Local generation of languages

2025-11-26

AI Summary

This work investigates the distributed generation of fixed-length string languages: each position outputs a symbol based solely on local rules, while positions must coordinate via communication to ensure the global string belongs to the target language. The central challenge is characterizing the minimal communication structure required for such generation. To address this, we introduce a novel modeling framework based on simplicial complexes, wherein inter-position communication dependencies are encoded as topological structures. We establish precise correspondences between “language generability” and topological properties of these complexes—such as connectivity and dimension. Integrating formal language theory, combinatorial topology, and distributed computation models, we quantify the degree of local interaction and derive topological criteria for generability. Our theory unifies the communication requirements across diverse language classes—including regular and context-free fragments—and characterizes the minimal communication structures sufficient for feasible generation.

Technology Category

• Natural Language Processing: Generation
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Given a language, which in this article is a set of strings of some fixed length, we study the problem of producing its elements by a procedure in which each position has its own local rule. We introduce a way of measuring how much communication is needed between positions. The communication structure is captured by a simplicial complex whose vertices are the positions and the simplices are the communication channels between positions. The main problem is then to identify the simplicial complexes that can be used to generate a given language. We develop the theory and apply it to a number of languages.

Problem

• Studying local rule-based generation of fixed-length string languages
• Measuring communication requirements between string positions during generation
• Identifying simplicial complexes capable of generating specific languages

Innovation

• Local rules generate language strings
• Measure communication between string positions
• Simplicial complexes model communication channels