Finite Sentence-Interface Control for Learning Bounded-Fan-Out Linear MCFGs under Fixed Monoid Typing

We study positive-data learning of bounded-fan-out linear multiple context-free grammars under a fixed explicit finite monoid homomorphism $h$. The main obstacle beyond the context-free case is that an MCFG nonterminal derives a tuple whose components may be placed in a surrounding sentence in different orders. We introduce sentence-interface types as finite external control objects for such tuple occurrences. A type records the permutation of tuple components in the final sentence together with the $h$-values of the boundary intervals between them. For reduced working binary linear nondeleting MCFG presentations whose string languages satisfy $(f,h)$-tuple substitutability, we build a typed refinement, a finite characteristic sample, and a canonical positive-data learner. Once the sample contains this characteristic sample and remains contained in the target language, the learner reconstructs the language exactly. Consequently, for fixed fan-out bound $f$ and fixed explicit $h$, the resulting class is identifiable in the limit from positive data. Moreover, the hypothesis associated with any given finite sample is constructible in polynomial time for fixed $f$ and fixed $h$, including output size. Thus sentence-interface control is the finite mechanism that lifts fixed-$h$ distributional reconstruction from context-free grammars to bounded-fan-out linear MCFGs.