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On the Theory of Average Case Complexity
 Journal of Computer and System Sciences
, 1997
"... This paper takes the next step in developing the theory of average case complexity initiated by Leonid A Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. ..."
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Cited by 125 (6 self)
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This paper takes the next step in developing the theory of average case complexity initiated by Leonid A Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. Our results include: ffl the equivalence of search and decision problems in the context of average case complexity; ffl an initial analysis of the structure of distributionalNP (i.e. NP problems coupled with "simple distributions") under reductions which preserve average polynomialtime; ffl a proof that if all of distributionalNP is in average polynomialtime then nondeterministic exponentialtime equals deterministic exponential time (i.e., a collapse in the worst case hierarchy); ffl definitions and basic theorems regarding other complexity classes such as average logspace. An exposition of the basic definitions suggested by Levin and suggestions for some alternative definitions ...
A Faster Distributed Protocol for Constructing a Minimum Spanning Tree
 In Proc. of the 15 th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... Abstract This paper studies the problem of constructing a minimumweight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art to ..."
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Cited by 43 (3 self)
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Abstract This paper studies the problem of constructing a minimumweight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today isa protocol with running time O(\Lambda (G) + pn * log * n) due to Kutten and Peleg [22], where \Lambda (G) denotesthe diameter of the graph G. Peleg and Rubinovich [29] have shown that ~\Omega (pn) time is required forconstructing an MST even on graphs of small diameter, and claimed that their result &quot;establishes the asymptotic nearoptimality &quot; of the protocol of [22].In this paper we refine this claim, and devise a protocol that constructs the MST in ~ O(u(G,!)+pn)rounds, where u(G,!) is the MSTradius of the graph. The ratio between the diameter and the MSTradius may be as large as \Theta ( n), and, consequently, on some inputs our protocol is faster than theprotocol of [22] by a factor of ~\Omega (p n). Also, on every input, the running time of our protocol is nevergreater than twice the running time of the protocol of [22].
An overview of computational complexity
 Communications of the ACM
, 1983
"... foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving P ..."
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foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that &quot;Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NPcompleteness. The ensuing exploration of the boundaries and nature of the NPcomplete class of problems has been one of the most active and important research activities in computer science for the last decade. Cook is well known for his influential results in fundamental areas of computer science. He has made significant contributions to complexity theory, to timespace tradeoffs in computation, and to logics for programming languages. His work is characterized by elegance and insights and has illuminated the very nature of computation.&quot; During 19701979, Cook did extensive work under grants from the
A system for incremental learning based on algorithmic probability
 Probability,” Proceedings of the Sixth Israeli Conference on Artificial Intelligence, Computer Vision and Pattern Recognition
, 1989
"... We have employed Algorithmic Probability Theory to construct a system for machine learning of great power and generality. The principal thrust of present research is the design of sequences of problems to train this system. Current programs for machine learning are limited in the kinds of concepts a ..."
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Cited by 22 (7 self)
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We have employed Algorithmic Probability Theory to construct a system for machine learning of great power and generality. The principal thrust of present research is the design of sequences of problems to train this system. Current programs for machine learning are limited in the kinds of concepts accessible to them, the kinds of problems they can learn to solve, and in the efficiency with which they learn — both in computation time needed and/or in amount of data needed for learning. Algorithmic Probability Theory provides a general model of the learning process that enables us to understand and surpass many of these limitations. Starting with a machine containing a small set of concepts, we use a carefully designed sequence of problems of increasing difficulty to bring the machine to a high level of problem solving skill. The use of training sequences of problems for machine knowledge acquisition promises to yield Expert Systems that will be easier to train and free of the brittleness that characterizes the narrow specialization of present day systems of this sort. It is also expected that the present research will give needed insight in the design of training sequences for human learning.
Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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Cited by 20 (9 self)
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
Notes on Levin's Theory of AverageCase Complexity
 Electronic Colloquium on Computational Complexity
, 1997
"... Abstract. In 1984, Leonid Levin initiated a theory of averagecase complexity. We provide an exposition of the basic definitions suggested by Levin, and discuss some of the considerations underlying these definitions. Keywords: Averagecase complexity, reductions. This survey is rooted in the author ..."
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Cited by 19 (2 self)
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Abstract. In 1984, Leonid Levin initiated a theory of averagecase complexity. We provide an exposition of the basic definitions suggested by Levin, and discuss some of the considerations underlying these definitions. Keywords: Averagecase complexity, reductions. This survey is rooted in the author’s (exposition and exploration) work [4], which was partially reproduded in [1]. An early version of this survey appeared as TR97058 of ECCC. Some of the perspective and conclusions were revised in light of a relatively recent work of Livne [21], but an attempt was made to preserve the spirit of the original survey. The author’s current perspective is better reflected in [7, Sec. 10.2] and [8], which advocate somewhat different definitional choices (e.g., focusing on typical rather than average performace of algorithms). 1
Autonomous theory building systems
, 1991
"... We are interested in very general systems which are programmed once and which from then on learn autonomously all sorts of things simply by observing a sequence of input data. In this preliminary note we give examples of techniques which apparently permit to deal with two basic aspects related ..."
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Cited by 13 (4 self)
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We are interested in very general systems which are programmed once and which from then on learn autonomously all sorts of things simply by observing a sequence of input data. In this preliminary note we give examples of techniques which apparently permit to deal with two basic aspects related
Progress in Incremental Machine Learning
, 2003
"... We will describe recent developments in a system for machine learning that we've been working on for some time (Sol 86, Sol 89). It is meant to be a "Scientist's Assistant" of great power and versatility in many areas of science and mathematics. It di#ers from other ambitious ..."
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Cited by 12 (1 self)
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We will describe recent developments in a system for machine learning that we've been working on for some time (Sol 86, Sol 89). It is meant to be a "Scientist's Assistant" of great power and versatility in many areas of science and mathematics. It di#ers from other ambitious work in this area in that we are not so much interested in knowledge itself, as we are in how it is acquired  how machines may learn. To start o#, the system will learn to solve two very general kinds of problems. Most, but perhaps not all problems in science and engineering are of these two kinds.