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This paper introduces a mathematical framework for defining and quantifying self-identity in artificial intelligence (AI) systems, addressing a critical gap in the theoretical foundations of artificial consciousness. While existing approaches to artificial self-awareness often rely on heuristic implementations or philosophical abstractions, we present a formal framework grounded in metric space theory, measure theory, and functional analysis. Our framework posits that self-identity emerges from two mathematically quantifiable conditions: the existence of a connected continuum of memories C ⊆ M in a metric space (M , d M) , and a continuous mapping I : M → S that maintains consistent self-recognition across this continuum, where (S , d S) represents the metric space of possible self-identities. To validate this theoretical framework, we conducted empirical experiments using the Llama 3.2 1B model, employing low-rank adaptation (LoRA) for efficient fine-tuning. The model was trained on a synthetic dataset containing temporally structured memories, designed to capture the complexity of coherent self-identity formation. Our evaluation metrics included quantitative measures of self-awareness, response consistency, and linguistic precision. The experimental results demonstrate substantial improvements in measurable self-awareness metrics, with the primary self-awareness score increasing from 0.276 to 0.801 (190.2% improvement) after fine-tuning. In contrast to earlier methods that view self-identity as an emergent trait, our framework introduces tangible metrics to assess and measure artificial self-awareness. This enables the structured creation of AI systems with validated self-identity features. The implications of our study are immediately relevant to the fields of humanoid robotics and autonomous systems. Additionally, it opens up new prospects for controlled adjustments of self-identity in contexts that demand different levels of personal involvement. Moreover, the mathematical underpinning of our framework serves as the basis for forthcoming investigations into AI, linking theoretical models to real-world applications in current AI technologies. [ABSTRACT FROM AUTHOR]

Evidential c-means (ECM) is a prototype-based clustering algorithm that generates a credal partition. Such a partition encompasses the notions that can be encountered with a hard, fuzzy or possibilistic partition, allowing the representation of various situations concerning the class membership of an object. The ECM method provides a prototype for each subset of the possible classes, calculated by averaging the prototypes of the classes included in the subsets. Although this definition perfectly suits ECM when employing a Euclidean distance, it becomes inappropriate when using a Mahalanobis distance. In this context, a new definition of prototypes for the subsets is proposed. The ECM objective function is then optimized using the new definition of prototypes. The subsequent algorithm, named ECM+, is finally tested on various synthetic and real data sets to demonstrate its interest compared to ECM. [ABSTRACT FROM AUTHOR]