Binary Addition on Cellular Automata

A cellular auto mata approach to fast addit ion of binary numbers is pr esented. This approach utilizes the intrinsically parallel nature of the cellular auto maton to add binary numbers represented as on/off pixels on the computer screen . Numbers nbit long are added serially pairwise until the final sum is ob tained . This approach can eas ily be genera lized to numbers of any base, but is const rained by the number of plan es available on the Cellular Automata Machin e. The concept of cellular auto mata was introduced by von Neumann and Ulam to mod el physical , chemical, an d biologi cal systems. Today cellular auto mata are used for several different purposes. One such application of cellul ar automata is in the area of computat ion. It has been shown that cellular aut omata can be used as general purpose compute rs and may therefore be used as general para digms for par allel computat ion [3]. This paper concentrates on the capabilit ies of cellular automata for bin ary addition . The approach given in this pap er to solving this pr oblem is intended to emphasize the simplicity of cellular automata in implementing elementary mathematical operations. The operation of bin ar y addit ion was implemented on a Cellul ar Automata Machine (CAM) designed by the Information Mechanics Group of the Massachusetts Institute of Technology. The rul es discussed in the pap er may assume a knowledge of the language const ruc ts used (see [1]). The approach given has certain unique features. The CAM screen may be filled up almost ent irely with bin ary numbers, thus allowing as many as 256 numbers, each having a length of up to 256 bits , t o be added. These limitations ar e solely machine depend ent , since CAM only allows a screen of 256 x 256 pixels. It must be mentioned here that the appro ach does not take care of overflow. To unders tand this approach, it might help to visualize the numbers as beads on an ab acus. The topmost pair of numbers are added first , and the sum obtained is then added to the number appearing beneath them . This goes on until all the numbers are exhausted, afte r which all that remains is the final sum of all the numbers . Henceforth the final sum is added with 0 until stoppe d. In adding two numbers , the corresponding 480 Bhavin Sheth , Prantik Nag, and Robert W. Hellwarth Figure 1: The numbers 1012 a nd 0112 are to be added. The row of ones on plane 2 can be seen clearly. T he numbe r 0012 lies in the row right below t he nu mber 1012 and on t he same row as t he ones on pla ne 2. Figure 2: The numbers are shown . P lane 2 is hidden by an appropr ia te color map . (See below for t he color map.) bit s are added in par allel and the carr ies are obtained . These carr ies are added to the par tial sum obtained in the next ste p. Once again carries will be obtained , and will be added to t he next higher bits in the following step. This procedure repeats 255 t imes, by which time the sum is obtained . A binary number is represented as a series of on /off pixels on the display. A binary '1' will app ear as a one on plane 0 and zeros on the other three planes. A binary '0' will be a zero on all four planes. A binary numb er is represented length-wise on the screen, with th e left-most bit being the most significant bit. Let us first examine the addit ion of two bin ary numb ers. The presence of th e row of ones on plane 2 serves to distinguish between the two numbers. After the completion of the first step, the number at the top disappears . This disappear ance of the number is taken care of by Binary Addition on Cellular Automat a Figure 3: The first step is executed. The corresponding bits are added and the carr ies are saved and will be added to the numb er in the following steps . P lane 0 at pr esent has 0110, while plane 1 has 0010, which will be added to the number 0110. Not e th at the carries will be shifted to the right by one bit, bu t will be added to the appropriate bits by the use of EAST.