Characterizing relative decidability in terms of model completeness

TL;DR

This paper characterizes when a complete theory is relatively decidable by linking it to conservative model complete extensions, confirming a conjecture and highlighting differences with incomplete theories.

Contribution

It proves a conjecture relating relative decidability to conservative model complete extensions for complete theories, and shows the characterization fails for incomplete theories.

Findings

Complete theories are relatively decidable iff they have a conservative model complete extension of a specific form.

The conjecture by Chubb, Miller, and Solomon is verified for complete theories.

No similar characterization applies to incomplete theories.

Abstract

A theory $T$ is said to be relatively decidable if for every model of $T$, one can compute the elementary diagram of that model from its atomic diagram together with $T$. We verify a conjecture of Chubb, Miller, and Solomon by showing that for complete theories $T$, $T$ is relatively decidable if and only if $T$ has a conservative model complete extension of the form $T \cup \{\varphi(\bar{c})\}$ where $T \models \exists \bar{x} \; \varphi(\bar{x})$. We also show that no such characterization works for incomplete theories.