Stochastic arithmetic: Addition and multiplication by scalars

Stochastic Arithmetic as a Model of Granular Computing

“…Such is the computation of the sum of a series. The number N of decimal digits thus obtained for system (26) with the initialization…”

The Stochastic Geometric Machine Model

“…If X 1 , X 2 ∈ S + , then ∀α ∈ R, X 1 + X 2 ∈ S + . In [4], it is shown that (S + , +, R, * ) is an s-space of monoid structure. Thus, for α, β ∈ R, X, X 1 , X 2 ∈ R + , it holds that…”

“…In the fields of roundoff error analysis and validity of numerical software, it makes possible to develop theoretical studies. In [4], an algebraic approach abstracted from the properties of stochastic arithmetic permits to reduce computation with stochastic numbers to computation in familiar vector spaces. For that, a stochastic number is conceived as a gaussian random variable (in the continuous space) with a known mean value (real number) and a known standard deviation (nonnegative real number).…”

“…The coherence space D → ∞ is defined as the least fixed point for the equation (3.1), representing finite processes. The completion procedure guarantees the existence of the least solution to the recursive 4 The notations π…”

“…Coherent subsets are conceived as linear traces of linear functions interpreting states and processes of the SGM. The notion of coherence between coherent subsets provides semantic interpretation for parallel and non-deterministic computations performed over array structure of stochastic numbers [4], represented by pairs of elements of the subspace of total objects of the Bi-structured Coherence Space of Rational Intervals, IQ [1], a constructive computational representation of the set of real numbers. The non-deterministic machine states are conceived as collections of linear functions (with singletons modelling deterministic states) from the coherence space R 3 , interpreting the memory positions related to the euclidian geometric space, to the coherence space of values (IQ), modelling both the set of mean values and the set of standard deviations of stochastic numbers.…”