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.

翻译：系列并联图是带有指定源顶点和目标顶点的二元边标记图，通过串行组合和并行组合构建。若一个图集合的成员性质仅取决于其在同态映射到有限代数下的像，则称该集合为可识别的。对于系列并联图，更一般地，对于有界树宽图，可识别性与计数单调二阶逻辑中的可定义性一致。尽管有这一强大的逻辑刻画，但可识别的系列并联图（及有界树宽图）集合的句法表示在简洁性和算法有效性方面仍未被充分理解。基于先前针对系列并联图引入的正则文法，我们证明了可识别集合具有简洁且有效的句法表示。主要贡献在于改进了有限识别子代数的构造，其大小为正则文法规模的单指数级，优于此前已知的双指数级界。由此证明，由正则文法表示的集合的交集与语言包含问题为ExpTime完全问题，从而改进了此前已知的2ExpTime上界。