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
Thoughts
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
Thoughts
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
Thoughts
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
Thoughts
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
Thoughts
Introduction In programming language theory, syntax and semantics form the twin pillars that define a language's structure and meaning. A program must be syntactically well-formed (obeying the grammar rules of the language) and semantically well-defined (having a clear meaning or behavior for each construct). When either pillar is
Thoughts
Introduction Modern thought in linguistics, philosophy, and computer science converges on a fundamental duality: syntax and semantics. Syntax refers to the formal structure or grammar of a language, while semantics concerns the meaning conveyed by that structure. In natural language, thinkers from Gottlob Frege to Ludwig Wittgenstein have grappled with
Database
Introduction Modern search systems often combine vector search (dense embedding retrieval) with traditional text search (BM25/TF-IDF) to improve result relevance. This is known as hybrid search, which leverages both semantic similarity and exact keyword matching. In practice, hybrid approaches have shown significant improvements in search quality across various domains