Assembly Theory provides a promising framework to explain the complexity of systems such as molecular structures and the origins of life, with broad applicability across various disciplines. In this study, we explore and consolidate different aspects of Assembly Theory by introducing a simplified Toy Model to simulate the autocatalytic formation of complex structures. This model abstracts the molecular formation process, focusing on the probabilistic control of catalysis rather than the intricate interactions found in organic chemistry. We establish a connection between probabilistic catalysis events and key principles of Assembly Theory, particularly the probability of a possible construction path in the formation of a complex object, and examine how the assembly of complex objects is impacted by the presence of autocatalysis. Our findings suggest that this presence of autocatalysis tends to favor longer consecutive construction sequences in environments with a low probability of catalysis, while this bias diminishes in environments with higher catalysis probabilities, highlighting the significant influence of environmental factors on the assembly of complex structures.

Assembly theory provides a promising framework to explain the complexity of systems such as molecular structures and the origins of life, with broad applicability across various disciplines. In this study, we explore and consolidate different aspects of assembly theory by introducing a simplified toy model to simulate the autocatalytic formation of complex structures. This model abstracts the molecular formation process, focusing on the probabilistic control of catalysis rather than the intricate interactions found in organic chemistry. We establish a connection between probabilistic catalysis events and key principles of assembly theory, particularly the probability of a possible construction path in the formation of a complex object, and examine how the assembly of complex objects is impacted by the presence of autocatalysis. Our findings suggest that autocatalysis tends to favor longer consecutive construction sequences in environments with low catalysis probability, while this bias diminishes in environments with higher catalysis probabilities, highlighting the significant influence of environmental factors on the assembly of complex structures.

We introduce a class of activation functions for neural networks based on the Blancmange curve and the Weierstrass function. These activation functions are designed to be qualitatively similar to standard functions such as tanh, while incorporating fractal, self-affine, and self-similar patterns. The goal is to force the neural network to produce non-smooth outputs, preventing excessive smoothing in predictions and thus increasing expressivity.We evaluate this new class of activation functions on standard classification problems and demonstrate how fractal properties of activation functions influence the expressivity of neural networks.