Undecidability of Adjacent Equality for Insertion, Shuffle, and Crossover Language Operations

[2608.27755] Undecidability of Adjacent Equality for Insertion, Shuffle, and Crossover Language Operations Skip to main content Search Submit Donate Log in Computer Science > Formal Languages and Automata Theory arXiv:2608.27755 (cs) [Submitted on 27 Aug 2026 ( v1 ), last revised 1 Oct 2026 (this version, v2)] Title: Undecidability of Adjacent Equality for Insertion, Shuffle, and Crossover Language Operations Authors: Charles E. Hughes View PDF Abstract: We study a family of language operations based on insertion, shuffle, and crossover and investigate the undecidability of adjacent equality together with finite convergence and associated spectrum questions. Insertion and shuffle operations on formal languages arise in formal language theory, models of concurrency, and biologically inspired computation. This paper studies a different question from the usual closure problem, specifically whether an increasing sequence of languages generated by repeated insertion, or by increasing the permitted degree of bounded shuffle, reaches an instance of adjacent equality after finitely many stages. We show that several such adjacent equality questions are undecidable. In particular, reaching such an adjacent equality event is undecidable for each of the following: iterated insertion of a regular language into a context-free language; bounded shuffle of a regular language with a context-free language as the bound increases; and the corresponding self-insertion and self-bounded-shuffle hierarchies for context-free languages. The new reductions proceed directly from the undecidability of context-free-language universality, using separator-delimited block constructions and, for self-operations, an absorbing regular language of guard violations. Earlier trace-based proofs relied on mortality and uniform halting. More generally, we investigate finite-stage equality and stabilization (persistent equality) in hierarchies generated by insertion and bounded shuffle. In addition to giving substantially simpler proofs of earlier undecidability results, we obtain general criteria for one-step equality, develop new reductions for self-insertion, and identify several open problems, including structural questions concerning insertion depth and degree whose resolution determines whether adjacent equality necessarily implies permanent stabilization. Subjects: Formal Languages and Automata Theory (cs.FL) MSC classes: 68Q45, 68Q04 ACM classes: F.4.3 Cite as: arXiv:2608.27755 [cs.FL] (or arXiv:2608.27755v2 [cs.FL] for this version) https://doi.org/10.48550/arXiv.2608.27755 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Charles Hughes [ view email ] [v1] Thu, 27 Aug 2026 22:35:14 UTC (258 KB) [v2] Thu, 1 Oct 2026 17:02:39 UTC (258 KB) Full-text links: Access Paper: View PDF view license Additional Features Audio Summary Current browse context: cs.FL < prev | next > new | recent | 2026-08 Change to browse by: cs References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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