average-case version of MCSP, the problem of deciding the minimum size of a circuit computing a given truth-table. Problems related to MCSP include those concerning time-bounded Kolmogorov complexity. MKTP, for example, asks for the KT-complexity of a given string. KT-complexity is a relaxation of circuit size, as it does away with the requirement that a short description of a string be interpreted as a boolean circuit. In this work, under assumptions of MKTP and the related problem MK t P being easy on average, we get learning algorithms for boolean functions in P / poly that work over any distribution D samplable by a family of polynomial-size circuits (given explicitly in the case of MKTP ), only use randomly drawn labeled examples from D , and are agnostic (do not require the target function to belong to the hypothesis class). Our results build upon the recent work of Hirahara and Nanashima (FOCS, 2021) who showed similar learning consequences but under a stronger assumption that NP is easy on average.
Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016)表明，对于P / poly中的布尔函数，在均匀分布和成员查询下，Razborov和Rudich (JCSS, 1997)意义上的自然属性的存在意味着Valiant意义上的PAC学习算法。从自然属性中获得不依赖于隶属度查询而是使用随机抽取的标记示例的学习算法仍然是一个开放的问题。自然属性可以理解为MCSP的平均情况版本，MCSP是决定计算给定真值表的电路的最小尺寸的问题。与MCSP相关的问题包括有时Kolmogorov复杂度问题。例如，MKTP要求给定字符串的kt复杂度。kt复杂度是电路大小的放松，因为它不需要将字符串的简短描述解释为布尔电路。在这项工作中，在MKTP和相关问题MK t P平均容易的假设下，我们得到了P / poly中布尔函数的学习算法，该算法可以在任何分布D上工作，这些分布D可由多项式大小的电路族采样(在MKTP的情况下明确给出)，仅使用从D中随机抽取的标记示例，并且是不可知的(不要求目标函数属于假设类)。我们的结果建立在Hirahara和Nanashima (FOCS, 2021)最近的工作基础上，他们显示了类似的学习结果，但在一个更强的假设下，即NP平均容易。
{"title":"Improved Learning from Kolmogorov Complexity","authors":"H. Goldberg, Valentine Kabanets","doi":"10.4230/LIPIcs.CCC.2023.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.12","url":null,"abstract":"Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016) showed that the existence of natural properties in the sense of Razborov and Rudich (JCSS, 1997) implies PAC learning algorithms in the sense of Valiant (Comm. ACM, 1984), for boolean functions in P / poly , under the uniform distribution and with membership queries. It is still an open problem to get from natural properties learning algorithms that do not rely on membership queries but rather use randomly drawn labeled examples. Natural properties may be understood as an average-case version of MCSP, the problem of deciding the minimum size of a circuit computing a given truth-table. Problems related to MCSP include those concerning time-bounded Kolmogorov complexity. MKTP, for example, asks for the KT-complexity of a given string. KT-complexity is a relaxation of circuit size, as it does away with the requirement that a short description of a string be interpreted as a boolean circuit. In this work, under assumptions of MKTP and the related problem MK t P being easy on average, we get learning algorithms for boolean functions in P / poly that work over any distribution D samplable by a family of polynomial-size circuits (given explicitly in the case of MKTP ), only use randomly drawn labeled examples from D , and are agnostic (do not require the target function to belong to the hypothesis class). Our results build upon the recent work of Hirahara and Nanashima (FOCS, 2021) who showed similar learning consequences but under a stronger assumption that NP is easy on average.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74434759","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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