Using Rough Set Theory to Find Minimal Log with Rule Generation

Data pre-processing is a major difficulty in the knowledge discovery process, especially feature selection on a large amount of data. In literature, various approaches have been suggested to overcome this difficulty. Unlike most approaches, Rough Set Theory (RST) can discover data de-pendency and reduce the attributes without the need for further information. In RST, the discernibility matrix is the mathematical foundation for computing such reducts. Although it proved its efficiency in feature selection, unfortunately it is computationally expensive on high dimensional data. Algorithm complexity is related to the search of the minimal subset of attributes, which requires computing an exponential number of possible subsets. To overcome this limitation, many RST enhancements have been proposed. Contrary to recent methods, this paper implements RST concepts in an iterated manner using R language. First, the dataset was partitioned into a smaller number of subsets and each subset processed independently to generate its own minimal attribute set. Within the iterations, only minimal elements in the discernibility matrix were considered. Finally, the iterated outputs were compared, and those common among all reducts formed the minimal one (Core attributes). A comparison with another novel proposed algorithm using three benchmark datasets was performed. The proposed approach showed its efficiency in calculating the same minimal attribute sets with less execution time.

A Combinatorial Approach to the Stirling Numbers of the First Kind with Higher Level

In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series.

Interior Operators Generated by Ideals in Complete Domains

This article presents some properties of a special class of interior operators generated by ideals. The mathematical framework is given by complete domains, namely complete posets in which the set of minimal elements is a basis. The first part of the paper presents some preliminary results; in the second part we present the novel interior operator denoted by G(i,I), an operator built starting from an interior operator i and an ideal I. Various properties of this operator are presented; in particular, the connection between the properties of the ideal I and the properties of the operator G(i,I). Two such properties (denoted by Pi and Qi) are extensively analyzed and characterized. Additionally, some characterizations for compact elements are presented.

Residuation in finite posets

When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.

Diagonal groups and arcs over groups

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have $$m+r$$ m + r partitions with $$r\ge 2$$ r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If $$m=2$$ m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $$m>2$$ m > 2 , things are more restricted. Any $$m+1$$ m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality $$m+r$$ m + r in the $$(m-1)$$ ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.

Examining transfer in the acquisition of the count/mass distinction in L2 English

We investigate whether second language (L2) learners of English rely on first language (L1) transfer and atomicity in the acquisition of the count/mass distinction by examining L1-French and L1-Chinese learners of English. Atomicity encodes whether a noun contains ‘atoms’ or minimal elements that retain the property of the noun. As a semantic universal, atomicity holds across languages. However, the count/mass status of nouns may differ cross-linguistically. Our results, which show difficulty on atomic mass nouns in both learner groups, support an argument that atomicity is used as a semantic universal in the L2. Our results also suggest that both count/mass status in the L1 and word frequency in the L2 impact performance, suggesting roles for both L1 lexical transfer and lexical frequency. In addition, learners had better performance on abstract as opposed to concrete atomic mass nouns, providing evidence consistent with a theory of the accessibility of atoms.

On the little secondary Bruhat order

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More on the Bruhat order for $(0,1)$-matrices, Linear Algebra Appl., 421:219--232, 2007) suggested the study of the secondary Bruhat order on ${\cal A}(R,S)$ but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes ${\cal A}(R,S)$. Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of ${\cal A}(R,S)$.

Zero morphemes in paradigms

This paper sheds a new light on the notion of zero morphemes in inflectional paradigms: on their formal definition (§ 1), on the way of counting them (§ 2–3) and on the way of conceptualizing them at a deeper, mathematical level (§ 4). We define (zero) morphemes in the language of cartesian set products and propose a method of counting them that applies the lexical relations of homophony, polysemy, allomorphy and synonymy to inflectional paradigms (§ 2). In this line, two homophonic or synonymous morphemes are different morphemes, while two polysemous and allomorphic morphemes count as one morpheme (§ 3). In analogy to the number zero in mathematics, zero morphemes can be thought of either as minimal elements in a totally ordered set or as neutral element in a set of opposites (§ 4). Implications for language acquisition are discussed in the conclusion (§ 5).

Two by two squares in set partitions

A partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].