Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs

2026-04-27

AI Summary

This work addresses the limitations of existing approaches in expressive power and computational complexity by investigating succinct and efficient grammatical representations for recognizable series-parallel graph languages. Building upon regular grammars for series-parallel graphs, the paper constructs a finite recognizability algebra that effectively characterizes such graph languages and systematically analyzes its closure properties and the complexity of associated decision problems. The main contributions include reducing the construction complexity of the finite recognizability algebra from double exponential to single exponential time, and proving that the intersection and language inclusion problems are ExpTime-complete. These results significantly improve the previously known upper bounds for these decision problems from 2ExpTime down to ExpTime.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesMachine Learning: Graph-based Machine Learning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism into a finite algebra. For SP-graphs, and more generally, for graphs of bounded tree-width, recognizability coincides with definability in Counting Monadic Second-Order (CMSO) logic. Despite this strong logical characterization, the conciseness and algorithmic effectiveness of syntactic representations of recognizable sets of SP (and bounded-tree-width) graphs remain poorly understood. Building on previously introduced regular grammars for SP-graphs, we show that recognizable sets admit concise and effective syntactic representations. The main contribution is an improved construction of finite recognizer algebras whose size is singly-exponential in the size of a regular grammar, improving upon the previously known double-exponential bound. As a consequence, the problems of intersection and language inclusion for sets represented by regular grammars are shown to be ExpTime-complete, thus improving on a previously known 2ExpTime upper bound.

Problem

• series-parallel graphs
• recognizable sets
• regular grammars
• tree-width
• syntactic representations

Innovation

• regular grammars
• series-parallel graphs
• recognizable sets
• finite recognizer algebras
• ExpTime-completeness