Query Feedback and Teaching for Stable and Certifiable Language Generation

Language generation in the limit studies whether a learner observing an enumeration of an unknown target language can eventually generate only valid, previously unseen elements. We study stability, introduced by Kalavasis, Mehrotra, and Velegkas (2025), which requires the generator's infinite output set to eventually remain unchanged, and introduce the stronger notion of certifiability, which requires the generator to determine from its observations that its current output is valid. Drawing on teaching sets in learning theory, we introduce a hierarchy of teaching dimensions for generation under three increasingly strict success criteria: correctness, stability, and certifiability. These dimensions measure the minimum number of selected examples whose appearance guarantees the corresponding success condition. We also study stability and certifiability in the membership-query model of Charikar and Pabbaraju (2025). We characterize stable and certifiable generation, both with and without query feedback, and establish their relationships with uniform generation and identification in the limit. In particular, we give a finite-witness characterization of stable generation, analogous to Angluin's tell-tale characterization of identification, resolving a gap left open by Kalavasis, Mehrotra, and Velegkas (2025). We further compare these generation notions with identification in the limit and contrast their teaching requirements on well-studied language families, including regular and erasing pattern languages.