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Courcelle’s Theorem states that on graphs $$\mathrm{G\$}$ of bounded tree-width with a given tree-decomposition of size $$t(G)\$$, graph properties 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.