Jose Iovino | Searcharxiv

arXiv2026

Approximability of deep computations

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.

math.LO↗

arXiv2019

Metastable convergence and logical compactness

The concept of metastable convergence was identified by Tao;it allows converting theorems about convergence into stronger theorems about uniform convergence. The Uniform Metastability Principle (UMP) states that if $T$ is a theorem about convergence, then the fact that $T$ is valid implies automatically that its (stronger) uniform version is valid, provided that $T$ can be stated in certain logical frameworks. In this paper we identify precisely the logical frameworks $L$ for which UMP holds. More precisely, we prove that the UMP holds for $L$ if and only if $L$ is a compact logic. We also prove a topological version of this equivalence. We conclude by proving new characterizations of logical compactness that yield additional information about the UMP.

math.LO↗