Abstract
AbstractMoore's interval arithmetic always provides the same results of arithmetic operations, e.g. [1, 3]+ [5, 9]= [6, 12]. But in real life problems, the operation result can be different, e.g. equal to [4, 7]. Therefore, real problems require more advanced arithmetic. The paper presents (on example of the division) an arithmetic of intervally-precisiated values (IPV-arithmetic) and its main advantages. Thanks to it, it is possible to process different tasks that people solve intuitively. The most important advantages are: existence of inverse elements of addition and multiplication, holding the distributivity law and the cancellation law of multiplication, possibility of achieving not only the solution span [x, x] but also the full, multidimensional solution and its cardinality distribution without using Monte Carlo method, possibility of achieving unique and complete solution sets of equations with unknowns, possibility of calculations with uncorrelated IPVs, possibility of calculations with fully c...
Highlights
An interval arithmetic defines basic operations on intervals: addition, subtraction, multiplication and division
The name interval arithmetic and other notions introduced by Moore are rather improper according to authors because they lead to a little incorrect understanding of this arithmetic and to incorrect approaches to calculation methods elaborated by many scientists
An interval is the basic model of an uncertain value and an interval arithmetic is used in higher forms of uncertainty processing, e.g. in the fuzzy arithmetic of type-1 and type-2, in the intuitionistic fuzzy arithmetic and the probabilistic arithmetic
Summary
An interval arithmetic defines basic operations on intervals: addition, subtraction, multiplication and division. Moore, who published his first book 1 on this subject in 1966 and the latest 2 in 2009 This arithmetic was called interval one, because with its use one can e.g. add two quantities a and b, which values are not precisely but only approximately known and their approximation or the precisiation has a form of an interval, e.g. a = [1, 3] and b = [3, 5]. Usually only precise knowledge of variables and parameter values is assumed in problem solving Such calculated results often considerably differ from real observations. The interval arithmetic has been used in the case of word-models in methods of Computing with Words 3,13,14,15,16,17,18,19 It is a very important branch of an artificial intelligence that conditions creation of the automatic thinking similar to the human one 20. Some basic concepts of this arithmetic were partly presented in 29,30,31,32,33,34,35 and in this paper we will focus on a division of IPVs
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