Approximability of deep computations
Samson Alva, Eduardo Dueñez, Jose Iovino, Claire Walton
TL;DR
This work develops a foundational framework for deep computations and ultracomputations by combining topology, model theory, and real-valued computation. It introduces countable-feature CSSs/CCSs, state-type spaces, and the notion of shards to study accumulation points (DCs/ucomps) and deep iterations/equilibria, establishing extendibility as a key criterion for definability. The main theoretical contributions include equivalences that connect computability of deep objects to uniform polynomial approximation and type-space extendibility, plus existence results for deep equilibria via Ellis–Numakura-type arguments. Collectively, the paper lays a rigorous groundwork for understanding when asymptotic, deep computations are effectively computable and how such equilibria arise in compositional computation structures. It also bridges foundational topology and model theory with practical considerations for neural-network-like deep computations through definability criteria and limit-exchange results.
Abstract
This is the first of a series of papers in which we study deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. In this first paper of the series, we characterize deep computations that are bona fide computable, and prove the existence of deep equilibria, which hitherto have been found only empirically in deep learning. A subsequent paper will study the complexity of ultracomputations. Our approach adapts and combines technology from topology of function spaces, structural Ramsey theory, topological dynamics, and model theory.