Polynomial Length Complexity for the Two-Layer C-RASP Fragment of Chen, Ma and Li
Year: 2026
Type: preprint
Abstract: A learner generalizes in length when it trains on short inputs and is correct on inputs of every length. Chen, Ma and Li (ICML 2025) made this quantitative through the length complexity of a class: the shortest input length at which every two distinct functions in the class already disagree. For their two-layer C-RASP fragment, with K first-layer heads and integer parameters of size at most T, they showed a bound of TO(K) and guessed a lower bound of the form Ω(TK) for K ≤ Θ(log T). (General two-layer C-RASP has no computable length bound; these results concern the restricted fragment.) We prove that two programs in this fragment that compute different functions always differ on an input of length at most (B+1)(T+1)(2B²+T²+T)+1 ≤ 5(K+1)³T⁴, where B=(K+1)T. The exponent of T no longer grows with K. Conversely, for every T ≥ 4 and 2 ≤ K ≤ T we construct pairs whose shortest distinguishing input has an exact closed-form length of at least (1/3)(K−1)T³. So for 2 ≤ K ≤ T the length complexity lies between Ω(KT³) and O(K³T⁴). The lower-bound construction matches all nine exact values we computed (2 ≤ K ≤ T, 4 ≤ T ≤ 8). Corollaries: the source paper's αO(1) bound for its minimum-complexity learner becomes O(α⁴ log³ α), against a lower bound of order α3/2; read as a statement about the class, the Ω(TK) guess fails for every fixed K ≥ 5 and for K = Θ(log T). Files: the paper (PDF), its LaTeX source, and a supplement with the exact solvers, the self-checking implementation of the upper-bound construction, all results, and SHA256SUMS. Code under the MIT license. Version 1.0 (2026-11-06): first release.
Authors: Baliga, Kundan, Xie, Aiden, Byju, Aditya
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225387252