Book ChapterDOI

# Conceptual Application of List Theory to Data Structures

01 Jan 2013-pp 551-560
TL;DR: Another application of the concept of position function to define lists is provided in defining data structures like Stack, Queue and Array to the context of fuzzy lists and intuitionistic fuzzy lists.
Abstract: Following the approach of defining a set through its characteristic function and a multiset (bag) through its count function, Tripathy, Ghosh and Jena ([3]) introduced the concept of position function to define lists. The new definition has much rigor than the earlier one used in computer science in general and functional programming ([2]) in particular. Several of the concepts in the form of operations, operators and properties have been established in a sequence of papers by Tripathy and his coauthors ([3, 6, 7, 8]. Also, the concepts of fuzzy lists ([4]) and that of intuitionistic fuzzy lists ([5]) have been defined and studied by them. Recently an application to develop list theoretic relational databases and operations on them has been put forth by Tripathy and Gantayat ([9]). In the present article we provide another application of this approach in defining data structures like Stack, Queue and Array. One of the major advantages of this approach is the ease in extending all the concepts for basic lists to the context of fuzzy lists and intuitionistic fuzzy lists. We also illustrate this approach in the present paper.
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Book ChapterDOI
01 Jan 2014
TL;DR: In this chapter, a list theory-based relational database model using position function approach is designed to illustrate how query processing can be realized for some of the relational algebraic operations.
Abstract: The concept of list is very important in functional programming and data structures in computer science. The classical definition of lists was redefined by Jena, Tripathy, and Ghosh (2001) by using the notion of position functions, which is an extension of the concept of count function of multisets and of characteristic function of sets. Several concepts related to lists have been defined from this new angle and properties are proved further in subsequent articles. In this chapter, the authors focus on crisp lists and present all the concepts and properties developed so far. Recently, the functional approach to realization of relational databases and realization of operations on them has been proposed. In this chapter, a list theory-based relational database model using position function approach is designed to illustrate how query processing can be realized for some of the relational algebraic operations. The authors also develop a list theoretic relational algebra (LRA) and realize analysis of Petri nets using this LRA.

1 citations

##### References
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Book
01 Aug 1996
TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

52,705 citations

Journal ArticleDOI
TL;DR: Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.

13,376 citations

### "Conceptual Application of List Theo..." refers background in this paper

• ...(i) Insertion at the beginning: B-INSERT(15, A) = cons (15, A) = [15, 1, 5, 3, 1, 6, 8, 9] (ii) Insertion at the end: E-INSERT(15, A) = A ╫ [15] = [1, 5, 3, 1, 6, 8, 9] ╫ [15] = [1, 5, 3, 1, 6, 8, 9, 15] (iii) Insertion at any position: Suppose i = 4, and x = 15....

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• ...INSERT(15, 4, A) = take(4 - 1, A) ╫ [15] ╫ drop(4 -1, A) = take(3, A) ╫ [15] ╫ drop(3, A) = [1, 5, 3] ╫ [15] ╫ [1, 6, 8, 9] = [1, 5, 3, 15, 1, 6, 8, 9] (iv) To find the next and previous element at the position i = 4: NEXT( 4, A) = hd(take(5, A) – take (4, A))...

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• ...1: Consider the array of numbers A, given by A = [1, 5, 3, 1, 6, 8, 9]....

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• ...The concept of intuitionistic fuzzy sets introduced by Atanassov ([1]) is an extension of the notion of fuzzy sets and is a better model than fuzzy sets to model impreciseness in data....

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Book
01 Jan 1988
TL;DR: This is a thorough introduction to the fundamental concepts of functional programming that includes a simple, yet coherent treatment of the Haskell class; a calculus of time complexity; and new coverage of monadic input-output.
Abstract: From the Publisher: This is a thorough introduction to the fundamental concepts of functional programming. The book clearly expounds the construction of functional programming as a process of mathematical calculation, but restricts itself to the mathematics relevant to actual program construction. It covers simple and abstract datatypes, numbers, lists, examples, trees, and efficiency. It includes a simple, yet coherent treatment of the Haskell class; a calculus of time complexity; and new coverage of monadic input-output.

884 citations

### "Conceptual Application of List Theo..." refers background in this paper

• ...The new definition has much rigor than the earlier one used in computer science in general and functional programming ([2]) in particular....

[...]

Journal ArticleDOI
TL;DR: The bag structure as a set-like object in which repeated elements are significant and the operation of selecting elements from a bag based upon their membership in a set is introduced.
Abstract: We introduce the bag structure as a set-like object in which repeated elements are significant. We discuss operations on bags such as intersection, union and addition. We introduce the operation of selecting elements from a bag based upon their membership in a set. We show the usefulness of the bag structure in relational data bases. We provide a definition for fuzzy bags. In these fuzzy bags the count of the number of elements itself becomes a crisp bags. We investigate a calculus for fuzzy bags.

695 citations

### "Conceptual Application of List Theo..." refers background or methods in this paper

• ...The concepts of fuzzy sets and fuzzy bags were introduced and studied by Zadeh [11] and Yager [10] respectively....

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• ...Similarly, the notion of a bag and its count function ([10]) are interchangeable concepts....

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• ...The concepts of fuzzy sets and fuzzy bags were introduced and studied by Zadeh [11] and Yager [10] respectively....

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Journal ArticleDOI
S. P. Jena
TL;DR: The concept of bag complement is redefined suitably and some theorems involving bag operations have beenestablished and many existing and new results have been established based upon this new definition.

93 citations

### "Conceptual Application of List Theo..." refers background in this paper

• ...INSERT(15, 4, A) = take(4 - 1, A) ╫ [15] ╫ drop(4 -1, A) = take(3, A) ╫ [15] ╫ drop(3, A) = [1, 5, 3] ╫ [15] ╫ [1, 6, 8, 9] = [1, 5, 3, 15, 1, 6, 8, 9] (iv) To find the next and previous element at the position i = 4: NEXT( 4, A) = hd(take(5, A) – take (4, A))...

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• ...This could be realized by Tripathy, Ghosh and Jena ([3]), where they introduced the notion of position function for lists....

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• ...1: Consider the array of numbers A, given by A = [1, 5, 3, 1, 6, 8, 9]....

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• ...= hd([1, 5, 3, 1, 6] – [1, 5, 3, 1]) = hd([6]) = 6 PREVIOUS(4, A) = hd(take(3, A) – take (2, A)) = hd([1, 5, 3] – [1, 5]) = hd([3]) = 3...

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• ...: Advances in Computing and Information Technology, AISC 177, pp.551-560. springerlink.com © Springer-Verlag Berlin Heidelberg 2012 (bag) through its count function, Tripathy, Ghosh and Jena introduced the concept of position function to define lists....

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