The nonexistence of unicorns and many-sorted L\"owenheim-Skolem theorems

TL;DR

This paper proves that unicorn theories, which are stably infinite and strongly finitely witnessable but not smooth, do not exist, and it establishes versions of the Löwenheim-Skolem theorem for many-sorted logic.

Contribution

It confirms the nonexistence of unicorn theories and extends Löwenheim-Skolem results to many-sorted logic, clarifying model-theoretic properties relevant to theory combination.

Findings

Unicorn theories do not exist.

Versions of Löwenheim-Skolem theorem for many-sorted logic are established.

Theories that are stably infinite and strongly finitely witnessable are indeed smooth.

Abstract

Stable infiniteness, strong finite witnessability, and smoothness are model-theoretic properties relevant to theory combination in satisfiability modulo theories. Theories that are strongly finitely witnessable and smooth are called strongly polite and can be effectively combined with other theories. Toledo, Zohar, and Barrett conjectured that stably infinite and strongly finitely witnessable theories are smooth and therefore strongly polite. They called counterexamples to this conjecture unicorn theories, as their existence seemed unlikely. We prove that, indeed, unicorns do not exist. We also prove versions of the L\"owenheim-Skolem theorem and the {\L}o\'s-Vaught test for many-sorted logic.