Arithmetic Operations with Self-Replicating Loops

Abstract

We present a possible collision-based implementation of arithmetic functions using a self-replicating cellular automaton capable of construction and computation. Our automaton makes use of some of the concepts developed by Langton for his self-replicating automaton, but provides the added advantage of being able to perform independent constructional and computational tasks along with self-replication. Our automaton is capable, like Langton’s automaton and with comparable complexity, of simple self-replication, but it also provides (at the cost, naturally, of increased complexity) the option of attaching to the automaton an executable program which will be duplicated and executed in each of the copies of the automaton. The arithmetic functions that we have implemented are performed by storing a dedicated program (sequence of states) on self-replicating loops, and letting the loops retrieve the operands, exchange data among themselves, and perform the calculations according to a set of rules. To determine the rules required for addition and multiplication, we exploited an existing algorithm for collision-based computation in the cellular automata environment and adapted it to exploit the features of self-replicating loops. This approach allowed us to study a variety of issues (synchronization, data exchange, etc.) related to the use of self-replicating machines for complex operations.