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

TL;DR

This paper develops a finite typed reconstruction theory for learning context-free languages under fixed monoid congruences, enabling exact reconstruction from positive data with polynomial efficiency.

Contribution

It introduces a finite typed reconstruction framework for context-free languages under monoid homomorphisms, proving learnability and polynomial bounds for the linear subclass.

Findings

Exact reconstruction from finite samples is possible for the class of languages considered.

The proposed hypothesis grammar can be constructed in polynomial time from positive data.

Polynomial bounds are established for the sample size and word length in the linear subclass.

Abstract

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.