Related papers: Interval Semantics for Standard Floating-Point Ari…
This paper presents a non-interleaving denotational semantics for the ?-calculus. The basic idea is to define a notion of test where the outcome is not only whether a given process passes a given test, but also in how many different ways it…
Logic programming with fixed-point definitions is a useful extension of traditional logic programming. Fixed-point definitions can capture simple model checking problems and closed-world assumptions. Its operational semantics is typically…
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise,…
Control and state estimation procedures need to be robust against imprecisely known parameters, uncertainty in initial conditions, and external disturbances. Interval methods and other set-based techniques form the basis for the…
In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine. While technique of avoiding overflow/underflow has practical application in the electrical…
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…
Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently.
The paper gives a brief account of the spaces of interval functions defined through the concepts of H-continuity, D-continuity and S-continuity. All three continuity concepts generalize the usual concept of continuity for real (point…
A numerical procedure and its MAPLE implementation capable of rigorously, albeit in a brute-force manner, proving specific strict one-variable inequalities in specific finite intervals is described. The procedure is useful, for instance, to…
Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball…
We provide the asymptotic distribution of the major indexes used in the statistical literature to quantify disparate treatment in machine learning. We aim at promoting the use of confidence intervals when testing the so-called group…
Many Engineering Problems could be mathematically described by Final Value Problem, which is the inverse problem of Initial Value Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the…
We study the characteristic function and moments of the integer-valued random variable $\lfloor X+\alpha\rfloor$, where $X$ is a continuous random variables. The results can be regarded as exact versions of Sheppard's correction. Rounded…
The majority of modern systems exhibit sophisticated concurrent behaviour, where several system components modify and observe the system state with fine-grained atomicity. Many systems (e.g., multi-core processors, real-time controllers)…
Floating-point addition on a finite-precision machine is not associative, so not all mathematically equivalent summations are computationally equivalent. Making this assumption can lead to numerical error in computations. Proper ordering…
A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time $-\infty$, called non-standardness. For example, two dyadic filtrations with trivial tail $\sigma$-field…
In this article, we first define the concept of ordered intervals, then introduce ordered fuzzy inner product and describe some of its properties.
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…
In recent years, half precision floating-point arithmetic has gained wide support in hardware and software stack thanks to the advance of artificial intelligence and machine learning applications. Operating at half precision can…