Universal Horn Sentences and the Joint Embedding Property

TL;DR

This paper proves that determining whether a universal Horn sentence with limited relation arity has the joint embedding property is undecidable, highlighting fundamental limits in model theory and computational logic.

Contribution

It establishes the undecidability of the joint embedding property for a broad class of universal Horn sentences in finite relational signatures.

Findings

Undecidability holds even for Horn sentences with relation arity at most two.

The result applies to finite models and the joint embedding property.

It advances understanding of the computational complexity in model-theoretic properties.

Abstract

The finite models of a universal sentence $\Phi$ in a finite relational signature are the age of a structure if and only if $\Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $\Phi$ has the joint embedding property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most two.