Programming
Abstract This essay presents a rigorous mathematical proof demonstrating that C++ template metaprogramming is Turing complete. By constructing a formal mapping between the components of a Turing machine and C++ template metaprogramming mechanisms, we establish that templates can simulate any computation performable by a Turing machine. Beyond the theoretical implications,
Thoughts
Introduction Our previous exploration of "The Mathematical Unity of Sigmoid, Perceptron, Logistic Regression, and Softmax" established the foundational equivalences between these core machine learning concepts. We demonstrated how sigmoid-activated perceptrons are mathematically identical to logistic regression, and how softmax functions generalize sigmoid to multi-class scenarios. This mathematical unity
Thoughts
Introduction In the landscape of machine learning and neural networks, certain mathematical formulations stand as foundational pillars upon which more complex architectures are built. Among these are the sigmoid function, the perceptron model, logistic regression, and the softmax function. While often introduced as separate concepts, these mathematical constructs share profound
Thoughts
1. Introduction The transformation of BM25 similarity scores into probability estimates represents a critical challenge in information retrieval systems. This process is essential for creating interpretable search results and enabling integration with probabilistic frameworks. While supervised learning approaches using query-document relevance pairs typically yield optimal results, practical implementations often face
Thoughts
Introduction In our previous exploration of probability transformations in vector search, we examined how softmax enables the normalization of disparate scoring systems into comparable probabilistic frameworks. This follow-up article delves deeper into the mathematical theory underpinning these transformations, with a specific focus on Bayesian probabilistic frameworks and their application to
Thoughts
Introduction The previous essays in this series have explored the computational nature of Large Language Models (LLMs) and their relationship to classical computation theory. In "Syntax, Semantics, and Computation: LLMs and Their Computational Boundaries," we established that LLMs themselves are not Turing-complete in the traditional sense due to
Thoughts
Introduction Martin Heidegger and Ludwig Wittgenstein represent two of the most influential philosophical voices of the 20th century, each offering profound insights into the nature of understanding, language, and human existence. While they share certain concerns—particularly a critique of traditional metaphysics and a focus on everyday practices—their approaches
Thoughts
Introduction In our previous exploration of "The Illusion of Direct Experience," we examined how human perception is not the immediate access to reality it appears to be, but rather a sophisticated neural construction. We argued that the brain's interpretive processes—translating sensory signals into coherent experiences—
Thoughts
Introduction The advent of Large Language Models (LLMs) has precipitated a fundamental reconsideration of what constitutes "understanding." A persistent critique of these systems centers on their lack of "direct sensory experience" of the world they describe through language. According to critics, LLMs merely process tokens—symbolic
Thoughts
Introduction The advent of Large Language Models (LLMs) has precipitated a profound philosophical quandary regarding the nature of understanding itself. These systems—trained on vast corpora of human-written text—demonstrate capabilities that would unquestionably be labeled as "understanding" if exhibited by humans. They can interpret nuanced requests, recognize
Thoughts
Abstract This research essay explores a novel approach to understanding Large Language Models (LLMs) through computational graph logging and differential analysis. We propose treating LLMs as complex mathematical functions and extracting their functional behavior by systematically logging kernel operations during inference. Our approach introduces two key innovations: (1) probabilistic kernel
Thoughts
Introduction The emergence of Large Language Models (LLMs) has prompted fundamental questions about their computational capabilities within the theoretical landscape of computer science. While these models demonstrate remarkable linguistic abilities, their precise classification within the hierarchy of computational systems remains an area of active exploration. Simultaneously, the Universal Approximation Theorem