Construction Defining Functionality:A Constructive Perspective on Functions through Their Generated Structures
Yumiko Nishiyama
TL;DR
This work introduces Construction Defining Functionality (CDF), a framework that reframes functions as generators of hierarchical structural spaces $S(f)$, decomposed into syntactic, semantic, and logical components $S_{\,\mathrm{syn}}(f)$, $S_{\,\mathrm{sem}}(f)$, and $S_{\,\mathrm{log}}(f)$. By linking these spaces to model-theoretic concepts and levels of logical expressibility, the authors establish a comprehensive classification that spans a hierarchical taxonomy of function types and a parallel axes-based analysis of the generated spaces. Key contributions include a formal definition of $S(f)$, a basic construction procedure, theoretical perspectives on function complexity via CDF, and a multi-faceted classification framework with both core and supplementary axes, plus a Master Table and analytical methods for evaluation. The work aims to provide a unifying lens across logic, computation, and spatial structures, with potential practical impact in areas such as AI, automated theorem proving, and formal analysis through a structured, extensible vocabulary for function-generated structures.
Abstract
In this work, we propose the concept of Construction Defining Functionality (CDF), which characterizes functions by the structural spaces they generate through iteration,recursion, and logical application. By viewing functions as generators of hierarchical structures, we formalize these generated structural spaces and organize a framework to classify and mathematically model their properties. The organized CDF framework captures the intrinsic constructive behaviors of functions via their generated structural spaces.