Universal Horn Sentences and the Joint Embedding Property
Manuel Bodirsky, Jakub Rydval, André Schrottenloher
TL;DR
This paper shows that the problem of determining whether the finite models of a universal sentence have the joint embedding property (JEP) is undecidable, even under the restriction that the sentence is Horn and the signature is binary. The authors develop a reduction from the universality problem for context-free grammars, constructing two components $Φ_{1}$ and $Φ_{2}$ so that $\llbracket{Φ_{1}\wedge Φ_{2}}\rrbracket^{<\omega}$ has the JEP if and only if the CFG is universal. The key technical device is encoding CFG derivations into Horn clauses over a carefully designed signature and exploiting connectedness of Horn clauses to guarantee JEP for the base part, while $Φ_{2}$ encodes potential JEP failures. The results delimit the boundary of decidability for model-theoretic properties of universal theories and have implications for CSPs and description logic formalisms that rely on ages definable by universal Horn sentences.
Abstract
The finite models of a universal sentence $Φ$ in a finite relational signature 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 of $Φ$ only contains relation symbols of arity at most two.