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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
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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—
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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
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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
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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
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Introduction In our previous explorations of programming language theory—from the philosophical underpinnings in "A Cross-Disciplinary Analysis," to the rigorous mathematical frameworks in "A Formal Perspective," to the practical implementations in "From Theory to Practice"—we have progressively bridged the gap between abstract theory
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Introduction The advent of Large Language Models (LLMs) has fundamentally altered our understanding of the relationship between natural language and computation. As established in "Syntax, Semantics, and Computation: LLMs and Their Computational Boundaries," these models occupy a unique position in the landscape of computational systems—neither fully Turing-complete
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Introduction Our previous discussions on syntax, semantics, and segfaults have traversed the spectrum from cross-disciplinary analysis to formal mathematical treatments to practical implementations. In "Syntax, Semantics, and Segfaults: A Cross-Disciplinary Analysis," we explored how structured rules of language (syntax) and their interpretation (semantics) govern both human communication and
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Introduction In our previous analyses, "Syntax, Semantics, and Segfaults: A Cross-Disciplinary Analysis" and "Syntax, Semantics, and Segfaults: A Formal Perspective," we explored the theoretical foundations that govern programming languages, from the philosophical underpinnings of meaning to the rigorous mathematical frameworks that ensure program correctness. We examined