Balance Machines: Computing = Balancing

We propose a natural computational model called a balance machine. The computational model consists of components that resemble ordinary physical balances, each with an intrinsic property to automatically balance the weights on their left, right pans. If we start with certain fixed weights (representing inputs) on some of the pans, then the balance– like components would vigorously try to balance themselves by filling the rest of the pans with suitable weights (representing the outputs). This balancing act can be viewed as a computation. We will show that the model allows us to construct those primitive (hardware) components that serve as the building blocks of a general purpose (universal) digital computer: logic gates, memory cells (flip-flops), and transmission lines. One of the key features of the balance machine is its “bidirectional” operation: both a function and its (partial) inverse can be computed spontaneously using the same machine. Some practical applications of the model are discussed. 1 Computing as a “Balancing Feat” A detailed account of the proposed model will be given in Section 2. In this section, we simply convey the essence of the model without delving into technical details. The computational model consists of components that resemble ordinary physical balances (see Figure 1), each with an intrinsic property to automatically balance the weights on their left, right pans. In other words, if we start with certain fixed weights (representing inputs) on some of the pans, then the balance– like components would vigorously try to balance themselves by filling the rest of the pans with suitable weights (representing the outputs). Roughly speaking, the proposed machine has a natural ability to load itself with (output) weights that “balance” the input. This balancing act can be viewed as a computation. There is just one rule that drives the whole computational process: the left and right pans of the individual balances should be made equal.Note that the machine is designed in such a way that the balancing act would happen automatically by virtue of physical laws (i.e., the machine is self-regulating). One of our aims is to show that all computations can be ultimately expressed using one primitive operation: 1 If the machine cannot (eventually) balance itself, it means that the particular instance does not have a solution. N. Jonoska et al. (Eds.): Molecular Computing (Head Festschrift), LNCS 2950, pp. 36–48, 2004. c © Springer-Verlag Berlin Heidelberg 2004 Balance Machines: Computing = Balancing 37 balancing. Armed with the computing = balancing intuition, we can see basic computational operations in a different light. In fact, an important result of this paper is that this sort of intuition suffices to conceptualize/implement any computation performed by a conventional computer. Fig. 1. Physical balance. The rest of the paper is organized as follows: Section 2 gives a brief introduction to the proposed computational model; Section 3 discusses a variety of examples showing how the model can be made to do basic computations; Section 4 is a brief note on the universality feature of the model; Section 5 reviews the notion of bilateral computing and discusses an application (solving the classic SAT problem); Section 6 concludes the paper. 2 The Balance Machine Model At the core of the proposed natural computational model are components that resemble a physical balance. In ancient times, the shopkeeper at a grocery store would place a standard weight in the left pan and would try to load the right pan with a commodity whose weight equals that on the left pan, typically through repeated attempts. The physical balance of our model, though, has an intrinsic self-regulating mechanism: it can automatically load (without human intervention) the right pan with an object whose weight equals the one on the left pan. See Figure 2 for a possible implementation of the self-regulating mechanism. In general, unlike the one in Figure 2, a balance machine may have more than just two pans. There are two types of pans: pans carrying fixed weights which remain unaltered by the computation and pans with variable (fluid) weights that are changed by activating the filler–spiller outlets. Some of the fixed weights represent the inputs, and some of the variable ones represent outputs. The inputs and outputs of balance machines are, by default, non–negative reals unless stated otherwise. The following steps comprise a typical computation by a given balance machine: 38 Joshua J. Arulanandham, Cristian S. Calude, and Michael J. Dinneen