On Frictional Mechanical Systems and Their Computational Power

In this paper we define a class of mechanical systems consisting of rigid objects (defined by linear or quadratic surface patches) connected by frictional contact linkages between surfaces. (This class of mechanisms is similar to the analytical engine developed by Babbage in the 1800s, except that we assume frictional surfaces instead of toothed gears.) We prove that a universal Turing Machine (TM) can be simulated by a (universal) frictional mechanical system in this class consisting of a constant number of parts. Our universal frictional mechanical system has the property that it can reach a distinguished final configuration through a sequence of legal movements if and only if the universal TM accepts the input string encoded by its initial configuration. There are two implications from this result. First, the robotic mover's problem is undecidable when there are frictional linkages. Second, a mechanical computer can be constructed that has the computational power of any conventional electronic computer and yet has only a constant number of mechanical parts. Previous constructions for mechanical computing devices (such as Babbage's analytical engine) either provided no general construction for finite state control or the control was provided by electronic devices (as was common in electromechanical computers such as Mark I subsequent to Turing's result). Our result seems to be the first to provide a general proof of the simulation of a universal TM via a purely mechanical mechanism. In addition, we discuss the universal frictional mechanical system in the context of an error model that allows an error up to $\epsilon$ in each mechanical operation. We first show that, for a universal TM M, a frictional mechanical system in this $\epsilon$-error model can be constructed such that, given any space bound S, the system can simulate the computation of M on any input string $\omega$ if M decides $\omega$ in space bound S, provided that $\epsilon < 2^{-cS}$ for some constant c. We also show that, for any universal TM M and space bound S, there exists a frictional mechanical system in the $\epsilon$-error model with $\epsilon = \Omega(1)$; it has O(S) parts and can simulate M on any input $\omega$ that M decides in space bound S.