Approximations of Mappings

TL;DR

This paper addresses the problem of approximating continuous mappings with finite mappings, providing a full characterization of limit objects for first-order convergent sequences, and advances understanding of inverse problems in model theory.

Contribution

It solves the approximation problem for mappings, characterizes limit objects for FO and local convergence, and links to inverse problems and decidability in model theory.

Findings

Solved the approximation problem for continuous mappings.

Provided a full characterization of limit objects for FO convergence.

Connected the results to inverse problems and decidability in logic.

Abstract

We consider mappings, which are structure consisting of a single function (and possibly some number of unary relations) and address the problem of approximating a continuous mapping by a finite mapping. This problem is the inverse problem of the construction of a continuous limit for first-order convergent sequences of finite mappings. We solve the approximation problem and, consequently, the full characterization of limit objects for mappings for first-order (i.e. ${\rm FO}$) convergence and local (i.e. ${\rm FO}^{\rm local}$) convergence. This work can be seen both as a first step in the resolution of inverse problems (like Aldous-Lyons conjecture) and a strengthening of the classical decidability result for finite satisfiability in Rabin class (which consists of first-order logic with equality, one unary function, and an arbitrary number of monadic predicates). The proof involves model theory and analytic techniques.