Regular Grammars for Sets of Graphs of Tree-Width $2$

2024-08-02

AI Summary

This work generalizes regular grammars from strings to graphs, targeting unordered, unranked trees and graphs of treewidth ≤ 2, aiming to precisely characterize their CMSO-definable (i.e., recognizable) subsets. Method: The authors first establish a rigorous formalization of regular graph grammars for the class of graphs with treewidth ≤ 2; they introduce *aperiodic parallel composition semigroups* and leverage tree decompositions, graph algebras, and finite semigroup theory. Contribution/Results: They prove that regular graph grammars exactly capture all CMSO-definable subsets of graphs of treewidth ≤ 2; establish an equivalence between MSO definability and aperiodic algebraic recognizability; and show that the graph language inclusion problem is in 2EXPTIME and EXPTIME-hard. This work unifies graph algebra, CMSO logic, tree decompositions, and finite semigroup theory, laying foundational groundwork for formal language theory over graph structures.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Regular word grammars are context-free grammars having restricted rules, that define all recognizable languages of words. This paper generalizes regular grammars from words to certain classes of graphs, by defining regular grammars for unordered unranked trees and graphs of tree-width $2$ at most. The qualifier ``regular'' is justified because these grammars define precisely the recognizable (equivalently, cmso-definable) sets of the respective graph classes. The proof of equivalence between regular and recognizable sets of graphs relies on the effective construction of a recognizer algebra of size doubly-exponential in the size of the grammar. This sets a woexptime upper bound on the (exptime-hard) problem of inclusion of a context-free language in a regular language, for graphs of tree-width $2$ at most. A further syntactic restriction of regular grammars suffices to capture precisely the mso-definable sets of graphs of tree-width $2$ at most, i.e., the sets defined by cmso formul{ae} without cardinality constraints. Moreover, we show that mso-definability coincides with recognizability by algebras having an aperiodic parallel composition semigroup, for each class of graphs defined by a bound on the tree-width.

Problem

• Generalizes regular grammars to graphs.
• Proves equivalence between regular and recognizable sets.
• Sets upper bounds for graph inclusion problems.

Innovation

• Generalizes regular grammars to graphs
• Defines recognizer algebra for graphs
• Establishes MSO-definability for graph classes