Abstract This essay extends the unified query algebra framework by incorporating two critical capabilities missing from the original formulation: general aggregation operations and hierarchical data structures. We demonstrate that while posting lists provide a powerful abstraction for many scenarios, they impose restrictions that prevent the framework from handling certain important
Abstract This supplementary essay extends the theoretical framework of unified query algebras across heterogeneous data paradigms by incorporating join operations and multiple field handling. We present a rigorous mathematical formulation that generalizes the posting list abstraction to accommodate joined document tuples while preserving the algebraic properties of the original framework.
1. Introduction: Formalizing Data Processing Paradigms In database system theory, we can formalize two fundamental paradigms that govern architectural decisions: the proactive (eager) paradigm and the reactive (lazy) paradigm. These approaches represent distinct computational models with profound implications for system behavior, performance characteristics, and theoretical properties. Definition 1.1: A
Abstract This research essay presents a formal algebraic framework that unifies operations across transaction processing, text retrieval, and vector search paradigms within a single mathematical structure. By establishing posting lists as a universal abstraction with well-defined algebraic properties, we develop a comprehensive theoretical foundation that preserves the expressivity of each
Introduction Modern applications increasingly demand database systems that seamlessly integrate traditional transaction processing with advanced search capabilities. This essay explores architectural innovations that enable efficient faceted search, hybrid vector-text querying with full boolean expressivity, and unified query optimization across heterogeneous paradigms. By examining both theoretical foundations and practical implementation strategies,
1. Introduction In the domain of similarity search for high-dimensional vectors, approximate nearest neighbor (ANN) algorithms have become indispensable for applications ranging from recommendation systems to image retrieval. Modern vector databases commonly employ sophisticated indexing methods, with HNSW (Hierarchical Navigable Small World) combined with IVF (Inverted File) and PQ (Product
1. Introduction In our previous explorations of probabilistic frameworks for information retrieval, we examined how transformations like softmax and sigmoid convert raw similarity scores into probabilities, enabling principled fusion of heterogeneous retrieval systems. While these transformations provide elegant mathematical foundations for ranking, they introduce a critical challenge when handling conjunctive
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,
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
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
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
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