Fundamenta Informaticae

We specify the frontier of decidability for fragments of the first-order theory of ordinal multiplication. We give a NEXPTIME lower bound for the complexity of the existential fragment of 〈 ω ω λ ;   × ,   ω ,   ω +1,   ω 2 + 1 〉 for every ordinal λ. Moreover, we prove (by reduction from Hilbert Tenth Problem) that the ∃*∀⁶-fragment of 〈 ω ω λ ; × 〉 is undecidable for every ordinal λ.

Distances based on morphological operations have shown good performance in a number of applications. Still, the existing erosion and dilation distance for gray scale images can not be computed in all situations. Furthermore, it is possible that dissimilarity between objects which are compared grows strongly, but the value of a mentioned distance does not change. We present a proposition with the necessary and sufficient conditions for computing these morphological distances and discuss drawbacks that they possess. In addition, we propose novel morphological distances, which can be computed in all situations and provide results with desirable properties. The applicability of novel morphological distances is presented in illustrative examples, including their applicability to real data.

In this short note we present an elementary matrix-constructive algorithmic proof of the Quillen-Suslin theorem for Ore extensions A := K[x; σ, δ], where K is a division ring, σ : K → K is a division ring automorphism and σ : K → K is a σ-derivation of K. It asserts that every finitely generated projective A-module is free. We construct a symbolic algorithm that computes the basis of a given finitely generated projective A-module. The algorithm is implemented in a computational package. Its efficiency is illustrated by four representative examples.

The objective of this manuscript is to present the concept of the complex interval-valued intuitionistic fuzzy (CIVIF) set, their algebraic operations and their corresponding aggregation operators, which can better represent the time-periodic problems and two-dimensional information in a single set. The proposed CIVIF set includes the characteristics of both complex intuitionistic fuzzy set, as well as the interval-valued intuitionistic fuzzy sets. Some of the basic operational laws and their properties have been investigated in details. Also, we have developed some new weighted and ordered weighted averaging and geometric aggregation operators with complex interval-valued intuitionistic fuzzy information. The proposed operations are the generalization of the operations of interval-valued intuitionistic fuzzy, complex fuzzy and complex intuitionistic fuzzy theories. Furthermore, a group decision-making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.

Color disagreement among histological images may affect the performance of computer-aided histological image analysis. So, one of the most important and challenging tasks in histological image analysis is to diminish the color variation among the images, maintaining the histological information contained in them. In this regard, the paper proposes a new circular clustering algorithm, termed as rough-fuzzy circular clustering. It integrates judiciously the merits of rough-fuzzy clustering and cosine distance. The rough-fuzzy circular clustering addresses the uncertainty due to vagueness and incompleteness in stain class definition, as well as overlapping nature of multiple contrasting histochemical stains. The proposed circular clustering algorithm incorporates saturation-weighted hue histogram, which considers both saturation and hue information of the given histological image. The efficacy of the proposed method, along with a comparison with other state-of-the-art methods, is demonstrated on publicly available hematoxylin and eosin stained fifty-eight benchmark histological images.

In this paper, we consider the dynamic version of covering the convex hull of a point set P in ℝ² by two congruent disks of minimum size. Here, the points can be added or deleted in the set P, and the objective is to maintain a data structure that, at any instant of time, can efficiently report two disks of minimum size whose union completely covers the boundary of the convex hull of the point set P. We show that maintaining a linear size data structure, we can report a radius r satisfying r ≤ 2r_opt at any query time, where r_opt is the optimum solution at that instant of time. For each insertion or deletion of a point in P, the update time of our data structure is O(log n). Our algorithm can be tailored to work in the restricted streaming model where only insertions are allowed, using constant work-space.

The problem studied in this paper has novelty in two ways: (i) it computes the covering of the convex hull of a point set P, which has lot of surveillance related applications, but not studied in the literature, and (ii) it also considers the dynamic version of the problem. In the dynamic setup, the extent measure problems are studied very little, and in particular, the k-center problem is not at all studied for any k ≥ 2.