Abstract: 

We prove structure theorems for stable real-valued functions with explicit efficient bounds, complementing earlier non-quantitative results of Chavarria, Conant, and Pillay obtained via ultraproducts. A key ingredient is a quantitative correspondence between order properties and binary trees encoded by a function, in direct analogy to Hodges' correspondence for graphs. In addition to driving the regularity results, this correspondence leads to resolutions of two open problems in statistical learning theory.

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