Andy Yang, Blerta Veseli, Corentin Barloy +5cs.FL cs.AI
Transformer-based language models are known to sometimes generalize to sequences longer than seen during training, but we lack a precise characterization of which tasks admit length generalization. It is not even known which regular languages transformers length-generalize on -- and this is a foundational class of languages. Our contributions are to establish the first complete characterization of which regular languages transformers length-generalize on and provide a decision algorithm running in polynomial time in the size of the language's syntactic monoid. These results rely on an effective characterization of the regular languages in C-RASP, a recently-established formalism that expresses which languages transformers length-generalize on. This characterization is challenging because classical tools like Krohn-Rhodes decomposition theory for finite semigroups are insufficient for C-RASP. Firstly, the basic building blocks of Krohn-Rhodes theory -- flip-flop and simple groups -- are not expressible in C-RASP. Secondly, the basic building block of C-RASP (unbounded counting) is not expressible by the finite semigroups of Krohn-Rhodes theory. Thus, length generalization on regular languages is controlled by an algebraic property that is invisible to classical finite decomposition theory. We generalize classical decomposition theory from finite semigroups to the infinite additive group on the integers, allowing us to characterize C-RASP in terms of iterated wreath products of the integers and derive a provable polynomial-time decision algorithm for regular language membership. Experiments across a broad test suite of regular languages confirm that our theory captures transformers' length-generalization behavior more accurately than existing classifications.
Streaming language-model output creates an enforcement boundary: a control that detects a prohibited pattern after releasing its completing chunk cannot recall it. We study a production policy in which each ordered family is the conjunction of two regular-language predicates. Incremental matching is classical. The problem is exact composition at release time across arbitrary chunk partitions, including end-of-prefix word boundaries that can change on extension. We define an ASCII-explicit policy grammar, compile each predicate to a persistent nondeterministic finite automaton (NFA), distinguish stable from provisional assertion state, apply document-order family priority, and check the decision before releasing each chunk. We show that the resulting monitor is release-boundary equivalent to an absorbing cumulative oracle for every policy in the declared grammar. Production Python and TypeScript implementations were evaluated on 101,653 partitioned cases; a public surrogate added 100,345 cases. Both campaigns produced zero oracle, cross-runtime, or intended-family mismatches. In a frozen neutral-output profile, the memoized incremental and native-regex cumulative slopes at 64-character chunks were 0.973 and 1.976. At 16,384 characters the incremental median was 30.2 ms versus 96.6 ms for native cumulative scanning at that chunk size. Native regex remained faster at 512-character chunks (12.4 versus 29.4 ms), exposing the constant-factor crossover rather than hiding it. A shared per-stream cache cap and 129-symbol alphabet bound optimization state; the campaign peaked at 364 of 4,096 without bypass. The result is policy conformance for a deterministic backstop, not evidence of semantic safety or policy completeness.