# Beyond Interval Systems: What Is Feasible and What Is Algorithmically Solvable?

## Abstract

In many real-life applications of interval computations, the desired quantities appear (in a good approximation to reality) as a solution to a system of interval linear equations. It is known that such systems are difficult to solve (NP-hard) but still algorithmically solvable. If instead of the (approximate) interval linear systems, we consider more realistic (and more general) formulations, will the corresponding problems still be algorithmically solvable? We consider three natural generalizations of interval linear systems: to conditions which are more general than linear systems, to multi-intervals instead of intervals, and to dynamics (differential and difference equations) instead of statics (linear and algebraic equations). We show that the problem is still algorithmically solvable for non-linear systems and even for more general conditions, and it is still solvable if we consider linear or non-linear systems with multi-intervals instead of intervals. However, generalized conditions with multi-interval uncertainty are already algorithmically unsolvable. For dynamics: difference equations are still algorithmically solvable, differential equations are, in general, unsolvable.

## Keywords

interval computations feasible algorithmically solvable## Preview

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