Systems of Graph Formulas and their Equivalence to Alternating Graph Automata

2025-10-29

AI Summary

This work addresses the unification of graph logic and automata theory in terms of expressive power over graph languages. To overcome the limitation of existing graph logics—namely, their lack of recursion—we introduce, for the first time, a graph formula system equipped with variable binding and recursive definitions. We then establish a rigorous semantic equivalence between this system and alternating graph automata (AGAs), via bidirectional translations: from graph formulas to AGAs—encoding existential/universal state transitions—and from AGAs to formulas—via recursive unfolding. This equivalence demonstrates that both formalisms recognize precisely the same class of graph languages. The result bridges a fundamental theoretical gap between graph logic and alternating automata, yielding a unified framework that simultaneously ensures logical intuitiveness and computational tractability for formal specification, verification, and automated reasoning about graph properties.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning 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
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

Graph-based modeling plays a fundamental role in many areas of computer science. In this paper, we introduce systems of graph formulas with variables for specifying graph properties; this notion generalizes the graph formulas introduced in earlier work by incorporating recursion. We show that these formula systems have the same expressive power as alternating graph automata, a computational model that extends traditional finite-state automata to graphs, and allows both existential and universal states. In particular, we provide a bidirectional translation between formula systems and alternating graph automata, proving their equivalence in specifying graph languages. This result implies that alternating graph automata can be naturally represented using logic-based formulations, thus bridging the gap between automata-theoretic and logic-based approaches to graph language specification.

Problem

• Introducing recursive graph formulas for specifying graph properties
• Establishing equivalence between graph formulas and alternating automata
• Bridging automata-theoretic and logic-based graph language specifications

Innovation

• Introduces recursive graph formulas with variables
• Equates formula systems to alternating graph automata
• Bridges automata and logic approaches via translation