Distributional Learning of Context-Free Languages under Fixed Finite-Monoid Typing

We study distributional learning of context-free languages under a fixed recognizable congruence $\sim_h$ given as the kernel of an explicit finite monoid homomorphism $h:\Sigma^*\to M$. For this fixed-$h$ setting, we develop a finite typed reconstruction theory for context-free $\sim_h$-substitutable languages. Starting from a reduced context-free grammar, we introduce a typed refinement that records both yield types and outer context types, show that the relevant structure is concentrated in a finite typed reconstruction basis, and prove that this basis is exposed by a finite observation set. Occurrences of the same nonterminal symbol may therefore have to be separated when their outer $h$-contexts differ. We then prove exact reconstruction from positive data. From any finite sample $K\subseteq\Sigma^*$, we construct a canonical hypothesis grammar $\hat G(K)$, and we show that once $K$ contains the finite observation set associated with the target typed grammar, $\hat G(K)$ generates the target language exactly. Consequently, for every explicit finite monoid homomorphism $h$, the class $\mathcal C_h^{\mathrm{cf}}$ of context-free $\sim_h$-substitutable languages is identifiable in the limit from positive data, with polynomial-time hypothesis construction and update. For the linear subclass $\mathcal C_h^{\mathrm{lin}}$, we further prove polynomial upper bounds on characteristic-sample size and word length. Thus the same learner gives a full polynomial time-and-data result for the linear subclass.