We establish that the two-dimensional ray tracing problem with thin lenses and plane mirrors is Turing-complete, thereby resolving an open question posed by Reif et al. in 1994 as to whether three-dimensional space is necessary for computational universality in optical systems. To this end, we consider the standard approximation of reflection and refraction, namely the ABCD model for paraxial optics, which describes ray propagation through lenses (refraction) via a 2 × 2 matrix, combined with the geometric reflection model for plane mirrors. In the absence of mirrors, two-dimensional ray tracing using any combination of lenses in this ABCD matrix model can be described by a single 2 × 2 matrix–vector product, where the matrix has real entries and determinant 1. Conversely, we show that any such matrix with determinant 1 can be represented as a composition of exactly three appropriately spaced thin lenses. When mirrors are combined with lenses, the ray tracing problem can be described by a flowchart using only two variables, which establishes Turing computability for rational-valued inputs, spaces and matrix entries. Building on this observation, we present a construction of ray tracing that simulates a reversible Turing machine. We begin with a restricted version of the reversible flowchart problem, in which only two variables and certain linear functions are permitted. We prove that this restricted variant is Turing-complete. We then show that such a flowchart admits a geometric realization using lenses and mirrors in our model, thereby establishing the main result: Turing-completeness of the two-dimensional ray tracing problem with ABCD-model lenses and mirrors.

We introduce and investigate the computational complexity of a novel physical problem known as the Pinball Wizard problem. It involves an idealized pinball moving through a maze composed of one-way gates (outswing doors), plane walls, parabolic walls, moving plane walls, and bumpers that cause acceleration or deceleration. Given the initial position and velocity of the pinball, the task is to decide whether it will hit a specified target point. By simulating a two-stack pushdown automaton, we show that the problem is Turing-complete - even in two-dimensional space. In our construction, each step of the automaton corresponds to a constant number of reflections. Thus, deciding the Pinball Wizard problem is at least as hard as the Halting problem. Furthermore, our construction allows bumpers to be replaced with moving walls. In this case, even a ball moving at constant speed - a so-called ray particle - can be used, demonstrating that the Ray Particle Tracing problem is also Turing-complete.

This contribution considers the problem of robust target localization using the possibly unreliable hybrid received signal strength and time-of-arrival measurements, in the Internet of Things (IoT) context. Traditional positioning approaches relying on either extra a priori error information for robustification or the computationally intensive convex programming techniques for optimization do not fit well into the IoT applications with limited computing resources. Such concerns, however, will jeopardize the straightforward applicability of many ready-made solutions to the IoT positioning services if left untreated. In this article, the problem is resolved in a different manner. We adopt here a Geman–McClure like loss function, which is much less sensitive to the biased sensor observations, in order to statistically robustify the$\ell _{2}$-sapce-based location estimator. A computationally attractive iterative message passing algorithm is then developed to conduct efficient optimization. Simulation results demonstrate the performance superiority of the proposed scheme over its competitors in various localization environments.