Mechanical Computing: The Computational Complexity of Physical Devices

Abstract

Glossary Computable Capable of being worked out by calculation, especially using a computer. Mechanism A machine or part of a machine that performs a particular task computation; the use of a computer for calculation. Simulation Used to denote both the modeling of a physical system by a computer and the modeling of the operation of a computer by a mechanical system; the difference will be clear from the context. Definition of the Subject and Its Importance Mechanical devices for computation appear to be largely displaced by the widespread use of microprocessor-based computers that are pervading almost all aspects of our lives. Nevertheless, mechanical devices for computation are of interest for at least three reasons: (a) Historical: The use of mechanical devices for computation is of central importance in the historical study of technologies, with a history dating to thousands of years and with surprising applications even in relatively recent times. (b) Technical and practical: The use of mechanical devices for computation persists and has not yet been completely displaced by widespread use of microprocessor-based computers. Mechanical computers have found applications in various emerging technologies at the microscale that combine mechanical functions with computational and control functions not feasible by purely electronic processing. Mechanical computers also have been demonstrated at the molecular scale and may also provide unique capabilities at that scale. The physical designs for these modern micro-and molecular-scale # Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Adamatzky (ed.), Unconventional Computing, https://doi. 35 mechanical computers may be based on the prior designs of the large-scale mechanical computers constructed in the past. (c) Impact of physical assumptions on complexity of motion planning, design, and simulation: The study of computation done by mechanical devices is also of central importance in providing lower bounds on the computational resources such as time and/or space required to simulate a mechanical system observing given physical laws. In particular, the problem of simulating the mechanical system can be shown to be computationally hard if a hard computational problem can be simulated by the mechanical system. A similar approach can be used to provide lower bounds on the computational resources required to solve various motion-planning tasks that arise in the field of robotics. Typically, a robotic motion-planning task is specified by a geometric description of the robot (or collection of robots) to be moved, its initial and final positions, the obstacles it is to avoid, as well as a model for the type of feasible motion and physical laws for the movement. The problem of planning such as robotic motion-planning task can be shown to be computationally hard if a hard computational problem can be simulated by the robotic motion-planning task.