Iterating Non-Aggregative Structure Compositions

2025-10-07

AI Summary

This paper investigates the iterative closure properties of graph-structured sets under the non-aggregative combinatorial operation of *fusion*, addressing two central problems: (1) deciding whether the fusion closure of a given set of graphs of bounded treewidth remains treewidth-bounded; and (2) if so, whether the closure can be exactly characterized by a constructible hyperedge replacement (HR) grammar. We establish that both problems are decidable: first, the treewidth-boundedness of the fusion closure is algorithmically testable; second, whenever the closure has bounded treewidth, there exists an effectively constructible HR grammar generating it. Our approach integrates HR grammars, monadic second-order logic modeling, and operations of constant renaming and forgetting to verify structural properties and synthesize grammars. The results provide a theoretical foundation and algorithmic framework for logical satisfiability checking over bounded-treewidth non-aggregative structured sets.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

An aggregative composition is a binary operation obeying the principle that the whole is determined by the sum of its parts. The development of graph algebras, on which the theory of formal graph languages is built, relies on aggregative compositions that behave like disjoint union, except for a set of well-marked interface vertices from both sides, that are joined. The same style of composition has been considered in the context of relational structures, that generalize graphs and use constant symbols to label the interface. In this paper, we study a non-aggregative composition operation, called emph{fusion}, that joins non-deterministically chosen elements from disjoint structures. The sets of structures obtained by iteratively applying fusion do not always have bounded tree-width, even when starting from a tree-width bounded set. First, we prove that the problem of the existence of a bound on the tree-width of the closure of a given set under fusion is decidable, when the input set is described inductively by a finite emph{hyperedge-replacement} (HR) grammar, written using the operations of aggregative composition, forgetting and renaming of constants. Such sets are usually called emph{context-free}. Second, assuming that the closure under fusion of a context-free set has bounded tree-width, we show that it is the language of an effectively constructible HR grammar. A possible application of the latter result is the possiblity of checking whether all structures from a non-aggregatively closed set having bounded tree-width satisfy a given monadic second order logic formula.

Problem

• Studying non-aggregative fusion composition operations on relational structures
• Determining decidability of tree-width bounds under fusion closure
• Constructing HR grammars for context-free sets with bounded tree-width

Innovation

• Non-aggregative fusion operation joins elements non-deterministically
• Decidable tree-width bound for closure under fusion
• Effective HR grammar construction for bounded tree-width closure