  Restricted latent class models (RLCMs) have recently gained prominence in
educational assessment, psychiatric evaluation, and medical diagnosis.
Different from conventional latent class models, restrictions on the RLCM model
parameters are imposed by a design matrix to respect practitioners' scientific
assumptions. The design matrix, called $Q$-matrix in the cognitive diagnosis
literature, is usually constructed by practitioners and domain experts, yet it
is subjective and could be misspecified. To address this problem, researchers
have proposed to estimate the $Q$-matrix from data. On the other hand, the
fundamental learnability issue of the $Q$-matrix and model parameters remains
underexplored and existing studies often impose stronger than needed or even
impractical conditions. This paper proposes sufficient and necessary conditions
for joint identifiability of the $Q$-matrix and the RLCM model parameters under
different types of RLCMs. The developed identifiability conditions only depend
on the design matrix and are easy to verify in practice.
