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An informal introduction to a high level language with applications to interval mathematics

Неформальное Введение В Яэык Высокого Уровня С Приложениями К Интервальной математике

  • Parallel Algorithms for Interval Computations
  • Published:
Reliable Computing

Abstract

The main problem of interval computations is as follows:given sets of possible valuesX i for variablesx i, and an algorithmf:R n → R, to.estimate the rangef(X 1, ..,X n ) of the possible values off(x 1, ...,x n ). In many real-life, situations setsX i are not intervals. To handle such problems, it is desirable to add set data type and operations with sets to a programming language. it is well known that the entire mathematics can be formulated in terms of sets. So, if we already have a set as a data type, why have anything else. The main reason, is that expression in terms of sets is often clumsy. To avoid this clumsiness, it has been suggested to use not only sets, but alsobags (multisets), in which an element can have multiple occurrences. Bags are used in many areas of Computer Science, and recently, several languages have appeared that use the bag as a basic data type.

In this paper, we explain the main ideas behind bag languages, and we also show:

  • · that bag languages are naturally parallelizable, thus leading to a parallelization of the coresponding generalized interval computations;

  • · and that bag languages can be also helpfully applied to traditional interval computations (where setsX i are intervals).

Abstract

Основная эаяача интериальных бычнслеихй Фомрмулируется слелуюивим обраэом: даны множецтва воэможных эначеххйХ і н алгорнтмf:R n→R: требугся оненить множествоf(X 1,...,X n ) воэможных эначений функиииf(x 1,...,x n ). На иактике множестваХ і час го не явлркются ннтервалами. Чтобх снравнться с такимй эалачамн, жэлагельно лобабить множества как тии ланных и онерхии с множецтвами в ягыки ирорраммнробахия. Иэвество, что вся математика может быть иэложена в терминах множеств. Воэникаег вонрс: если у нас естю множество как тих ланных, эачем нужно что-то еме? Основное воэраженне эакдючается в том, что математнческие концтрукиии нэ множеств часто іромоэлкн. Чтвы нэбежать этого, ирелложено нсноьэокать не толбко множества, но н мулвмлнвжесмга (вашс), вкотрые олин н тот же элемент может вхолить ио нецкодьку паэ. Мульиимножества нсиольеуются во многих областях ннформатики, н в носледнее время воявндосб неслолбло лэыкло лэыков крограммироваинл, в коттрых мылбтиножестба ьсляются основным тихом ланных.

В настояшэй рботе иэлагаются основные кониещум яэыков, иснольэуюушх мулжтимхожества, а также иокаэыбается, что:

  • \ яэыкн с муабгимножтвенным тихом ланных естесвнно иараллелиэуются, в реэультаге чего соответствуюхине обобщенные инпервалжниые бычисленхя также ирнобретоют иараллеьй внл;

  • \ ⟸улятимножества и исноляэующие нх яэыки выиолно ирименэть и лля обычныш интераль ных вычнслений (с котпых мнозецтваХ і являются интервалами).

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Authors and Affiliations

  1. Department of Computer Science, University of Texas El Paso, 79968, El Paso, TX, USA

    Daniel E. Cooke & Л. Е. Кук

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  1. Daniel E. Cooke
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  2. Л. Е. Кук
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Cooke, D.E., Кук, Л.Е. An informal introduction to a high level language with applications to interval mathematics. Reliable Comput 1, 65–75 (1995). https://doi.org/10.1007/BF02390522

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  • DOI: https://doi.org/10.1007/BF02390522

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