Co-evolution of self-replication and function in a digital primordial soup

While traditional evolutionary algorithms hard-code reproduction, self-replication can emerge spontaneously within digital ``primordial soups''. This paper investigates the co-evolution of this emergent self-replication alongside problem-solving capabilities. We initialize a population of random 32-byte Z80 assembly programs, requiring self-replication to arise purely through random assembly-level mutations and pairwise program interactions. To link these behaviors, we introduce a task-based validation step: correctly evaluating a polynomial raises a program's interaction probability above a baseline rate. Our experiments yield four primary findings. First, self-replication and mathematical problem-solving successfully co-evolve from initial randomness. Second, the pressure to compute accelerates the emergence of compact, robust reproductive architectures that preserve memory for task execution. Third, applying metabolic constraints increases the likelihood that programs evolve conditional halting, terminating early during validation while bypassing the halt during interaction to execute block-copy replication. Finally, when programs are partitioned into spatial task niches, spontaneous self-replication generates an emergent learning curriculum, utilizing simple solutions as stepping stones toward complex polynomials. Altogether, these results demonstrate an interactive feedback loop: environmental task demands actively shape the physical architecture of self-replication, while spontaneous replication alters the evolutionary trajectory of functional problem-solving.

Density-embedding layers: a general framework for adaptive receptive fields

The effectiveness and performance of artificial neural networks, particularly for visual tasks, depends in crucial ways on the receptive field of neurons. The receptive field itself depends on the interplay between several architectural aspects, including sparsity, pooling, and activation functions. In recent literature there are several ad hoc proposals trying to make receptive fields more flexible and adaptive to data. For instance, different parameterizations of convolutional and pooling layers have been proposed to increase their adaptivity. In this paper, we propose the novel theoretical framework of density-embedded layers, generalizing the transformation represented by a neuron. Specifically, the affine transformation applied on the input is replaced by a scalar product of the input, suitably represented as a piecewise constant function, with a density function associated with the neuron. This density is shown to describe directly the receptive field of the neuron. Crucially, by suitably representing such a density as a linear combination of a parametric family of functions, we can efficiently train the densities by means of any automatic differentiation system, making it adaptable to the problem at hand, and computationally efficient to evaluate. This framework captures and generalizes recent methods, allowing a fine tuning of the receptive field. In the paper, we define some novel layers and we experimentally validate them on the classic MNIST dataset.