On the nature of two-bit multiplier landscapes

The two-bit multiplier is a simple electronic circuit, small enough to be evolvable, and practically useful for the implementation of many digital systems. In this paper we study the structure of the two-bit multiplier fitness landscapes generated by circuit evolution on an idealised model of a field-programmable gate array. The two-bit multiplier landscapes are challenging. The difficulty in studying these landscapes stems from the genotype representation which allows us to evolve the functionality and connectivity of an array of logic cells. Here, the genotypes are simply strings defined over two completely different alphabets. This makes the study of the corresponding landscapes much more involved. We outline a model for studying the two-bit multiplier landscapes and estimate the amplitudes derived from the Fourier transform of these landscape. We show that the two-bit multiplier landscapes can be characterised in terms of subspaces, determined by the interactions between the genotype partitions.