Related papers: The Unum Number Format: Mathematical Foundations, …
This paper argues that the operations of a 'Universal Turing Machine' (UTM) and equivalent mechanisms such as the 'Post Canonical System' (PCS) - which are widely accepted as definitions of the concept of `computing' - may be interpreted as…
Weighted model counting computes the sum of the rational-valued weights associated with the satisfying assignments for a Boolean formula, where the weight of an assignment is given by the product of the weights assigned to the positive and…
A unifying approach to software and hardware design generated by ideas of Idempotent Mathematics is discussed. The so-called idempotent correspondence principle for algorithms, programs and hardware units is described. A software project…
Analysis on fractals is a growing field, with hints of potential for widespread applicability across all of STEM. One of the most heavily researched type of fractals are the nested fractals, fractal shapes defined by virtue of being made of…
We show how information on the uniformity properties of a point set employed in numerical multidimensional integration can be used to improve the error estimate over the usual Monte Carlo one. We introduce a new measure of (non-)uniformity…
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
Unitary Fourier transform lies at the core of the multitudinous computational and metrological algorithms. Here we show experimentally how the unitary Fourier transform-based phase estimation protocol, used namely in quantum metrology, can…
Unified models (UMs) hold promise for their ability to understand and generate content across heterogeneous modalities. Compared to merely generating visual content, the use of UMs for interleaved cross-modal reasoning is more promising and…
Unitary best approximation to the exponential function on an interval on the imaginary axis has been introduced recently. In the present work two algorithms are considered to compute this best approximant: an algorithm based on rational…
To deal with uncertainty in reasoning, interval-valued logic has been developed. But uniform intervals cannot capture the difference in degrees of belief for different values in the interval. To salvage the problem triangular and…
From a theoretical point of view, finding the solution set of a system of inequalities in only two variables is easy. However, if we want to get rigorous bounds on this set with floating point arithmetic, in all possible cases, then things…
Level-index arithmetic appeared in the 1980s. One of its principal purposes is to abolish the issues caused by underflows and overflows in floating point. However, level-index arithmetic does not expand the set of numbers but spaces out the…
We outline a method of calculation of partition functions of orientable manifolds with fluctuating metric and perform the calculation for the specific case of the unit interval.
The introduction of posit reopened the debate about the utility of IEEE754 in specific domains. In this context, we propose a high-level language (Scala) library that aims to reduce the effort of designing and testing new number…
The concept of measurement is discussed. It is argued that counting process in mathematics is also measurement which requires a basic unit. The idea of scale is put forward. The basic unit itself, which are composed of the infinitesimal of…
The correct computation of orbits of discrete dynamical systems on the interval is considered. Therefore, an arbitrary-precision floating-point approach based on automatic error analysis is chosen and a general algorithm is presented. The…
This work establishes a rigorous theoretical foundation for analyzing deep learning systems by leveraging Infinite Time Turing Machines (ITTMs), which extend classical computation into transfinite ordinal steps. Using ITTMs, we reinterpret…
Simple floating point operations like addition or multiplication on normalized floating point values can be computed by current AMD and Intel processors in three to five cycles. This is different for denormalized numbers, which appear when…
Intermittently powered devices enable new applications in harsh or inaccessible environments, such as space or in-body implants, but also introduce problems in programmability and correctness. Researchers have developed programming models…
This article is meant to give a lucid and widely accessible, self-contained account of a novel way of performing arithmetic operations on fuzzy intervals. Based on two formulae of generalized inversion (the first in close analogy to the…