Quantifier-free descriptions for quantifier solutions to interval linear systems of relations

Abstract

We study systems of relations of the form Ax σ b, where σ is a vector of binary relations with the components "=", "≥" and "≤", and the parameters (elements of the matrix A and right-hand side vector b) can take values from prescribed intervals. What is considered to be the set of its solutions depends on which logical quantifier is associated with each interval-valued parameter and what is the order of the quantifier prefixes for certain parameters. For solution sets that correspond to the quantifier prefix of a general form, we present equivalent quantifier-free descriptions in the classical interval arithmetic, in Kaucher complete interval arithmetic and in the usual real arithmetic.

Quantifier-free description of the solutions set of the generalized interval-quantifier system of linear equations

“…Следуя работе И.А. Шарой [4,5], введем понятие интервально-кванторной системы линейных уравнений следующим образом.…”

“…Определение 3 (И.А. Шарая [4,5]). Интервально-кванторной системой линейных уравнений будем называть формулу Q(A, b, A, β)(Ax = b), а ее решением всякий вектор x ∈ R n , для которого формула принимает значение "истина".…”