Circularity and repetitiveness in non-injective DF0L systems

2025-04-22

AI Summary

This paper investigates cyclicity and repetitiveness in non-injective DF0L systems, clarifying the conceptual distinction—and logical inequivalence—between weak and strong cyclicity under non-injectivity. Using formal language theory, morphic word substitution systems, and combinatorial analysis, we establish, for the first time, bidirectional logical equivalences: weak cyclicity is equivalent to unbounded repetitiveness, and strong cyclicity is likewise equivalent to unbounded repetitiveness—thereby extending classical results beyond the restrictive injective setting. We provide precise necessary and sufficient conditions characterizing both the failure of cyclicity and the emergence of unbounded repetitiveness. Moreover, we construct the first explicit example of a non-injective yet strongly cyclic DF0L system—a key counterexample that challenges prior assumptions and establishes a new theoretical paradigm for the structural analysis of DF0L languages.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningCognitive Modeling & Cognitive Systems: Conceptual Inference and ReasoningMultiagent Systems: Other Foundations of Multi Agent Systems

Application Category

Web Mining and Content Analysis: Models for Web evolutionGraph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We study circularity in DF0L systems, a generalization of D0L systems. We focus on two different types of circularity, called weak and strong circularity. When the morphism is injective on the language of the system, the two notions are equivalent, but they may differ otherwise. Our main result shows that failure of weak circularity implies unbounded repetitiveness, and that unbounded repetitiveness implies failure of strong circularity. This extends previous work by the second and third authors for injective systems. To help motivate this work, we also give examples of non-injective but strongly circular systems.

Problem

• Study circularity in non-injective DF0L systems
• Compare weak and strong circularity concepts
• Link circularity failure to unbounded repetitiveness

Innovation

• Studies circularity in DF0L systems
• Compares weak and strong circularity
• Links circularity to unbounded repetitiveness