Data Representation and Computer Arithmetic

The work is devoted to the study of the properties of the left-shift operator $Q_{s}^{*}$-representation of real numbers and the study of the type of distribution of the sequences produced by the corresponding operator. The $Q_{s}^{*}$-representation of real numbers is a natural generalization of the classical s-representation and is topologically similar to the latter. E. Borel's classic result that almost all numbers are s-normal was over time translated into the terms of uniformly distributed sequences produced by the left-shift operator of the digits of the corresponding representation. It was proved that a number is s-normal only when the corresponding sequence generated by this number in the sense of the left shift operator is uniformly distributed. Despite the topological similarity between the $Q_{s}^{*}$-representation of real numbers and the classical s-representation, proving similar results for the former requires fundamentally new approaches that include the use of the apparatus of ergodic theory. The absence of the effect of metric transitivity of the appearance of digits, which is characteristic of the classical s-representation, does not allow the use of appropriate approaches to the $Q_{s}^{*}$-representation. The construction of normal numbers in various representation systems is a separate non-trivial problem and is the subject of many studies. In many cases, criteria for the normality of numbers, which can have a continuous structure (similar to the classical criteria of uniform distribution of the sequence) or a discrete structure, are useful for constructing the corresponding numbers. This paper presents generalizations of discrete criteria for the normality of numbers, which applied both to the classical s-representation and to the $Q_{s}$-representation of real numbers (the latter is a partial case of the $Q_{s}^{*}$-representation).

We develop the concept of minimum weight binary continued fraction representation of a rational number as an extension of minimum weight binary radix representation of an integer. The relation of these representations to the attainment of optimum efficiency in the shift and add or subtract model of binary computer arithmetic is discussed.

This article reports on a study of high-school and of technologist students (prospective engineers and economists) understanding of real numbers. Our study was based on written response to a properly designed questionnaire and on interviews taken from students. The quantitative results of our experiment showed an almost complete failure of the technologist students to deal with processes connected to geometric constructions of incommensurable magnitudes. However, this didn’t prevent them in answering satisfactorily the other questions. The superiority of their correct answers with respect to those of high-school students, although the majority of them correspond to mediocre graduates of the secondary education, was evident in most cases. This is a strong indication that the age and the width of mathematical knowledge of the individual play an important role for the better understanding of the real numbers. The results of our experiment suggest also that the ability to transfer in comfort among several representations of real numbers helps students in obtaining a better understanding of them. A theoretical explanation about this is obtained through the adoption of the conceptual framework of dimensions of knowledge, introduced by Tirosh et al. (1998) for studying the comprehension of rational numbers. Following in part the idea of generic decomposition of the APOS analysis (Weller et al. 2009) we suggest a possible order for development of understanding the real numbers by students when teaching them at school. Some questions open to further research are also mentioned at the end of the paper. Keywords: Real, rational, irrational, algebraic and transcendental numbers, fractions, decimals, representations of real numbers.

A number system specifies the representation of real numbers by symbolic sequences, so its key element is the value mapping \(\Phi : \Sigma \rightarrow \overline{\mathbb {R}}\). Mobius number systems are based on representations of real numbers by sequences of Mobius transformations, so the alphabet of the subshift \(\Sigma \) consists of the symbols of the transformations. We have several means how to define suitable subshifts \(\Sigma \) and suitable value mappings \(\Phi \).

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.

Topological properties of real number representations

For the last 25 years, occupancy grids have been intensively used as a well-understood framework for many robotic applications, such as path planning or obstacle avoidance. They offer a unifying framework for multiple heterogeneous sensor integration using a probabilistic representation of sensor data. This integration is computed through Bayesian techniques or evidence combination approaches, both requiring high computation workload using real number representation. In critical application domains, it is challenging to fuse data coming out of several sensors in real-time using constrained embedded platforms. In this paper, we propose a revised theoretical formulation of multi-sensor fusion using only integer arithmetic. We apply this novel framework to compute occupancy grid by only using integer numbers to represent probabilities. Compared to the state-of-the-art solutions, our fusion framework enables implementation on platforms with no floating-point support. Our experiments demonstrate that fusion of real automotive data from a 4-scans LIDAR can be integrated into a microcontroller without a floating-point unit. Our approach opens the perspective for microcontroller or even for hardware block based on ASIC or FPGA to support occupancy grid applications with real-time performance.

Modern trends in the development of the theory and practice of scientific research show that one of the main elements of determining the reliability of research results is computer modeling. With the development of high-level programming languages and the availability of technical means, a significant number of researchers are using modeling software as a research tool. With the help of simulation systems, decisions are made that affect human lives, the directions of development of industries and the prospects of the state. Experts in the relevant industries, based on the results of computer calculations, do not take into account the peculiarities of the accuracy of calculations in a computer with binary numbers and the influence of binary arithmetic on the results of simulation. In the article, the authors used an approach to determine the sources of computer modeling errors using fragments of program code. With the help of fragments of the program code the differences of results of decimal and binary calculations with features of representation of binary numbers with a floating point are shown. It was found that the result of calculations is most influenced by errors of number denormalization, cyclicity errors of of sum operation and rounding errors depending on the description of the accuracy of representation of variables directly in the program code. The use of this approach makes it possible to exclude or determine the ranges of numerical values, the representation of which in binary form, can lead to distortion of the simulation results already during the implementation of software products. Keywords: floating point representations, numerical simulation values, mathematical model, computer architecture, simulation errors, rounding errors, denormalization errors.

The purpose of this paper is to summarize proposed specifications for floating-point arithmetic and elementary functions. The topics considered are: the base of the number system, precision control, number representation, arithmetic operations, mother basic operations, elementary functions, and exception handling. The possibility of doing without fixed-point arithmetic is also discussed. The specifications are intended to be entirely at the level of a programming language such as Fortran. The emphasis is on convenience and simplicity from the user's point of view. The specifications are not complete every detail, but it is intended that they be complete in spirit — some further details, especially syntatic details, would have to be provided, but the proposals are otherwise relatively complete.

We analyse the algorithmic properties of programs induced by the choice of the representation of data in lambda-calculus. From a logical point of view there are two canonical ways of defining the data types: the iterative one and the recursive one. Both define the same mathematical object, but we show that they have a completely different algorithmic content. The essential of the difference appears in the operational properties of two programs: the predecessor and the addition on the type of unary natural numbers (for the type of lists this would be the programs cdr and append). The results we prove in this paper state a fundamental duality between the iterative and recursive representation of data in lambda-calculus. For the iterative representation of natural numbers (Church numerals) there is a "one-step" addition, but we prove in §3 that there is no "one-step" predecessor (by "one-step" we mean "whose computation requires only number of reduction steps bounded by a constant"). For the recursive representation of natural numbers we have the converse situation: there is a "one-step" predecessor but we prove in §4 that there is no "one-step" addition. For simplicity, we state these results for the type of natural numbers, but they hold in fact for all the usual data types defined as multisorted term algebras. Their practical significance for programming, may be, appears clearer on the type of lists where the predecessor is replaced by the cdr and the addition by append. In §5, we briefly present a new representation of natural numbers for which we have both, a "one-step" predecessor and a "one-step" addition.

Gathering a group of mobile agents is a fundamental task in the field of distributed and mobile systems. This can be made drastically more difficult to achieve when some agents are subject to faults, especially the Byzantine ones that are known as being the worst faults to handle. In this paper we study, from a deterministic point of view, the task of Byzantine gathering in a network modeled as a graph. In other words, despite the presence of Byzantine agents, all the other (good) agents, starting from {possibly} different nodes and applying the same deterministic algorithm, have to meet at the same node in finite time and stop moving. An adversary chooses the initial nodes of the agents (the number of agents may be larger than the number of nodes) and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label. The agents move in synchronous rounds and can communicate with each other only when located at the same node. Within the team, f of the agents are Byzantine. A Byzantine agent acts in an unpredictable and arbitrary way. For example, it can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. Besides its label, which corresponds to a local knowledge, an agent is assigned some global knowledge denoted by GK that is common to all agents. In literature, the Byzantine gathering problem has been analyzed in arbitrary n-node graphs by considering the scenario when GK=(n,f) and the scenario when GK=f. In the first (resp. second) scenario, it has been shown that the minimum number of good agents guaranteeing deterministic gathering of all of them is f+1 (resp. f+2). However, for both these scenarios, all the existing deterministic algorithms, whether or not they are optimal in terms of required number of good agents, have the major disadvantage of having a time complexity that is exponential in n and L, where L is the value of the largest label belonging to a good agent. In this paper, we seek to design a deterministic solution for Byzantine gathering that makes a concession on the proportion of Byzantine agents within the team, but that offers a significantly lower complexity. We also seek to use a global knowledge whose the length of the binary representation (that we also call size) is small. In this respect, assuming that the agents are in a strong team i.e., a team in which the number of good agents is at least some prescribed value that is quadratic in f, we give positive and negative results. On the positive side, we show an algorithm that solves Byzantine gathering with all strong teams in all graphs of size at most n, for any integers n and f, in a time polynomial in n and the length |l_{min}| of the binary representation of the smallest label of a good agent. The algorithm works using a global knowledge of size O(log log log n), which is of optimal order of magnitude in our context to reach a time complexity that is polynomial in n and |l_{min}|. Indeed, on the negative side, we show that there is no deterministic algorithm solving Byzantine gathering with all strong teams, in all graphs of size at most n, in a time polynomial in n and |l_{min}| and using a global knowledge of size o(log log log n).

The algorithms.- Questions of rationality and irrationality.- Some concepts and tools from probability theory.- Metric theory of representations with stochastically independent digits.- A survey on the applications of ergodic theory to series representations.- Metric theory in the general case.- The hausdorff dimension of certain sets related to series expansions.- Miscellaneous topics.

A Class of Weak Bernoulli Transformations Associated with Representations of Real Numbers

Representations for real numbers

We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof for the correctness of a corecursive program for the average function with regard to the signed digit stream representation. We implemented this proof in the interactive proof system Minlog. Thus, reliable, corecursive functions for real computation can be guaranteed, which is very helpful in formal software development for real numbers.