Universal Horn Sentences and the Joint Embedding Property

Abstract

The finite models of a universal sentence Φ are the age of a structure if and only if Φ has the joint embedding property. We prove that the computational problem whether a given universal sentence Φ has the joint embedding property is undecidable, even if Φ is additionally Horn and the signature is binary.

Deciding Atomicity of Subword-Closed Languages

“…In particular, in [3] Braunfeld has shown that this question is undecidable for hereditary classes of graphs defined by finitely many forbidden induced subgraphs. One more undecidability result appeared in [2], where Bodirsky et al have shown that the joint embedding property is undecidable for the class of all finite models of a given universal Horn sentence. On the other hand, several positive results have been obtained by McDevitt and Ruškuc in [9], where the authors studied classes of words and permutations closed under taking consecutive subwords, also known as factors, and consecutive subpermutations.…”