Interval-valued computing is a kind of massively parallel computing. It operates on specific subsets of the interval [0,1) – unions of subintervals. They serve as basic data units and are called interval-values. It was established in [9], by a rather simple observation, that interval-valued computing, as a digital computing model, has computing power equivalent to Turing machines. However, this equivalence involves an unlimited number of interval-valued variables. In [14], the equivalence with Turing machines is established using a simulation that uses only a fixed number of interval-valued variables and this number depends only on the number of states of the Turing machine – in a logarithmic way. The simulation given there allows us to extend interval-valued computations into infinite length to capture the computing power of red-green Turing machines. In this extension of [14], based on the quasi-periodic techniques used in the simulations in that paper, a reformulation of the interval-valued computations is given, named circular interval-valued computers. This reformulation enforces the finiteness of the number of used interval-valued variables by building the finiteness into the syntax rules.

In this study, a tomography reconstruction problem of binary images is considered on the isometric grid. On this grid, the triangle pixels have two types of orientations, accordingly, the authors call them delta or nabla shape pixels. The proposed reconstruction method uses data of projections of three natural directions. They are the lane directions of the triangular tessellation (these directions are somewhat analogous to row/column directions on the rectangular grids). The projection ray, penetrating through a grid lane, now not passing through the middle of pixels (i.e. through the middle line of triangle shape pixels), as usually taken, but little bit shifted from the middle parallel to the lane. This method provides the exact information about the number of nabla and delta shape triangle pixels in each lane of the image. This additional information is included in the reconstruction process to improve the quality of reconstruction. They formulate the suggested model into an energy‐minimisation problem and apply a gradient‐based approach for its minimisation. They show and analyse various experimental results on test images. The presented approach shows both better quality reconstructions and shorter running time than the earlier approaches.

Mathematical morphology provides various tools for image analysis. The two basic operations, dilation and erosion, are based on translations with the help of a given structural element (another image of the grid). In contrast to the case of discrete subgroups of $\mathbb R^n$, the triangular grid is not closed under translations; therefore, we use a restriction for the structural elements. Namely, we allow only those trixels (triangle pixels) to be in the structural elements which represent vectors such that the grid is closed under translations by these vectors. We prove that both strict dilation and erosion have nice properties. Strict opening and closing have also been defined by combining strict dilation and erosion.

[Abstract Not Available]

Interval-valued computing is a new computing paradigm that is based on manipulations of interval-values. Interval-values are finite unions of intervals on the unit interval [0, 1) so this kind of computing can be considered as a continuous space machine like optical computing [25]. Based on the mas sive parallelism of this paradigm, various intractable problems can be solved efficiently, i.e., by polynomial number of steps. In this paper, the well-known complexity classes, NP and coNP are addressed. A specific subclass of polynomial size interval-valued computations is proven to characterize NP, that is, exactly languages with non-deterministically polynomial time complexity can be decided by interval-valued computations of this subclass. This specific subclass of interval-valued computations does not use any of the shift operators, moreover the product operator is used only in the starting section of the computation. Due to the fact that interval-valued computing is a deterministic model of computing, an analogue result can be established for the class coNP.

Short circuit, short cut, or by other name, lazy evaluations play important roles in various fields of computer science including logic, hardware design, programming, decision making. In this paper, one of the best known and used fuzzy logic systems, the product logic is considered. The evaluation of lots of formulae can be quickened by various pruning techniques by discovering which remaining part of the formula has no influence on the final result for various reasons. The presented techniques can be seen as generalizations of short circuit evaluations in Boolean logic and also of alpha-beta pruning of game trees.

In this paper, we consider the binary tomography reconstruction problem. A new approach is proposed what exploits a possibility provided by the natural structure of the triangular grid, which is not available in the case of the classical square grid. In contrast to the square grid, in the case of the triangular grid information need for the reconstruction of the unknown image is increasing when not only one, but two projections are used by lanes. In this way, the number of Δ and ∇ shaped pixels per lane can be determined. We propose this type of projection approach and call it dense projections. The reconstruction is based on three projection directions by the lane directions of the grid (they are analogous to row and column directions on the square grid). Our algorithm is deterministic and uses energy minimization technique to find (near) optimal solution in a reasonable time. The experimental evaluation of the new method, using regular hexagon shaped test images, is given. Comparison with reconstructions based on the square grid is also considered.

Chain-code picture systems are Lindenmayer-systems over special alphabets. The generated words are interpreted as chain-code pictures containing vertices and edges of the regular triangular grid. The finiteness of various chain-code picture languages are investigated based on properties of length-contracting, length-constant and length-expanding sD0L parallel rewriting systems. The case of shuffled axioms is also explored.

A new combinatorial 3-coordinate system for cells in the face centered cubic grid is presented, and some of its properties are detailed. Three independent coordinates are used to address the voxels (rhombic dodecahedra), their faces (rhombs), their edges and the points at their corners. The incidence (boundary and co-boundary) and adjacency relations between the cells can easily be captured by these coordinate values. The new coordinate system can effectively by applied in various image processing morphological and topological operations.

Binary tomography is a hard and challenging task in image processing. The usage of non-traditional grids may have several benefits in this task also, due to the fact that their symmetric properties differ from the properties of the square grid. There are various approaches to do binary tomography, e.g., network flow algorithms, genetic and memetic algorithms, simulated annealing. On the triangular grid tomography algorithms are based on 3 and 6 projection directions. The first algorithms have used projections by lanes (orthogonal to coordinate axes). In this paper, we present an algorithm that uses 3 projections parallel to the coordinate axes. By our experimental results it can be seen that this newly proposed algorithm is effective to get good results in a short time.

In this article, we investigate several variants of Watson–Crick automata in which both heads read the doubled DNA strand from 5′ to 3′ direction. Some versions of these automata recognize exactly the linear context-free languages. Deterministic automata are not so powerful: the language class of two-deterministic linear languages is defined by them. It is shown that all fixed-linear, and specially, all even linear languages are in this class. A hierarchy of the accepted languages is presented using the all-final, simple, 1-limited and no-state versions and combined restrictions. By full-reading of both strands, some languages that are not context-free can also be accepted.

We consider stateless counter machines which mix the features of one-head counter machines and a special type of two-head Watson-Crick automata (WK-automata). Our Watson-Crick counter machines are biologically motivated. They have two heads that read the input starting from the two extremes. The reading process is finished when there are no more symbols between the heads, i.e., every letter of the input is processed by either head. Depending on whether the heads are required to advance at each move, we distinguish between realtime and non-realtime machines. If every counter makes at most k alternations between nondecreasing and decreasing modes in every computation, then the machine is k-reversal. It is reversal bounded if it is k-reversal for some k. In this paper we concentrate on the properties of both deterministic and nondeterministic stateless WK-automata with reversal bounded counters.

5′ → 3′ WK-automata are Watson-Crick automata whose two heads start on opposite ends of the input word and always run in opposite directions. One full reading in both directions is called a run. We prove that the expressive power of these automata in

In this paper we present some methods for indexing and segmenting colour images. The proposed procedures are based on well-known algorithms, but now we use digital distance functions generated by neighbourhood sequences to measure distance between colours. The application of such distance functions is quite natural and descriptive, since the colour coordinates of the pixels are nonnegative integers. An additional interesting property of neighbourhood sequences is that they do not generate metric in general, so we can obtain many distance functions in this way. We describe our methods for RGB images in detail, but other image representations also could be considered. Moreover, the proposed methods can be applied in arbitrary dimension without any difficulties.