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

Abstract

We study positive-data learning of context-free languages whose nonempty factors are typed by a fixed morphism $s:{\Sigma }^{+}\to S$ into a finite semigroup. A polynomial-time set-driven operator reconstructs each substitutable target from finite positive witnesses; a separate conservative learner identifies nonempty targets in Gold's sense. The encoded characteristic-data size is polynomial in grammar size and a typed shortest-yield parameter. Ordinary grammar thickness suffices when the typing is determined by a fixed prefix-suffix window and a finite-group value; for linear targets, linear grammar size alone suffices. Conversely, a fixed two-element semilattice typing yields finite targets with grammars of size and ordinary thickness both $O(n)$ that require exponentially long characteristic samples. The lower bound applies to any set-driven operator admitting characteristic samples for the fixed-typing class. Retyping closure, examples beyond fixed windows, and a Dyck-language obstruction delineate the framework's scope.

arXiv comment26 pages. v5: substantially revised version. Typing is now by a finite-semigroup morphism on nonempty factors (title changed accordingly); the learning algorithm is corrected by separating a set-driven reconstruction operator from a conservative Gold learner; new exponential lower bound on characteristic data for a two-element semilattice typing; manuscript reorganized and shortened