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Counting sub-multisets of fixed cardinality [article]

Sebastiano Ferraris, Alex Mendelson, Gerardo Ballesio, Tom Vercauteren
2015 arXiv   pre-print
This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements.  ...  This is also the number of distinct samples of a given size that may be produced by sampling without replacement from a finite population partitioned into subsets, in the case where items belonging to  ...  We wish to determine the number of multisets of cardinality n that may be considered submulitisets of a multiset S of cardinality N and dimension k.  ... 
arXiv:1511.06142v1 fatcat:hkbndxnnjrfjhoo27qok5xue4e

Multiset Theories of Cardinality-based Feature Diagrams [article]

Aliakbar Safilian, Tom Maibaum
2016 arXiv   pre-print
The most common characterization of the semantics of a cardinality-based diagram is the set of flat multisets over features satisfying the constraints.  ...  We address this problem by proposing another multiset theory for the cardinality-based feature diagram, called the hierarchical theory of the diagram.  ...  Definition 1 (Cardinality-based Feature Diagrams) A cardinality-based feature diagram (CFD) is a 5-tuple D = (F, r, ↑ , G, C), where: (i) T = (F, r, ↑ ) is a tree with set F of nodes (called features),  ... 
arXiv:1601.06242v1 fatcat:ob6isumqbjd4vkoukfdmnmdmge

Scalar and fuzzy cardinalities of crisp and fuzzy multisets

Jaume Casasnovas, Francesc Rosselló
2009 International Journal of Intelligent Systems  
In this paper we define in an axiomatic way scalar and fuzzy cardinalities of finite crisp and fuzzy multisets, and we obtain explicit descriptions for them.  ...  Scalar cardinalities of finite multisets over ]0, 1] We introduce and discuss in this section the notion of scalar cardinality of crisp finite multisets on ]0, 1].  ...  As far as cardinalities of multisets goes, an extension to fuzzy multisets of Zadeh's original definition of the cardinality of fuzzy sets has already been introduced [1, 4, 27] .  ... 
doi:10.1002/int.20350 fatcat:ofoodesznzcdjlh2iijeprwxn4

Computational Model for Cardinality Bounded Multiset Space

Adeku Musa Ibrahim, Absalom El-Shamir Ezugwu, Aminu Abdulsalami
2012 International Journal of Applied Mathematical Research  
In this paper, we develop and implement an efficient algorithm to generate the elements of a cardinality bounded multiset space for and the frequency tables for various values of and .  ...  In [1] , manual generation of the elements of multiset space and mathematical models for computing the corresponding frequency numbers were presented.  ...  Let denote the set of all multisets each having m objects occurring with multiplicities at most n times, including 0. We call , a cardinality bounded multiset space for [1] .  ... 
doi:10.14419/ijamr.v1i3.146 fatcat:2fqhgo37evhx5gqb4xlwngvskm

A Comparison of Lex Bounds for Multiset Variables in Constraint Programming [article]

Yat-Chiu Law and Jimmy Ho-Man Lee and May Hiu-Chun Woo and Toby Walsh
2011 arXiv   pre-print
These representations integrate together information about the cardinality, variety (number of distinct elements in the multiset), and position in some total ordering.  ...  Set and multiset variables in constraint programming have typically been represented using subset bounds.  ...  Toby Walsh is funded by the Australian Department of Broadband, Communications and the Digital Economy, the ARC, and the Asian Office of Aerospace Research and Development through grant AOARD-104123.  ... 
arXiv:1106.5890v1 fatcat:jw4pztpjrrfevh5arty7zvef3e

Dedekind multisets and function shells

Wayne D. Blizard
1993 Theoretical Computer Science  
The cardinality of a multiset is, therefore, the sum of the multiplicities of its elements. We denote the cardinality of a classical set X by /X 1 and the cardinality of a multiset M by C(M).  ...  and the algebra of cardinal-valued multisets.  ...  The degree of "many-one character" off (its distance from "injectiveness" measured by comparing ldomfl to Irange(f gives a measure of the degree of multiplicity in the multiset M, (its distance from "setness  ... 
doi:10.1016/0304-3975(93)90351-s fatcat:z2fhswnfffblperofuciuiseaa

Variety Reasoning for Multiset Constraint Propagation

Yat Chiu Law, Jimmy Ho-Man Lee, May H. C. Woo
2009 International Joint Conference on Artificial Intelligence  
Multiset variables are a generalization of set variables by allowing the elements to have repetitions. In this paper, we generalize cardinality reasoning for multiset variables.  ...  The rules interact varieties with the traditional components of multiset variables (such as cardinalities) to obtain stronger propagation.  ...  Multiset Variable with Cardinality and Variety We define a multiset variable S consisting of three components.  ... 
dblp:conf/ijcai/LawLW09 fatcat:mef6ova5mzguxg7qdubuiej2ju

Multiset variable representations and constraint propagation

Y. C. Law, J. H. M. Lee, T. Walsh, M. H. C. Woo
2013 Constraints  
The rules interact varieties with the traditional components of multiset variables (such as cardinalities) to obtain stronger propagation.  ...  Experimental results show that performing variety reasoning on top of cardinality reasoning can effectively reduce more search space and achieve better runtime in solving some multiset CSPs.  ...  Toby Walsh is supported by the Australian Government's Department of Broadband, Communications and the Digital Economy, the Australian Research Council, and the Asian Office of Aerospace Research and Development  ... 
doi:10.1007/s10601-012-9138-7 fatcat:bru5ipqarfcszcldvcfexamgky

Efficient Automated Reasoning About Sets and Multisets with Cardinality Constraints [chapter]

Ruzica Piskac
2020 Lecture Notes in Computer Science  
In this presentation, we outline an efficient decision procedure for reasoning about multisets with cardinality constraints.  ...  Finally, we report on our prototype tool which can efficiently reason about sets and multisets formulas with cardinality constraints.  ...  This presentation is based on the previously published results on reasoning about multisets with cardinality constraints [6, [9] [10] [11] .  ... 
doi:10.1007/978-3-030-51074-9_1 fatcat:t5rmotq7ofdibd72emsnuhrity

Universal and Determined Constructors of Multisets of Objects [article]

Dmytro Terletskyi
2015 arXiv   pre-print
Such constructors are very useful in cases of very big cardinalities of multisets, because they give us an opportunity to calculate a multiplicity of each object and cardinality of multiset before its  ...  Within COOST, the author proposes universal constructor of multisets of objects that gives us a possibility to create arbitrary multisets of objects.  ...  That makes them very useful in cases of very big cardinalities of multisets.  ... 
arXiv:1510.04188v1 fatcat:kinkrs72nzcy3g6aapwch3yflq

Disjoint, Partition and Intersection Constraints for Set and Multiset Variables [chapter]

Christian Bessiere, Emmanuel Hebrard, Brahim Hnich, Toby Walsh
2004 Lecture Notes in Computer Science  
on the cardinality of a set or multiset.  ...  restricting the cardinality of each set or multiset variable (columns).  ...  ¢ £ Intersection constraints The Disjoint constraint restricts the pair-wise intersection of any two set or multiset variables to the empty set.  ... 
doi:10.1007/978-3-540-30201-8_13 fatcat:oz7j4lncl5aj7haueplvpbzbsm

Representation Learning with Multisets [article]

Vasco Portilheiro
2019 arXiv   pre-print
We propose training this model on a novel task: predicting the size of the symmetric difference (or intersection) between pairs of multisets.  ...  We formalize this problem via a measure theoretic definition of multisets, and obtain a theoretically-motivated learning model.  ...  CARDINALITY PREDICTION We examine the performance of six kinds of models -multisets, unrestricted multisets, and DeepSets, each with or without learned multiset operations -on the tasks of predicting cardinality  ... 
arXiv:1911.08577v1 fatcat:65wvvzpoere3vcwurfb4wuoyqa

The H-index can be easily manipulated [article]

Bart de Keijzer, Krzysztof R. Apt
2013 arXiv   pre-print
This suggests the need for an alternative scientific performance measure that is resistant to this type of manipulation.  ...  Further, if a critical number x ∈ S appears in a multiset T ∈ S of cardinality exceeding 2, then we can split T in any way so that x is put in a multiset T ′ of cardinality 2.  ...  Indeed, by Property 3, every critical number is in a set of cardinality 2. Now, let x be a critical number and let {x, y} ∈ S be the multiset of cardinality 2 that contains x.  ... 
arXiv:1304.2557v3 fatcat:zp4zwfjsnrfwvox4b3uvp6mlkq

Basic operations for fuzzy multisets

Á. Riesgo, P. Alonso, I. Díaz, S. Montes
2018 International Journal of Approximate Reasoning  
Among the various existing generalisations of the concept, we find fuzzy multisets. In this case, membership values are multisets in [0, 1] rather than single values.  ...  Moreover, we show how the hesitant operations can be worked out from an extension of the fuzzy multiset operations and investigate the important role that the concepts of order and sorting sequences play  ...  Acknowledgments This work was partially supported by the Spanish Ministry of Science and Technology under projects TIN2014-59543-P and TIN-2017-87600-P.  ... 
doi:10.1016/j.ijar.2018.06.008 fatcat:r5mhweocu5csxjgvc7yibfdzoq

Generalizations of Fuzzy Multisets for Including Infiniteness [chapter]

Sadaaki Miyamoto
2001 Lecture Notes in Computer Science  
First, a class of fuzzy multisets which have an infinite membership set for an element of the universe and finite cardinality is introduced.  ...  It is shown that the sum, union, intersection as well as most t-norm and conorm operations for two sets in this class except the drastic sum keep the property of the finite cardinality of the derived set  ...  A ∩ B ∈ F M 0 (X), Fuzzy multisets of finite cardinality A basic measure of a fuzzy set F is its cardinality defined by |F | = x∈X µ F (x).  ... 
doi:10.1007/3-540-45548-5_32 fatcat:5rty67qveffvbo2fgtbc7uiyk4
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