无逻辑的Courcelle定理扩展
Extensions of Courcelle's Theorem without Logic
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中文总结 AI 辅助
本文推广了Courcelle定理，用连接矩阵推广形式的纯组合假设替代原单二阶逻辑可定义性假设，阐明了此类定理中逻辑的作用。
中文摘要 AI 辅助
Courcelle定理指出，对于树宽至多为k的图G，若给定大小为t(G)的树分解，可在t(G)的大小的线性时间内检查在单二阶逻辑中可定义的图性质P。受L. Lovász使用连接矩阵而非逻辑的研究启发，本文给出了Courcelle定理的推广版本，将可定义性假设替换为使用连接矩阵推广形式的纯组合假设。该研究阐明了此类定理中逻辑的作用，并展示了其纯组合假设。
英文摘要
Courcelle's Theorem states that on graphs $G$ of tree-width at most $k$ with a given tree-decomposition of size $t(G)$, graph properties $\mathcal{P}$ definable in Monadic Second Order Logic can be checked in linear time in the size of $t(G)$. Inspired by L. Lovász' work using connection matrices instead of logic, we give a generalized version of Courcelle's theorem which replaces the definability hypothesis by a purely combinatorial hypothesis using a generalization of connection matrices. This paper clarifies the role of logic in such theorems and displays their purely combinatorial assumption.
