We present an adaptation of model-based verification, via model checking pushdown systems, to semantics-based verification. First we introduce the algebraic notion of pushdown system specifications (PSS) and adapt a model checking algorithm for this new notion. We instantiate pushdown system specifications in the $\mathbb{K}$ framework by means of Shylock, a relevant PSS example. We show why $\mathbb{K}$ is a suitable environment for the pushdown system specifications and we give a methodology for defining the PSS in $\mathbb{K}$. Finally, we give a parametric $\mathbb{K}$ specification for model checking pushdown system specifications based on the adapted model checking algorithm for PSS.