Characterize the power of non-trivial prompts in prompted generation

Characterize, in full generality, the effect of imposing the non-trivial prompt restriction in the adversarial prompted language-generation model. Specifically, given a countable collection of languages {L1, L2, L3, ...} over a countable set U and an unknown target language K in this collection, where in each step t the adversary supplies a prompt p_t that has at least one continuation c_t with p_t c_t ∈ K − S_t, determine precisely the conditions under which prompted generation from K in the limit is achievable under this restriction.

Background

The paper extends its generation-in-the-limit framework to a prompted setting where, at each step, the adversary provides both a new sample string from the target language K and a prompt p_t that the algorithm must complete so that the concatenation lies in K − S_t. The authors prove a positive result for robust prompts (those that admit arbitrarily long continuations in every candidate language), showing prompted generation in the limit is achievable without extra assumptions.

They then consider a weaker assumption—non-trivial prompts—requiring only that each prompt p_t have at least one valid continuation in K − S_t. Under this weaker assumption, they establish a positive result using an algorithm augmented with the ability to decide regular subset queries, and they note that for context-free languages this augmentation is not needed because the relevant decision problems are decidable. However, they explicitly leave the fully general characterization of what is achievable under the non-trivial prompt restriction as an open question.