The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich [49] showed a lower bound of ˜Ω(D + √ n) for this problem, where D is the hop-diameter of G. Whether or not this problem can be solved in O(n) time when D is relatively small is a major open question. Despite intensive research [10, 17, 33, 41, 45] that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this article, we answer this question in the affirmative. We devise an algorithm that requires O((n log n)5/6) time, for D = O(√ n log n), and O(D1/3 ⋅ (n log n)2/3) time, for larger D. This running time is sublinear in n in almost the entire range of parameters, specifically, for D = o(n/ log2 n). We also generalize our result in two directions. One is when edges have bandwidth b ≥ 1, and the other is the s-sources shortest paths problem. For both problems, our algorithm provides bounds that improve upon the previous state-of-the-art in almost the entire range of parameters. In particular, we provide an all-pairs shortest paths algorithm that requires O(n5/3 ⋅ log 2/3 n) time, even for b = 1, for all values of D. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our distributed algorithm computes a hopset G′′ of a skeleton graph G′ of G without first computing G′ itself. We then conduct a Bellman-Ford exploration in G′ ∪ G′′, while computing the required edges of G′ on the fly. As a result, our distributed algorithm computes exactly those edges of G′ that it really needs, rather than computing approximately the entire G′.
{"title":"Distributed Exact Shortest Paths in Sublinear Time","authors":"Michael Elkin","doi":"10.1145/3387161","DOIUrl":"https://doi.org/10.1145/3387161","url":null,"abstract":"The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich [49] showed a lower bound of ˜Ω(D + √ n) for this problem, where D is the hop-diameter of G. Whether or not this problem can be solved in O(n) time when D is relatively small is a major open question. Despite intensive research [10, 17, 33, 41, 45] that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this article, we answer this question in the affirmative. We devise an algorithm that requires O((n log n)5/6) time, for D = O(√ n log n), and O(D1/3 ⋅ (n log n)2/3) time, for larger D. This running time is sublinear in n in almost the entire range of parameters, specifically, for D = o(n/ log2 n). We also generalize our result in two directions. One is when edges have bandwidth b ≥ 1, and the other is the s-sources shortest paths problem. For both problems, our algorithm provides bounds that improve upon the previous state-of-the-art in almost the entire range of parameters. In particular, we provide an all-pairs shortest paths algorithm that requires O(n5/3 ⋅ log 2/3 n) time, even for b = 1, for all values of D. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our distributed algorithm computes a hopset G′′ of a skeleton graph G′ of G without first computing G′ itself. We then conduct a Bellman-Ford exploration in G′ ∪ G′′, while computing the required edges of G′ on the fly. As a result, our distributed algorithm computes exactly those edges of G′ that it really needs, rather than computing approximately the entire G′.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"1 1","pages":"1 - 36"},"PeriodicalIF":0.0,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75019123","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}
Olaf Beyersdorff, Ilario Bonacina, Leroy Chew, J. Pich
We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.
{"title":"Frege Systems for Quantified Boolean Logic","authors":"Olaf Beyersdorff, Ilario Bonacina, Leroy Chew, J. Pich","doi":"10.1145/3381881","DOIUrl":"https://doi.org/10.1145/3381881","url":null,"abstract":"We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"31 1","pages":"1 - 36"},"PeriodicalIF":0.0,"publicationDate":"2020-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91081782","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 distributed minimum spanning tree (MST) problem is one of the most central and fundamental problems in distributed graph algorithms. Kutten and Peleg devised an algorithm with running time O(D + √n ⋅ log* n), where D is the hop diameter of the input n-vertex m-edge graph, and with message complexity O(m + n3/2). Peleg and Rubinovich showed that the running time of the algorithm of Kutten and Peleg is essentially tight and asked if one can achieve near-optimal running time together with near-optimal message complexity. In a recent breakthrough, Pandurangan et al. answered this question in the affirmative and devised a randomized algorithm with time Õ(D+ √ n) and message complexity Õ(m). They asked if such a simultaneous time- and message optimality can be achieved by a deterministic algorithm. In this article, building on the work of Pandurangan et al., we answer this question in the affirmative and devise a deterministic algorithm that computes MST in time O((D + √ n) ⋅ log n) using O(m ⋅ log n + n log n cdot log* n) messages. The polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of Pandurangan et al. In addition, our algorithm and its analysis are very simple and self-contained as opposed to rather complicated previous sublinear-time algorithms. Finally, we use our new algorithm to devise a randomized MST algorithm with running time Õ(μ (G,ω) + √ n) and message complexity Õ(|E|), where μ-radius μ (G,ω) ≤ D is a graph parameter, which is typically much smaller than D. This improves a previous bound from Elkin.
分布式最小生成树问题是分布式图算法中最核心、最基本的问题之一。Kutten和Peleg设计了一种运行时间为O(D +√n⋅log* n)的算法，其中D为输入n顶点m边图的跳直径，消息复杂度为O(m + n3/2)。Peleg和Rubinovich表明Kutten和Peleg算法的运行时间本质上是紧的，并询问是否可以在接近最优的消息复杂度下实现接近最优的运行时间。在最近的一项突破中，Pandurangan等人肯定地回答了这个问题，并设计了一个随机算法，时间为Õ(D+√n)，消息复杂度为Õ(m)。他们问，这种同时的时间和消息最优性是否可以通过确定性算法实现。在本文中，基于Pandurangan等人的工作，我们肯定地回答了这个问题，并设计了一种确定性算法，该算法使用O(m⋅log n + n log n cdot log* n)条消息计算时间为O((D +√n)⋅log n)的MST。我们算法的时间复杂度和消息复杂度的多对数因子明显小于Pandurangan等人的结果中各自的因子。此外，我们的算法及其分析相对于以往复杂的亚线性时间算法来说非常简单和独立。最后，我们利用新算法设计了一个随机化的MST算法，其运行时间为Õ(μ (G，ω) +√n)，消息复杂度为Õ(|E|)，其中μ-半径μ (G，ω)≤D是一个图参数，通常比D小得多。
{"title":"A Simple Deterministic Distributed MST Algorithm with Near-Optimal Time and Message Complexities","authors":"Michael Elkin","doi":"10.1145/3380546","DOIUrl":"https://doi.org/10.1145/3380546","url":null,"abstract":"The distributed minimum spanning tree (MST) problem is one of the most central and fundamental problems in distributed graph algorithms. Kutten and Peleg devised an algorithm with running time O(D + √n ⋅ log* n), where D is the hop diameter of the input n-vertex m-edge graph, and with message complexity O(m + n3/2). Peleg and Rubinovich showed that the running time of the algorithm of Kutten and Peleg is essentially tight and asked if one can achieve near-optimal running time together with near-optimal message complexity. In a recent breakthrough, Pandurangan et al. answered this question in the affirmative and devised a randomized algorithm with time Õ(D+ √ n) and message complexity Õ(m). They asked if such a simultaneous time- and message optimality can be achieved by a deterministic algorithm. In this article, building on the work of Pandurangan et al., we answer this question in the affirmative and devise a deterministic algorithm that computes MST in time O((D + √ n) ⋅ log n) using O(m ⋅ log n + n log n cdot log* n) messages. The polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of Pandurangan et al. In addition, our algorithm and its analysis are very simple and self-contained as opposed to rather complicated previous sublinear-time algorithms. Finally, we use our new algorithm to devise a randomized MST algorithm with running time Õ(μ (G,ω) + √ n) and message complexity Õ(|E|), where μ-radius μ (G,ω) ≤ D is a graph parameter, which is typically much smaller than D. This improves a previous bound from Elkin.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"20 1","pages":"1 - 15"},"PeriodicalIF":0.0,"publicationDate":"2020-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84449271","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}
G. Gottlob, Matthias Lanzinger, R. Pichler, Igor Razgon
Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H), its generalized hypertree width ghw(H), and its fractional hypertree width fhw(H), respectively. It is known that hw(H)≤ k can be checked in polynomial time for fixed k, while checking ghw(H)≤ k is NP-complete for k ≥ 3. The complexity of checking fhw(H)≤ k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H)≤ k is NP-complete, even for k=2. The same construction allows us to prove also the NP-completeness of checking ghw(H)≤ k for k=2. After that, we identify meaningful restrictions that make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw.
{"title":"Complexity Analysis of Generalized and Fractional Hypertree Decompositions","authors":"G. Gottlob, Matthias Lanzinger, R. Pichler, Igor Razgon","doi":"10.1145/3457374","DOIUrl":"https://doi.org/10.1145/3457374","url":null,"abstract":"Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H), its generalized hypertree width ghw(H), and its fractional hypertree width fhw(H), respectively. It is known that hw(H)≤ k can be checked in polynomial time for fixed k, while checking ghw(H)≤ k is NP-complete for k ≥ 3. The complexity of checking fhw(H)≤ k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H)≤ k is NP-complete, even for k=2. The same construction allows us to prove also the NP-completeness of checking ghw(H)≤ k for k=2. After that, we identify meaningful restrictions that make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"37 1","pages":"1 - 50"},"PeriodicalIF":0.0,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78992028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a generalized network design (GND) problem, a set of resources are assigned (non-exclusively) to multiple requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource-specific cost function. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current article advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) while the existing results are restricted to routing requests in undirected graphs, identifying the resources with the graph’s edges, the current article presents a generic approximation framework that yields approximation results for a much wider family of requests (including various types of Steiner tree and Steiner forest requests) in both directed and undirected graphs, where the resources can be identified with either the edges or the vertices; (2) while the existing results assume that a request contributes the same weight to each resource it uses, our approximation framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) while most of the existing approximation algorithms are based on convex programming, our approximation framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current article is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources’ technology and does not grow with the number of resources, the number of requests, or their weights. The design of our approximation framework relies heavily on Roughgarden’s smoothness toolbox [43], thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.
{"title":"Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency","authors":"Y. Emek, S. Kutten, R. Lavi, Yangguang Shi","doi":"10.1145/3377387","DOIUrl":"https://doi.org/10.1145/3377387","url":null,"abstract":"In a generalized network design (GND) problem, a set of resources are assigned (non-exclusively) to multiple requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource-specific cost function. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current article advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) while the existing results are restricted to routing requests in undirected graphs, identifying the resources with the graph’s edges, the current article presents a generic approximation framework that yields approximation results for a much wider family of requests (including various types of Steiner tree and Steiner forest requests) in both directed and undirected graphs, where the resources can be identified with either the edges or the vertices; (2) while the existing results assume that a request contributes the same weight to each resource it uses, our approximation framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) while most of the existing approximation algorithms are based on convex programming, our approximation framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current article is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources’ technology and does not grow with the number of resources, the number of requests, or their weights. The design of our approximation framework relies heavily on Roughgarden’s smoothness toolbox [43], thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"8 1","pages":"1 - 33"},"PeriodicalIF":0.0,"publicationDate":"2020-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75855321","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}
Preferences are a fundamental ingredient in a variety of fields, ranging from economics to computer science, for deciding the best choices among possible alternatives. Contexts provide another important aspect to be considered in the selection of the best choices, since, very often, preferences are affected by context. In particular, the problem of preference propagation from more generic to more specific contexts naturally arises. Such a problem has only been addressed in a very limited way and always resorts to practical, ad hoc approaches. To fill this gap, in this article, we analyze preference propagation in a principled way and adopt an abstract context model without making any specific assumptions on how preferences are stated. Our framework only requires that the contexts form a partially ordered set and that preferences define a strict partial order on the objects of interest. We first formalize the basic properties that any propagation process should satisfy. We then introduce an algebraic model for preference propagation that relies on two abstract operators for combining preferences, and, under mild assumptions, we prove that the only possible interpretations for such operators are the well-known Pareto and Prioritized composition. We then study several propagation methods based on such operators and precisely characterize them in terms of the stated properties. We finally identify a method meeting all the requirements, on the basis of which we provide an efficient algorithm for preference propagation.
{"title":"Foundations of Context-aware Preference Propagation","authors":"P. Ciaccia, D. Martinenghi, Riccardo Torlone","doi":"10.1145/3375713","DOIUrl":"https://doi.org/10.1145/3375713","url":null,"abstract":"Preferences are a fundamental ingredient in a variety of fields, ranging from economics to computer science, for deciding the best choices among possible alternatives. Contexts provide another important aspect to be considered in the selection of the best choices, since, very often, preferences are affected by context. In particular, the problem of preference propagation from more generic to more specific contexts naturally arises. Such a problem has only been addressed in a very limited way and always resorts to practical, ad hoc approaches. To fill this gap, in this article, we analyze preference propagation in a principled way and adopt an abstract context model without making any specific assumptions on how preferences are stated. Our framework only requires that the contexts form a partially ordered set and that preferences define a strict partial order on the objects of interest. We first formalize the basic properties that any propagation process should satisfy. We then introduce an algebraic model for preference propagation that relies on two abstract operators for combining preferences, and, under mild assumptions, we prove that the only possible interpretations for such operators are the well-known Pareto and Prioritized composition. We then study several propagation methods based on such operators and precisely characterize them in terms of the stated properties. We finally identify a method meeting all the requirements, on the basis of which we provide an efficient algorithm for preference propagation.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"40 1","pages":"1 - 43"},"PeriodicalIF":0.0,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74699930","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 Invited Articles section of this issue consists of two articles. The article “The WeisfeilerLeman Dimension of Planar Graphs is at most 3,” by Sandra Kiefer, Ilia Ponomarenko, and Pascal Schweitzer, was invited from the 32nd Annual ACM/ IEEE Symposium on Logic in Computer Science (LICS’17). The article “Computing the Geometric Intersection Number of Curves,” by Vincent Despré and Francis Lazarus, won best paper award at the 33rd International Symposium on Computational Geometry (SoCG 2017). We want to thank the LICS’17 and SoCG’17 Program Committees for their help in selecting these invited articles and editors Nachum Dershowitz and Jean-Daniel Boissonnat for handling the articles.
本期特邀文章部分由两篇文章组成。由Sandra Kiefer, Ilia Ponomarenko和Pascal Schweitzer撰写的文章“平面图的WeisfeilerLeman维数最多为3”被邀请参加第32届ACM/ IEEE计算机科学逻辑研讨会(LICS ' 17)。Vincent despr和Francis Lazarus的论文《计算曲线的几何相交数》(Computing The Geometric Intersection Number of Curves)在第33届计算几何国际学术研讨会(SoCG 2017)上获得了最佳论文奖。我们要感谢LICS ' 17和SoCG ' 17项目委员会在选择这些受邀文章和编辑Nachum Dershowitz和Jean-Daniel Boissonnat处理这些文章方面的帮助。
{"title":"Invited Articles Foreword","authors":"É. Tardos","doi":"10.1145/3371337","DOIUrl":"https://doi.org/10.1145/3371337","url":null,"abstract":"The Invited Articles section of this issue consists of two articles. The article “The WeisfeilerLeman Dimension of Planar Graphs is at most 3,” by Sandra Kiefer, Ilia Ponomarenko, and Pascal Schweitzer, was invited from the 32nd Annual ACM/ IEEE Symposium on Logic in Computer Science (LICS’17). The article “Computing the Geometric Intersection Number of Curves,” by Vincent Despré and Francis Lazarus, won best paper award at the 33rd International Symposium on Computational Geometry (SoCG 2017). We want to thank the LICS’17 and SoCG’17 Program Committees for their help in selecting these invited articles and editors Nachum Dershowitz and Jean-Daniel Boissonnat for handling the articles.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"29 1","pages":"1 - 1"},"PeriodicalIF":0.0,"publicationDate":"2019-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83981397","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 learning problems involving arbitrary classes of functions F, underlying measures μ, and targets Y. Because proper learning procedures, i.e., procedures that are only allowed to select functions in F, tend to perform poorly unless the problem satisfies some additional structural property (e.g., that F is convex), we consider unrestricted learning procedures that are free to choose functions outside the given class. We present a new unrestricted procedure whose sample complexity is almost the best that one can hope for and holds for (almost) any problem, including heavy-tailed situations. Moreover, the sample complexity coincides with what one could expect if F were convex, even when F is not. And if F is convex, then the unrestricted procedure turns out to be proper.
{"title":"An Unrestricted Learning Procedure","authors":"S. Mendelson","doi":"10.1145/3361699","DOIUrl":"https://doi.org/10.1145/3361699","url":null,"abstract":"We study learning problems involving arbitrary classes of functions F, underlying measures μ, and targets Y. Because proper learning procedures, i.e., procedures that are only allowed to select functions in F, tend to perform poorly unless the problem satisfies some additional structural property (e.g., that F is convex), we consider unrestricted learning procedures that are free to choose functions outside the given class. We present a new unrestricted procedure whose sample complexity is almost the best that one can hope for and holds for (almost) any problem, including heavy-tailed situations. Moreover, the sample complexity coincides with what one could expect if F were convex, even when F is not. And if F is convex, then the unrestricted procedure turns out to be proper.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"134 1","pages":"1 - 42"},"PeriodicalIF":0.0,"publicationDate":"2019-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75511194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In first-order term unification, variables represent well-formed terms over a given signature, and we are to solve equations built using function symbols from the signature and such variables; this problem is well-known to be decidable (in linear time). In second-order term unification, the variables take arguments (i.e., other terms) and a substitution uses those arguments an arbitrary number of times; for instance, an equation f(X(c),X(c)) = X(f(c,c)) has a solution X = •, where • is a special symbol denoting the place in which the argument is substituted. Under this substitution, both sides evaluate to f(c,c). There are other solutions, for instance X = f(•,•), which evaluates both sides tof(f(c,c),f(c,c)); in general, a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct. Second-order unification is in general undecidable. Context unification is a natural problem in between first- and second-order unification—we deal with equations over terms, the variables take arguments, but we restrict the set of solutions: The argument is used exactly once. Formally, contexts are terms with exactly one occurrence of the special symbol • and in context unification, we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. For instance, when the aforementioned equation f(X(c),X(c)) = X(f(c,c)) is treated as a context unification problem, then it has exactly one solution: X = •. Other substitutions that are solutions of it as an instance of the second-unification problem, say X = f(•, •), are not valid, as • is used more than once. Context unification also generalizes satisfiability of word equations, which is decidable (in PSPACE). The decidability status of context unification remained unknown for almost two decades. In this article, we show that context unification is in PSPACE (in EXPTIME , when tree regular constraints are also allowed). Those results are obtained by extending the recently developed recompression technique, which was previously defined for strings and used to obtain a new PSPACE algorithm for satisfiability of word equations. In this article, the technique is generalized to trees, and the corresponding algorithm is generalized from word equations to context unification. The idea of recompression is to apply simple compression rules (replacing pairs of neighboring function symbols) to the solution of the context equation; to this end, we appropriately modify the equation (without the knowledge of the actual solution) so compressing the solution can be simulated by compressing parts of the equation. It is shown that if the compression operations are appropriately chosen, then the size of the instance is polynomial during the whole algorithm, thus giving a PSPACE-upper bound.
{"title":"Deciding Context Unification","authors":"Artur Jeż","doi":"10.1145/3356904","DOIUrl":"https://doi.org/10.1145/3356904","url":null,"abstract":"In first-order term unification, variables represent well-formed terms over a given signature, and we are to solve equations built using function symbols from the signature and such variables; this problem is well-known to be decidable (in linear time). In second-order term unification, the variables take arguments (i.e., other terms) and a substitution uses those arguments an arbitrary number of times; for instance, an equation f(X(c),X(c)) = X(f(c,c)) has a solution X = •, where • is a special symbol denoting the place in which the argument is substituted. Under this substitution, both sides evaluate to f(c,c). There are other solutions, for instance X = f(•,•), which evaluates both sides tof(f(c,c),f(c,c)); in general, a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct. Second-order unification is in general undecidable. Context unification is a natural problem in between first- and second-order unification—we deal with equations over terms, the variables take arguments, but we restrict the set of solutions: The argument is used exactly once. Formally, contexts are terms with exactly one occurrence of the special symbol • and in context unification, we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. For instance, when the aforementioned equation f(X(c),X(c)) = X(f(c,c)) is treated as a context unification problem, then it has exactly one solution: X = •. Other substitutions that are solutions of it as an instance of the second-unification problem, say X = f(•, •), are not valid, as • is used more than once. Context unification also generalizes satisfiability of word equations, which is decidable (in PSPACE). The decidability status of context unification remained unknown for almost two decades. In this article, we show that context unification is in PSPACE (in EXPTIME , when tree regular constraints are also allowed). Those results are obtained by extending the recently developed recompression technique, which was previously defined for strings and used to obtain a new PSPACE algorithm for satisfiability of word equations. In this article, the technique is generalized to trees, and the corresponding algorithm is generalized from word equations to context unification. The idea of recompression is to apply simple compression rules (replacing pairs of neighboring function symbols) to the solution of the context equation; to this end, we appropriately modify the equation (without the knowledge of the actual solution) so compressing the solution can be simulated by compressing parts of the equation. It is shown that if the compression operations are appropriately chosen, then the size of the instance is polynomial during the whole algorithm, thus giving a PSPACE-upper bound.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"17 1","pages":"1 - 45"},"PeriodicalIF":0.0,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82955434","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}
Reasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, which are frequently used to mechanize mathematical results and carry out program verification efforts, equality is appallingly syntactic, and as a result, exploiting equivalences is cumbersome at best. Parametricity and univalence are two major concepts that have been explored in the literature to transport programs and proofs across type equivalences, but they fall short of achieving seamless, automatic transport. This work first clarifies the limitations of these two concepts when considered in isolation and then devises a fruitful marriage between both. The resulting concept, called univalent parametricity, is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences. Our approach handles both type and term dependency, as well as type-level computation. In addition to the theory of univalent parametricity, we present a lightweight framework implemented in the Coq proof assistant that allows the user to transparently transfer definitions and theorems for a type to an equivalent one, as if they were equal. For instance, this makes it possible to conveniently switch between an easy-to-reason-about representation and a computationally efficient representation as soon as they are proven equivalent. The combination of parametricity and univalence supports transport à la carte: basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences between functions over these types, in order to be able to transport more programs and proofs, as well as to yield more efficient terms. We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq. This work paves the way to easier-to-use proof assistants by supporting seamless programming and proving modulo equivalences.
{"title":"The Marriage of Univalence and Parametricity","authors":"Nicolas Tabareau, É. Tanter, Matthieu Sozeau","doi":"10.1145/3429979","DOIUrl":"https://doi.org/10.1145/3429979","url":null,"abstract":"Reasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, which are frequently used to mechanize mathematical results and carry out program verification efforts, equality is appallingly syntactic, and as a result, exploiting equivalences is cumbersome at best. Parametricity and univalence are two major concepts that have been explored in the literature to transport programs and proofs across type equivalences, but they fall short of achieving seamless, automatic transport. This work first clarifies the limitations of these two concepts when considered in isolation and then devises a fruitful marriage between both. The resulting concept, called univalent parametricity, is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences. Our approach handles both type and term dependency, as well as type-level computation. In addition to the theory of univalent parametricity, we present a lightweight framework implemented in the Coq proof assistant that allows the user to transparently transfer definitions and theorems for a type to an equivalent one, as if they were equal. For instance, this makes it possible to conveniently switch between an easy-to-reason-about representation and a computationally efficient representation as soon as they are proven equivalent. The combination of parametricity and univalence supports transport à la carte: basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences between functions over these types, in order to be able to transport more programs and proofs, as well as to yield more efficient terms. We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq. This work paves the way to easier-to-use proof assistants by supporting seamless programming and proving modulo equivalences.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"47 1","pages":"1 - 44"},"PeriodicalIF":0.0,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73465350","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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