AbstractThis note addresses the input and output of intervals in the sense of intervalarithmetic and interval constraints. The most obvious, and so far most widely usednotation, for intervals has drawbacks that we remedy with a new notation that wepropose to call factored notation. It is more compact and allows one to ﬁnd a goodtrade-oﬀ between interval width and ease of reading. We describe how such a trade-oﬀ can be based on the information yield (in the sense of information theory) of thelast decimal shown. 1 Introduction Once upon a time, it was a matter of professional ethics among computers never to writea meaningless decimal. Since then computers have become machines and thereby lost anyform of ethics, professional or otherwise. The human computers of yore were helped intheir ethical behaviour by the fact that it took eﬀort to write spurious decimals. Now thesituation is reversed: the lazy way is to use the default precision of the I O library function.As a result it is common to see ﬁfteen decimals, all but three of which are meaningless.Of course interval arithmetic is not guilty of such negligence. After all, the very raisond’ˆetre of the subject is to be explicit about the precision of computed results. Yet, eveninterval arithmetic is plagued by phoney decimals, albeit in a more subtle way. Just as con-ventional computation often needs more care in the presentation of computational results,the most obvious interval notation with default precision needs improvement.As a bounded interval has two bounds, say, l and u, the most straightforward notationis something like [l,u]. Written like this, it may not be immediately obvious what is wrongwith writing it that way. But when confronted with a real-life consequence
The standard notation in the Numerica book @cite_2 solves the scanning problem in an interesting way. It uses the idea of the @math notation, but writes instead @math . This variation has the advantage of not introducing new notation. The reason why we still prefer factored notation is clear from the @math example ), which, if rewritten as @math becomes @math Although it is attractive not to introduce special-purpose notation, there is so much redundancy here that the factored alternative: @math seems worth the new notation.
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We consider document listing on string collections, that is, finding in which strings a given pattern appears. In particular, we focus on repetitive collections: a collection of size @math over alphabet @math is composed of @math copies of a string of size @math , and @math edits are applied on ranges of copies. We introduce the first document listing index with size @math , precisely @math bits, and with useful worst-case time guarantees: Given a pattern of length @math , the index reports the @math strings where it appears in time @math , for any constant @math (and tells in time @math if @math ). Our technique is to augment a range data structure that is commonly used on grammar-based indexes, so that instead of retrieving all the pattern occurrences, it computes useful summaries on them. We show that the idea has independent interest: we introduce the first grammar-based index that, on a text @math with a grammar of size @math , uses @math bits and counts the number of occurrences of a pattern @math in time @math , for any constant @math . We also give the first index using @math bits, where @math is parsed by Lempel-Ziv into @math phrases, counting occurrences in time @math . This paper describes our system designed for the NLPCC 2015 shared task on Chinese word segmentation WS and POS tagging for Weibo Text. We treat WS and POS tagging as two separate tasks and use a cascaded approach. Our major focus is how to effectively exploit multiple heterogeneous data to boost performance of statistical models. This work considers three sets of heterogeneous data, i.e., Weibo @math , 10K sentences, Penn Chinese Treebank 7.0 @math , 50K, and People's Daily @math , 280K. For WS, we adopt the recently proposed coupled sequence labeling to combine @math , @math , and @math , boosting F1 score from @math baseline model trained on only @math to @math @math . For POS tagging, we adopt an ensemble approach combining coupled sequence labeling and the guide-feature based method, since the three datasets have three different annotation standards. First, we convert @math into the annotation style of @math based on coupled sequence labeling, denoted by @math . Then, we merge CTB7 and @math to train a POS tagger, denoted by @math , which is further used to produce guide features on @math . Finally, the tagging F1 score is improved from 87.93 to 88.99 +1.06 . A traditional counterexample to a linear-time safety property shows the values of all signals at all times prior to the error. However, some signals may not be critical to causing the failure. A succinct explanation may help human understanding as well as speed up algorithms that have to analyze many such traces. In Bounded Model Checking (BMC), a counterexample is constructed from a satisfying assignment to a Boolean formula, typically in CNF. Modern SAT solvers usually assign values to all variables when the input formula is satisfiable. Deriving minimal satisfying assignments from such complete assignments does not lead to concise explanations of counterexamples because of how CNF formulae are derived from the models. Hence, we formulate the extraction of a succinct counterexample as the problem of finding a minimal assignment that, together with the Boolean formula describing the model, implies an objective. We present a two-stage algorithm for this problem, such that the result of each stage contributes to identify the “interesting” events that cause the failure. We demonstrate the effectiveness of our approach with an example and with experimental results. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math .