2026 , Volume 31, № 2, p.49-75

The paper addresses interval systems of equations in which interval parameters can carry different types of uncertainty (corresponding to different logical quantifiers) and which act in arbitrary order. The so-called quantifier solution sets arise as solutions for such interval systems of equations, corresponding to multi-stage games or multi-stage decision-making processes under interval uncertainty.

A convenient equivalent language based on the concept of interval-quantifier equation systems is proposed for describing interval systems of equations with different types of interval uncertainty. In particular, multi-stage interval systems of equations are considered.

The main result of the paper is analytical quantifier-free characterizations and descriptions of the quantifier solution sets for interval systems of linear algebraic equations, including multi-stage ones. On their basis, it is possible to further construct computational methods for recognizing and estimating the quantifier solution sets. These descriptions generalize the previously proposed characterizations for sets of AE-solutions of interval linear systems that correspond to one-step decision-making procedures under interval uncertainty.

In addition to describing the quantifier solution sets in the form of interval relations using interval arithmetic, more familiar descriptions are also derived in the form of vector inequalities with modules in arithmetic vector spaces. They are similar to the well-known Oettli – Prager inequality for the united solution set of interval linear systems of equations. The benefit of such descriptions is that they can be used to involve traditional non-interval (point) numerical methods of analysis and optimization in solving interval problems.

Generalizations of the main results of the work to interval equations in partially ordered linear spaces are considered.