Computable functors and effective interpretability

TL;DR

This paper establishes an equivalence between two notions of reducibility in structures, connecting a syntactical interpretability concept with a computational Medvedev reducibility, and extends this to effective bi-interpretability and class reductions.

Contribution

It introduces a new equivalence between effective interpretability and Medvedev reducibility, advancing the understanding of structure reducibility in computable model theory.

Findings

Proves the equivalence of two reducibility notions.

Extends results to effective bi-interpretability.

Applies to reductions between classes of structures.

Abstract

Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a strengthening of the well-known Medvedev reducibility. We extend our result to effective bi-interpretability and also to effective reductions between classes of structures.