Topic

# Sequence

About: Sequence is a research topic. Over the lifetime, 20478 publications have been published within this topic receiving 314110 citations. The topic is also known as: ordered list.

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TL;DR: The proposed concept of compressibility is shown to play a role analogous to that of entropy in classical information theory where one deals with probabilistic ensembles of sequences rather than with individual sequences.

Abstract: Compressibility of individual sequences by the class of generalized finite-state information-lossless encoders is investigated. These encoders can operate in a variable-rate mode as well as a fixed-rate one, and they allow for any finite-state scheme of variable-length-to-variable-length coding. For every individual infinite sequence x a quantity \rho(x) is defined, called the compressibility of x , which is shown to be the asymptotically attainable lower bound on the compression ratio that can be achieved for x by any finite-state encoder. This is demonstrated by means of a constructive coding theorem and its converse that, apart from their asymptotic significance, also provide useful performance criteria for finite and practical data-compression tasks. The proposed concept of compressibility is also shown to play a role analogous to that of entropy in classical information theory where one deals with probabilistic ensembles of sequences rather than with individual sequences. While the definition of \rho(x) allows a different machine for each different sequence to be compressed, the constructive coding theorem leads to a universal algorithm that is asymptotically optimal for all sequences.

3,753 citations

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TL;DR: In this paper, the EM algorithm converges to a local maximum or a stationary value of the (incomplete-data) likelihood function under conditions that are applicable to many practical situations.

Abstract: Two convergence aspects of the EM algorithm are studied: (i) does the EM algorithm find a local maximum or a stationary value of the (incomplete-data) likelihood function? (ii) does the sequence of parameter estimates generated by EM converge? Several convergence results are obtained under conditions that are applicable to many practical situations Two useful special cases are: (a) if the unobserved complete-data specification can be described by a curved exponential family with compact parameter space, all the limit points of any EM sequence are stationary points of the likelihood function; (b) if the likelihood function is unimodal and a certain differentiability condition is satisfied, then any EM sequence converges to the unique maximum likelihood estimate A list of key properties of the algorithm is included

3,414 citations

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TL;DR: In this paper, the authors describe a problem solver called STRIPS that attempts to find a sequence of operators in a space of world models to transform a given initial world model in which a given goal formula can be proven to be true.

Abstract: We describe a new problem solver called STRIPS that attempts to find a sequence of operators in a space of world models to transform a given initial world model in which a given goal formula can be proven to be true. STRIPS represents a world model as an arbitrary collection in first-order predicate calculus formulas and is designed to work with models consisting of large numbers of formula. It employs a resolution theorem prover to answer questions of particular models and uses means-ends analysis to guide it to the desired goal-satisfying model.

2,883 citations

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TL;DR: A new approach to the problem of evaluating the complexity ("randomness") of finite sequences is presented, related to the number of steps in a self-delimiting production process by which a given sequence is presumed to be generated.

Abstract: A new approach to the problem of evaluating the complexity ("randomness") of finite sequences is presented. The proposed complexity measure is related to the number of steps in a self-delimiting production process by which a given sequence is presumed to be generated. It is further related to the number of distinct substrings and the rate of their occurrence along the sequence. The derived properties of the proposed measure are discussed and motivated in conjunction with other well-established complexity criteria.

2,473 citations

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01 Mar 1953

TL;DR: In this article, it is shown that the Schauder fixpoint theorem can play a somewhat analogous role in the theory of divergent iteration processes, and that the same methods can be used to prove that a given problem has a solution.

Abstract: Due largely to the works of Cesaro, Fejer, and Toeplitz, mean value methods have become famous in the summation of divergent series. The purpose of this paper is to show that the same methods can play a somewhat analogous role in the theory of divergent iteration processes. We shall consider iteration from the limited but nevertheless important point of view of an applied mathematician trying to use a method of successive approximations on some boundary value problem which may be either linear or nonlinear. It is now widely known that the Schauder fixpoint theorem [1] is a powerful method for proving existence theorems. If one wishes to use it to prove that a given problem has a solution, he proceeds by associating with the problem a convex compact set E in some Banach space, and a continuous transformation T which carries E into itself. Schauder's theorem asserts that T must have at least one fixpoint, say p, in E. If E and T have been appropriately chosen, it can then usually be shown that any such fixpoint must be a solution of the original problem and conversely. Mathematical literature since about 1935 abounds with illustrations of this technique. We mention here only [2] and [3] which contain the genesis of the present work. Let us then begin with a convex compact set E in a Banach space, and a continuous transformation T carrying E into itself. The problem which we shall consider is that of constructing in E a sequence of elements { x,n } that converge to a fixpoint of T. Ordinarily one starts by choosing more or less arbitrarily an initial point xi in E and then considering the successive iterates { xn } of xi under T, where

2,199 citations