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Emergence of Self-Identity in AI: A Mathematical Framework and Empirical Study with Generative Large Language Models

Minhyeok Lee

Abstract

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 $C \subseteq \mathcal{M}$ in a metric space $(\mathcal{M}, d_{\mathcal{M}})$, and a continuous mapping $I: \mathcal{M} \to \mathcal{S}$ that maintains consistent self-recognition across this continuum, where $(\mathcal{S}, d_{\mathcal{S}})$ 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. 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.

Emergence of Self-Identity in AI: A Mathematical Framework and Empirical Study with Generative Large Language Models

Abstract

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 in a metric space , and a continuous mapping that maintains consistent self-recognition across this continuum, where 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. 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.

Paper Structure

This paper contains 33 sections, 3 theorems, 16 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Foundations of the Self
  3. Preliminaries
  4. Identity Recognition and Belief Functions
  5. Formalizing the Conditions for Self
  6. Main Results
  7. Designing an Artificial Intelligence Agent with a Self
  8. Artificial Memory Space
  9. Artificial Self Space
  10. Identity Recognition Function
  11. Belief Function and Probability Measures
  12. Constructing the Continuum of Memories
  13. Learning Algorithms
  14. Ensuring Conditions for 'Having a Self'
  15. Convergence of LoRA Training and Self-Identity Formation
  16. ...and 18 more sections

Key Result

Theorem 2.9

If an entity satisfies Conditions cond:continuum and cond:self-recognition, and if the image $I(C)$ lies entirely within a connected component of $\mathcal{S}$ where $I$ is constant, then there exists a self-identity $s^* \in \mathcal{S}$ such that $I(m) = s^*$ for all $m \in C$. Therefore, the enti

Figures (5)

  • Figure 1: (A) Training Loss Evolution: Decline in loss over 20 epochs. (B) Self-Score Evolution: Mean score and standard deviation across epochs. (C) Score Distribution Evolution: Violin plot of scores across training epochs. (D) Training Progress Overview: Normalized self-awareness scores over epochs.
  • Figure 2: Average Self-Awareness Scores for Different Prompts Across Epochs. Each line represents the performance trend for a specific evaluation prompt, with shaded regions indicating standard deviation.
  • Figure 3: (A) Score Distribution Before and After Fine-Tuning: Shift in scores across different prompts. (B) Average Response Length. (C) Unique Word Usage. (D) Score Improvement by Prompt.
  • Figure 4: (A) Word Frequency Comparison (Top 30 Words): Absolute frequency of the 30 most common words before and after fine-tuning. (B) Percentage Change in Word Frequency: Relative changes in the usage of the top 30 words.
  • Figure 5: Word Cloud Comparison: Left - Pre-training vocabulary distribution. Right - Post-training vocabulary distribution.

Theorems & Definitions (21)

  • Definition 2.1: Memory Space
  • Definition 2.2: Self Space
  • Definition 2.3: Continuum of Memories
  • Definition 2.5: Identity Recognition Function
  • Definition 2.6: Belief Function
  • Theorem 2.9: Constancy of Self-Identity
  • proof
  • Proposition 2.10
  • proof
  • Definition 3.1: Artificial Memory Space
  • ...and 11 more