Among the different variants of Genetic Programming (GP), Geometric Semantic GP (GSGP) has proved to be both efficient and effective in finding good solutions. The fact that the operators of GSGP operate on the semantics of the individuals in a clear way provides guarantees on the way the search is performed. GSGP is not, however, free from limitations like the premature convergence of the population to a small - and possibly sub-optimal-area of the search space. One reason for this issue could be the fact that good individuals can quickly "spread" in the population suppressing the emergence of competition. To mitigate this problem, we impose a cellular automata (CA) inspired communication topology over GSGP. In CAs a collection of agents (as finite state automata) are positioned in a n-dimensional periodic grid and communicates only locally with the automata in their neighbourhoods. Similarly, we assign a location to each individual on an n-dimensional grid and the entire evolution for an individual will happen locally by considering, for each individual, only the individuals in its neighbourhood. Specifically, we present an algorithm in which, for each generation, a subset of the neighbourhood of each individual is sampled and the selection for the given cell in the grid is performed by extracting the two best individuals of this subset, which are employed as parents for the Geometric Semantic Crossover. We compare this cellular GSGP (cGSGP) approach with standard GSGP on eight regression problems, showing that it can provide better solutions than GSGP. Moreover, by analyzing convergence rates, we show that the improvement is observable regardless of the number of executed generations. As a side effect, we additionally show that combining a small-neighbourhood-based cellular spatial structure with GSGP helps in producing smaller solutions. Finally, we measure the spatial autocorrelation of the population by adopting the Moran's I coefficient to provide an overview of the diversity, showing that our cellular spatial structure helps in providing better diversity during the early stages of the evolution
