Our paper on Matroid secretary via labeling schemes was accepted at IPCO! In this work, we introduce an algorithmic framework for analyzing greedy-like algorithms for the matroid secretary problem by constructing a language associated with the matroid. Using this approach, we improve the state-of-the-art guarantee for laminar matroids by determining the exact probability-competitiveness of the greedy-improving algorithm. Additionally, we achieve a better competitive factor for graphic matroids compared to the current best-known result.