Stephen Alstrup, Haim Kaplan, M. Thorup, Uri Zwick
We describe a way of assigning labels to the vertices of any undirected graph on up to n vertices, each composed of n/2+O(1) bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an n/2+O(log n) bound of Moon. As a consequence, we obtain an induced-universal graph for n-vertex graphs containing only O(2n/2) vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs.
{"title":"Adjacency Labeling Schemes and Induced-Universal Graphs","authors":"Stephen Alstrup, Haim Kaplan, M. Thorup, Uri Zwick","doi":"10.1145/2746539.2746545","DOIUrl":"https://doi.org/10.1145/2746539.2746545","url":null,"abstract":"We describe a way of assigning labels to the vertices of any undirected graph on up to n vertices, each composed of n/2+O(1) bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an n/2+O(log n) bound of Moon. As a consequence, we obtain an induced-universal graph for n-vertex graphs containing only O(2n/2) vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84541718","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}
Bregman divergences are important distance measures that are used in applications such as computer vision, text mining, and speech processing, and are a focus of interest in machine learning due to their information-theoretic properties. There has been extensive study of algorithms for clustering and near neighbor search with respect to these divergences. In all cases, the guarantees depend not just on the data size n and dimensionality d, but also on a structure constant μ ≥ 1 that depends solely on a generating convex function φ and can grow without bound independently. In general, this μ parametrizes the degree to which a given divergence is "asymmetric". In this paper, we provide the first evidence that this dependence on μ might be intrinsic. We focus on the problem of ac{ann} search for Bregman divergences. We show that under the cell probe model, any non-adaptive data structure (like locality-sensitive hashing) for c-approximate near-neighbor search that admits r probes must use space Ω(dn1 + μ/c r). In contrast for LSH under l1 the best bound is Ω(dn1+ 1/cr). Our results interpolate between known lower bounds both for LSH-based ANN under l1 as well as the generally harder Partial Match problem (in non-adaptive settings). The bounds match the former when μ is small and the latter when μ is Ω(d/log n). This further strengthens the intuition that Partial Match corresponds to an "asymmetric" version of ANN, as well as opening up the possibility of a new line of attack for lower bounds on Partial Match. Our new tool is a directed variant of the standard boolean noise operator. We prove a generalization of the Bonami-Beckner hypercontractivity inequality (restricted to certain subsets of the Hamming cube), and use this to prove the desired directed isoperimetric inequality that we use in our data structure lower bound.
{"title":"A Directed Isoperimetric Inequality with application to Bregman Near Neighbor Lower Bounds","authors":"A. Abdullah, Suresh Venkatasubramanian","doi":"10.1145/2746539.2746595","DOIUrl":"https://doi.org/10.1145/2746539.2746595","url":null,"abstract":"Bregman divergences are important distance measures that are used in applications such as computer vision, text mining, and speech processing, and are a focus of interest in machine learning due to their information-theoretic properties. There has been extensive study of algorithms for clustering and near neighbor search with respect to these divergences. In all cases, the guarantees depend not just on the data size n and dimensionality d, but also on a structure constant μ ≥ 1 that depends solely on a generating convex function φ and can grow without bound independently. In general, this μ parametrizes the degree to which a given divergence is \"asymmetric\". In this paper, we provide the first evidence that this dependence on μ might be intrinsic. We focus on the problem of ac{ann} search for Bregman divergences. We show that under the cell probe model, any non-adaptive data structure (like locality-sensitive hashing) for c-approximate near-neighbor search that admits r probes must use space Ω(dn1 + μ/c r). In contrast for LSH under l1 the best bound is Ω(dn1+ 1/cr). Our results interpolate between known lower bounds both for LSH-based ANN under l1 as well as the generally harder Partial Match problem (in non-adaptive settings). The bounds match the former when μ is small and the latter when μ is Ω(d/log n). This further strengthens the intuition that Partial Match corresponds to an \"asymmetric\" version of ANN, as well as opening up the possibility of a new line of attack for lower bounds on Partial Match. Our new tool is a directed variant of the standard boolean noise operator. We prove a generalization of the Bonami-Beckner hypercontractivity inequality (restricted to certain subsets of the Hamming cube), and use this to prove the desired directed isoperimetric inequality that we use in our data structure lower bound.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77238581","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 consider some classical flow-time minimization problems in the unrelated machines setting. In this setting, there is a set of m machines and a set of n jobs, and each job j has a machine dependent processing time of pij on machine i. The flow-time of a job is the amount of time the job spends in a system (its completion time minus its arrival time), and is one of the most natural measure of quality of service. We show the following two results: an $O(min(log2 n, log n log P)) approximation algorithm for minimizing the total flow-time, and an O(log n) approximation for minimizing the maximum flow-time. Here P is the ratio of maximum to minimum job size. These are the first known poly-logarithmic guarantees for both the problems.
考虑了一些经典的不相关机器设置下的流时间最小化问题。在此设置中，有一组m台机器和一组n个作业，每个作业j在机器i上的处理时间pij与机器相关。作业的流时间是作业在系统中花费的时间(其完成时间减去其到达时间)，是最自然的服务质量度量之一。我们展示了以下两个结果:一个$O(min(log2 n, log n log P))近似算法用于最小化总流时间，一个$O(log n)近似算法用于最小化最大流时间。这里P是最大作业大小与最小作业大小的比值。这是已知的第一个针对这两个问题的多对数保证。
{"title":"Minimizing Flow-Time on Unrelated Machines","authors":"N. Bansal, Janardhan Kulkarni","doi":"10.1145/2746539.2746601","DOIUrl":"https://doi.org/10.1145/2746539.2746601","url":null,"abstract":"We consider some classical flow-time minimization problems in the unrelated machines setting. In this setting, there is a set of m machines and a set of n jobs, and each job j has a machine dependent processing time of pij on machine i. The flow-time of a job is the amount of time the job spends in a system (its completion time minus its arrival time), and is one of the most natural measure of quality of service. We show the following two results: an $O(min(log2 n, log n log P)) approximation algorithm for minimizing the total flow-time, and an O(log n) approximation for minimizing the maximum flow-time. Here P is the ratio of maximum to minimum job size. These are the first known poly-logarithmic guarantees for both the problems.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73803590","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 problem of identifying a planted assignment given a random k-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with O(n log n) clauses, there are distributions over clauses for which the best known efficient algorithms require nk/2 clauses. We propose and study a unified model for planted k-SAT, which captures well-known special cases. An instance is described by a planted assignment σ and a distribution on clauses with k literals. We define its distribution complexity as the largest r for which the distribution is not r-wise independent (1 ≤ r ≤ k for any distribution with a planted assignment). Our main result is an unconditional lower bound, tight up to logarithmic factors, of Ω(nr/2) clauses for statistical algorithms, matching the known upper bound (which, as we show, can be implemented using a statistical algorithm). Since known approaches for problems over distributions have statistical analogues (spectral, MCMC, gradient-based, convex optimization etc.), this lower bound provides a rigorous explanation of the observed algorithmic gap. The proof introduces a new general technique for the analysis of statistical algorithms. It also points to a geometric paring phenomenon in the space of all planted assignments. We describe consequences of our lower bounds to Feige's refutation hypothesis and to lower bounds on general convex programs that solve planted k-SAT. Our bounds also extend to the planted k-CSP model, defined by Goldreich as a candidate for one-way function, and therefore provide concrete evidence for the security of Goldreich's one-way function and the associated pseudorandom generator when used with a sufficiently hard predicate.
{"title":"On the Complexity of Random Satisfiability Problems with Planted Solutions","authors":"Vitaly Feldman, Will Perkins, S. Vempala","doi":"10.1145/2746539.2746577","DOIUrl":"https://doi.org/10.1145/2746539.2746577","url":null,"abstract":"The problem of identifying a planted assignment given a random k-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with O(n log n) clauses, there are distributions over clauses for which the best known efficient algorithms require nk/2 clauses. We propose and study a unified model for planted k-SAT, which captures well-known special cases. An instance is described by a planted assignment σ and a distribution on clauses with k literals. We define its distribution complexity as the largest r for which the distribution is not r-wise independent (1 ≤ r ≤ k for any distribution with a planted assignment). Our main result is an unconditional lower bound, tight up to logarithmic factors, of Ω(nr/2) clauses for statistical algorithms, matching the known upper bound (which, as we show, can be implemented using a statistical algorithm). Since known approaches for problems over distributions have statistical analogues (spectral, MCMC, gradient-based, convex optimization etc.), this lower bound provides a rigorous explanation of the observed algorithmic gap. The proof introduces a new general technique for the analysis of statistical algorithms. It also points to a geometric paring phenomenon in the space of all planted assignments. We describe consequences of our lower bounds to Feige's refutation hypothesis and to lower bounds on general convex programs that solve planted k-SAT. Our bounds also extend to the planted k-CSP model, defined by Goldreich as a candidate for one-way function, and therefore provide concrete evidence for the security of Goldreich's one-way function and the associated pseudorandom generator when used with a sufficiently hard predicate.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82688796","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}
Jakub Lacki, Jakub Ocwieja, Marcin Pilipczuk, P. Sankowski, Anna Zych
In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an n-vertex graph G=(V,E,w) with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set S ⊆ V, which changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). Our task here is twofold. We want to support updates in sublinear o(n) time, and keep the approximation factor of the algorithm as small as possible. We show that we can maintain a (6+ε)-approximate Steiner tree of a general graph in ~O(√n log D) time per terminal addition or removal. Here, strecz denotes the stretch of the metric induced by G. For planar graphs we achieve the same running time and the approximation ratio of (2+ε). Moreover, we show faster algorithms for incremental and decremental scenarios. Finally, we show that if we allow higher approximation ratio, even more efficient algorithms are possible. In particular we show a polylogarithmic time (4+ε)-approximate algorithm for planar graphs. One of the main building blocks of our algorithms are dynamic distance oracles for vertex-labeled graphs, which are of independent interest. We also improve and use the online algorithms for the Steiner tree problem.
{"title":"The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree","authors":"Jakub Lacki, Jakub Ocwieja, Marcin Pilipczuk, P. Sankowski, Anna Zych","doi":"10.1145/2746539.2746615","DOIUrl":"https://doi.org/10.1145/2746539.2746615","url":null,"abstract":"In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an n-vertex graph G=(V,E,w) with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set S ⊆ V, which changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). Our task here is twofold. We want to support updates in sublinear o(n) time, and keep the approximation factor of the algorithm as small as possible. We show that we can maintain a (6+ε)-approximate Steiner tree of a general graph in ~O(√n log D) time per terminal addition or removal. Here, strecz denotes the stretch of the metric induced by G. For planar graphs we achieve the same running time and the approximation ratio of (2+ε). Moreover, we show faster algorithms for incremental and decremental scenarios. Finally, we show that if we allow higher approximation ratio, even more efficient algorithms are possible. In particular we show a polylogarithmic time (4+ε)-approximate algorithm for planar graphs. One of the main building blocks of our algorithms are dynamic distance oracles for vertex-labeled graphs, which are of independent interest. We also improve and use the online algorithms for the Steiner tree problem.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84718122","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}
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for k = Θ(√n). In this paper we study the complexity of the planted clique problem under algorithms from the Sum-Of-Squares hierarchy. We prove the first average case lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the SOS hierarchy cannot find a planted k-clique unless k ≥ (√n/log n)1/rCr. Thus, for any constant number of rounds planted cliques of size no(1) cannot be found by this powerful class of algorithms. This is shown via an integrability gap for the natural formulation of maximum clique problem on random graphs for SOS and Lasserre hierarchies, which in turn follow from degree lower bounds for the Positivestellensatz proof system. We follow the usual recipe for such proofs. First, we introduce a natural "dual certificate" (also known as a "vector-solution" or "pseudo-expectation") for the given system of polynomial equations representing the problem for every fixed input graph. Then we show that the matrix associated with this dual certificate is PSD (positive semi-definite) with high probability over the choice of the input graph.This requires the use of certain tools. One is the theory of association schemes, and in particular the eigenspaces and eigenvalues of the Johnson scheme. Another is a combinatorial method we develop to compute (via traces) norm bounds for certain random matrices whose entries are highly dependent; we hope this method will be useful elsewhere.
{"title":"Sum-of-squares Lower Bounds for Planted Clique","authors":"R. Meka, Aaron Potechin, A. Wigderson","doi":"10.1145/2746539.2746600","DOIUrl":"https://doi.org/10.1145/2746539.2746600","url":null,"abstract":"Finding cliques in random graphs and the closely related \"planted\" clique variant, where a clique of size k is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for k = Θ(√n). In this paper we study the complexity of the planted clique problem under algorithms from the Sum-Of-Squares hierarchy. We prove the first average case lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the SOS hierarchy cannot find a planted k-clique unless k ≥ (√n/log n)1/rCr. Thus, for any constant number of rounds planted cliques of size no(1) cannot be found by this powerful class of algorithms. This is shown via an integrability gap for the natural formulation of maximum clique problem on random graphs for SOS and Lasserre hierarchies, which in turn follow from degree lower bounds for the Positivestellensatz proof system. We follow the usual recipe for such proofs. First, we introduce a natural \"dual certificate\" (also known as a \"vector-solution\" or \"pseudo-expectation\") for the given system of polynomial equations representing the problem for every fixed input graph. Then we show that the matrix associated with this dual certificate is PSD (positive semi-definite) with high probability over the choice of the input graph.This requires the use of certain tools. One is the theory of association schemes, and in particular the eigenspaces and eigenvalues of the Johnson scheme. Another is a combinatorial method we develop to compute (via traces) norm bounds for certain random matrices whose entries are highly dependent; we hope this method will be useful elsewhere.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76420996","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 consider a generalization, which we call the Shannon switching game on vertices, of a familiar board game called HEX. We show that determining who wins such a game if each player plays perfectly is very hard; in fact, it is as hard as carrying out any polynomial-space-bounded computation. This result suggests that the theory of combinatorial games is difficult.
{"title":"A combinatorial problem which is complete in polynomial space","authors":"S. Even, R. Tarjan","doi":"10.1145/800116.803754","DOIUrl":"https://doi.org/10.1145/800116.803754","url":null,"abstract":"We consider a generalization, which we call the Shannon switching game on vertices, of a familiar board game called HEX. We show that determining who wins such a game if each player plays perfectly is very hard; in fact, it is as hard as carrying out any polynomial-space-bounded computation. This result suggests that the theory of combinatorial games is difficult.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1975-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77977198","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 discuss the problem of generating code for a wide class of machines, restricting ourselves to the computation of expression trees. After defining a broad class of machines and discussing the properties of optimal programs on these machines, we derive a necessary and sufficient condition which can be used to prove the optimality of any code generation algorithm for expression trees on this class. We then present a dynamic programming algorithm which produces optimal code for any machine in the class; this algorithm runs in time which is linearly proportional to the number of vertices in an expression tree.
{"title":"Optimal code generation for expression trees","authors":"A. Aho, Stephen C. Johnson","doi":"10.1145/800116.803770","DOIUrl":"https://doi.org/10.1145/800116.803770","url":null,"abstract":"We discuss the problem of generating code for a wide class of machines, restricting ourselves to the computation of expression trees. After defining a broad class of machines and discussing the properties of optimal programs on these machines, we derive a necessary and sufficient condition which can be used to prove the optimality of any code generation algorithm for expression trees on this class. We then present a dynamic programming algorithm which produces optimal code for any machine in the class; this algorithm runs in time which is linearly proportional to the number of vertices in an expression tree.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1975-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76068705","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 paper is about mathematical problems in programming language semantics and their influence on recursive function theory. We define a notion of computability on continuous higher types (for all types) and show its equivalence to effective operators. This result shows that our computable operators can model mathematically (i.e. extensionally) everything that can be done in an operational semantics. These new recursion theoretic concepts which are appropriate to semantics also allow us to construct Scott models for the &lgr;-calculus which contain all and only computable elements. Depending on the choice of the initial cpo, our general theory yields a theory for either strictly determinate or else arbitrary non-deterministic objects (parallelism). The formal theory is developed in part II of this paper. Part I gives motivation and comparison with related work.
{"title":"Computability concepts for programming language semantics","authors":"H. Egli, R. Constable","doi":"10.1145/800116.803757","DOIUrl":"https://doi.org/10.1145/800116.803757","url":null,"abstract":"This paper is about mathematical problems in programming language semantics and their influence on recursive function theory. We define a notion of computability on continuous higher types (for all types) and show its equivalence to effective operators. This result shows that our computable operators can model mathematically (i.e. extensionally) everything that can be done in an operational semantics. These new recursion theoretic concepts which are appropriate to semantics also allow us to construct Scott models for the &lgr;-calculus which contain all and only computable elements. Depending on the choice of the initial cpo, our general theory yields a theory for either strictly determinate or else arbitrary non-deterministic objects (parallelism). The formal theory is developed in part II of this paper. Part I gives motivation and comparison with related work.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1975-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88731995","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 this paper a new fixedpoint approach towards the semantics of recursive programs is presented. The fixedpoint defined by a recursive program under this semantics contains, in some sense, the maximal amount of “interesting” information which can be extracted from the program. This optimal fixedpoint (which always uniquely exists) may be strictly more defined than the program's least fixedpoint. We consider both the theoretical and the computational aspects of the approach, as well as some techniques for proving properties of the optimal fixedpoint of a given recursive program.
{"title":"The optimal fixedpoint of recursive programs","authors":"Z. Manna, A. Shamir","doi":"10.1145/800116.803769","DOIUrl":"https://doi.org/10.1145/800116.803769","url":null,"abstract":"In this paper a new fixedpoint approach towards the semantics of recursive programs is presented. The fixedpoint defined by a recursive program under this semantics contains, in some sense, the maximal amount of “interesting” information which can be extracted from the program. This optimal fixedpoint (which always uniquely exists) may be strictly more defined than the program's least fixedpoint. We consider both the theoretical and the computational aspects of the approach, as well as some techniques for proving properties of the optimal fixedpoint of a given recursive program.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1975-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91301545","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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