A Positional Realizability Theory of Length Generalization in Single-Head Attention
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Why does length generalization (LG) depend so sharply on the positional encoding (PE)? Relative schemes extrapolate fixed offsets, absolute schemes fix positions, and neither handles growing offsets. We give a realizability theory for a deliberately controlled abstraction, a single positional-only attention head, that derives this well-documented empirical pattern and says precisely where it holds and breaks. The device is a simulation transfer operator T_L that re-evaluates a selected length-L_0-realizing positional kernel on the longer grid, with a scalar α_L [0,1] read off before any training from (Φ, t_n) alone. Here α_L serves as the computable witness of that taxonomy's derivation, not as a stronger empirical predictor. Our results are a relative-head deep-row characterization for kernels with a strict unique global maximizer (Theorem 2), an absolute/relative duality that derives the pattern (Proposition 4), a learned-absolute underdetermination result, and a static expressivity bound; a sufficiency theorem (Theorem 1) bridges α_L to LG, and the general deterministic case is a precise open conjecture (correcting a tempting but invalid "outside-colspan" argument). Experiments confirm the structure: in the controlled single-head regime α_L exactly separates the taxonomy's generalizing and non-generalizing (PE, task) cells (2625 trained-model measurements; stable at vocab 64/d=128). Together, these results turn the received PE taxonomy into a positional-kernel transfer theorem, and identify the content and depth mechanisms that escape it. The theory concerns the positional channel alone: tasks whose targets depend on token content, including the standard length-generalization suite such as addition and sorting, fall outside Theorem 2.
Autores
Jiasong Han, Yunkai Chai, Jintao Wang
Categorias
O overfeed.news indexa e aponta. Publicamos um trecho curto — o artigo completo fica em arXiv AI Papers.
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