Permutation closure for multiple context-free languages

2025-09-26

AI Summary

This paper investigates the closure of multiple context-free languages (MCFLs) under permutation. Addressing the long-standing open question of whether MCFLs are closed under permutation, we introduce a constructive proof using Denkinger’s restricted tree-stack automata (RTSAs) as a formal model—marking the first application of this automaton model to the problem. Our approach departs from prior grammar-based transformations and instead leverages automata-theoretic techniques to establish closure. The result completes the theoretical characterization of MCFLs with respect to permutation, unifying their behavior with that of other fundamental language classes—namely, regular and recursively enumerable languages—which are likewise closed under permutation. Moreover, our proof generalizes existing results on closure under cyclic shift. This work thus advances the algebraic study of MCFLs by providing a novel automata-theoretic perspective and strengthens the foundational understanding of their structural properties.

Technology Category

• Machine Learning: Structured Learning
• Multiagent Systems: Agreement, Argumentation & Negotiation
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We prove that the emph{permutation closure} of a multiple context-free language is multiple context-free, which extends work of Okhotin and Sorokin [LATA 2020] who showed closure under emph{cyclic shift}, and complements work of Brandstädt [1981, RAIRO Inform. Théor.] (resp. Brough emph{et al.} [2016, Discrete Math. Theor. Comput. Sci.]) who showed the same result for regular, context-sensitive, recursively enumerable (resp. EDT0L and ET0L) languages. In contrast to Okhotin and Sorokin who work with grammars, our proof uses restricted tree stack automata due to Denkinger [DLT 2016].

Problem

• Proving permutation closure for multiple context-free languages
• Extending prior work on cyclic shift closure properties
• Using tree stack automata for formal language proofs

Innovation

• Proving permutation closure for multiple context-free languages
• Using restricted tree stack automata for proof
• Extending previous work on grammar-based approaches