The field of artificial intelligence and cognitive modeling is rapidly advancing, with a focus on developing more efficient and effective methods for generating complex sequences, learning algorithms, and modeling human cognition. Recent research has explored the use of non-Markovian rules and neural networks to generate de Bruijn sequences, which has led to the development of novel methodologies for generating these sequences. Additionally, studies have investigated the ability of neural networks to generalize and extrapolate to new data, with findings suggesting that these networks can learn to solve tasks and generalize to new situations. The use of group theoretic analysis has also been applied to the study of base addition and symmetry learning, with results showing that simple neural networks can achieve radical generalization with the right input format and carry function. Furthermore, biologically realistic models of the brain have been developed to simulate language acquisition, with findings demonstrating that these models can learn the semantics of words, their syntactic role, and the word order of a language. Notable papers in this area include: A paper on rule-based generation of de Bruijn sequences, which introduces a novel methodology that combines derived properties and a neural network-based classifier to generate these sequences. A paper on algorithm development in neural networks, which undertakes a case study of the learning dynamics of recurrent neural networks trained on the streaming parity task and finds an implicit representational merger effect that can be interpreted as the construction of a finite automaton. A paper on a group theoretic analysis of base addition, which investigates the capacity of neural networks to discover and implement symmetry functions and finds that learning speed is closely correlated with carry function structure. A paper on simulated language acquisition in a biologically realistic model of the brain, which introduces a simple mathematical formulation of basic principles of neuroscience and implements a simulated neuromorphic system that can learn the semantics of words and generate novel sentences.