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arXiv · 2604.24151

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

Marius Bozga·Radu Iosif·Florian Zuleger
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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.

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Marius Bozga, Radu Iosif, Florian Zuleger. 2026-06-29. Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs. https://arxiv.org/abs/2604.24151

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