Universal 2-State 24-Neighborhood Asynchronous Cellular Automaton with Inner-Independent Open Rule

Abstract

This paper proposes a computationally universal two-dimensional square-lattice asynchronous cellular automaton, in which cells have merely two states. The transition rule of a cell is specified by the pattern of the cells or its rotation-symmetric or reflection-symmetric rules included in distances 1 and 2 Moore neighborhood, which does not include its own cell (inner-independent). In a former model, we defined only the rules where the neighboring patterns are taken into account in computation. In the current model, the transition rule is an open rule, which is defined completely in terms of updating 0 or 1. Universality of the model is proven through the construction of a glider and three circuit primitives on the cell space, which are universal for the class of Delay-Insensitive circuits. Correct operation of the circuit is proven through a validity check algorithm which checks all generated patterns from an initial configuration to a final configuration of the operation. As a result, the number of rules to update to 1 is found to be 1618, not including the rotation-symmetric and reflection-symmetric versions of the rules. We show how to determine the open rules of the model, and how to define the energy function to obtain an energy evolution during the operation of the circuit.