Complex-Valued Interval Computations are NP-Hard Even for Single Use Expressions

Abstract

In practice, after a measurement, we often only determine the interval containing the actual (unknown) value of the measured quantity. It is known that in such cases, checking whether the measurement results are consistent with a given hypothesis about the relation between quantities is, in general, NP-hard. However, there is an important case when this checking problem is feasible: the case of single-use expressions, i.e., expressions in which each variable occurs only once. Such expressions are ubiquitous in physics, e.g., Ohm’s law $V=I\cdot R$, formula for the kinetic energy $E=(1/2)\cdot m\cdot v^2$, formula for the gravitational force $F=G\cdot m_1\cdot m_2\cdot r^{-2}$, etc. In some important physical situations, quantities are complex-valued. A natural question is whether for complex-valued quantities, feasible checking algorithms are possible for single-use expressions. We prove that in the complex-valued case, computing the exact range is NP-hard even for single-use expressions. Moreover, it is NP-hard even for such simple expressions as the product $f(z_1,\ldots ,z_n)=z_1\cdot \ldots \cdot z_n$.