The multiplicative coalescent is a Markov process taking values in ordered @math . It is a mean-field process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (1998) each extreme eternal version @math of the multiplicative coalescent was described in three different ways. One of these specifications matches the (marginal) law of @math to that of the ordered excursion lengths above past minima of @math , where @math is a certain Levy-type process which (modulo shift and scaling) has infinitesimal drift @math at time @math . Using a modification of the breadth-first-walk construction from Aldous (1997) and Aldous and Limic (1998), and some new insight from the thesis by Uribe (2007), this work settles an open problem (3) from Aldous (1997), in the more general context of Aldous and Limic (1998). Informally speaking, @math is entirely encoded by @math , and contrary to Aldous' original intuition, the evolution of time for @math does correspond to the linear increase in the constant part of the drift of @math . In the "standard multiplicative coalescent" context of Aldous (1997), this result was first announced by Armendariz in 2001, and obtained in a recent preprint by Broutin and Marckert, who simultaneously account for the process of excess edge counts (or marks). The novel argument presented here is based on a sequence of relatively elementary observations. Some of its components (for example, the new dynamic random graph construction via "simultaneous" breadth-first walks) are of independent interest, and may be useful for obtaining more sophisticated asymptotic results on near critical random graphs and related processes.
The arguments presented in the sequel are elementary in part due to direct applications of a non-trivial result from @cite_10 , Section 2.6 (depending on @cite_10 , Section 2.5) in Section (more precisely, Corollary ). In comparison, (a) @cite_50 also rely on the convergence results of @cite_14 in the standard setting, as well as additional estimates proved in @cite_8 , and (b) the analysis done in @cite_33 , Sections 4 and 5 seems to be a formal analogue of that in @cite_10 , Sections 2.5-2.6.
@cite_14 @cite_33 @cite_8 @cite_50 @cite_10
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Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. We propose a new task in the field of computational argumentation in which we investigate qualitative properties of Web arguments, namely their convincingness. We cast the problem as relation classification, where a pair of arguments having the same stance to the same prompt is judged. We annotate a large datasets of 16k pairs of arguments over 32 topics and investigate whether the relation “A is more convincing than B” exhibits properties of total ordering; these findings are used as global constraints for cleaning the crowdsourced data. We propose two tasks: (1) predicting which argument from an argument pair is more convincing and (2) ranking all arguments to the topic based on their convincingness. We experiment with feature-rich SVM and bidirectional LSTM and obtain 0.76-0.78 accuracy and 0.35-0.40 Spearman’s correlation in a cross-topic evaluation. We release the newly created corpus UKPConvArg1 and the experimental software under open licenses. Researchers in both machine translation (e.g., 1990) and bilingual lexicography (e.g., Klavans and Tzoukermann 1990) have recently become interested in studying bilingual corpora, bodies of text such as the Canadian Hansards (parliamentary proceedings), which are available in multiple languages (such as French and English). One useful step is to align the sentences, that is, to identify correspondences between sentences in one language and sentences in the other language.This paper will describe a method and a program (align) for aligning sentences based on a simple statistical model of character lengths. The program uses the fact that longer sentences in one language tend to be translated into longer sentences in the other language, and that shorter sentences tend to be translated into shorter sentences. A probabilistic score is assigned to each proposed correspondence of sentences, based on the scaled difference of lengths of the two sentences (in characters) and the variance of this difference. This probabilistic score is used in a dynamic programming framework to find the maximum likelihood alignment of sentences.It is remarkable that such a simple approach works as well as it does. An evaluation was performed based on a trilingual corpus of economic reports issued by the Union Bank of Switzerland (UBS) in English, French, and German. The method correctly aligned all but 4 of the sentences. Moreover, it is possible to extract a large subcorpus that has a much smaller error rate. By selecting the best-scoring 80 of the alignments, the error rate is reduced from 4 to 0.7 . There were more errors on the English-French subcorpus than on the English-German subcorpus, showing that error rates will depend on the corpus considered; however, both were small enough to hope that the method will be useful for many language pairs.To further research on bilingual corpora, a much larger sample of Canadian Hansards (approximately 90 million words, half in English and and half in French) has been aligned with the align program and will be available through the Data Collection Initiative of the Association for Computational Linguistics (ACL DCI). In addition, in order to facilitate replication of the align program, an appendix is provided with detailed c-code of the more difficult core of the align program.