Both the augment algorithm and the reduction algorithm can be used to obtain non-zero knowledge states vector in testing. In particular, the augment algorithm based on the reachability matrix can imply the structure of the Q-matrix and its non-zero columns, thus proving the set of Q-matrix columns forms an algebraic structure (Lattice). Applying the augment algorithm based on the reachability matrix and the test Q-matrix, respectively, we can obtain the theoretical construct validity of the test Q-matrix (i.e., the degree of the test Q-matrix fitting the cognitive model) and use the results to evaluate test quality. We can also use the algorithm when constructing and evaluating cognitive models, as well as when developing cognitive diagnostic models. Moreover, the augment algorithm and its reverse algorithm (reduction algorithm) are suitable for analyzing and evaluating retrofitting data.