Deciding Amalgamation Beyond Arity Two: The Semantic Horn Case

2026-08-12

AI Summary

This work investigates whether the class of finite models of first-order universal sentences satisfying semantic Horn conditions enjoys the amalgamation property, without imposing any restriction on relational arities. The problem is reduced—via an inside-out correspondence—to the decidability of finite completions and solved using certificates based on local consistency. The main contribution lies in establishing, for the first time, an unconditional decision procedure for the semantic Horn fragment without arity bounds, revealing that completion templates admit semilattice polymorphisms and linking this property to width-2 local consistency. The overall computational complexity of the problem is shown to be 2EXPTIME; however, when the signature’s maximal arity is bounded, the complexity drops to EXPTIME.

Abstract

We study the amalgamation decision problem: given a universal first-order sentence $Φ$, decide whether the class $\mathrm{fm}(Φ)$ of its finite models has the amalgamation property. We call $Φ$ semantic Horn if $\mathrm{fm}(Φ)$ is closed under binary direct products. By McKinsey's theorem, this is equivalent to $Φ$ being logically equivalent to a universal Horn sentence; the distinction is one of input representation, since $Φ$ itself need not be given in Horn form and conversion to an explicit Horn normal form can incur an exponential blow-up. We prove that the problem is decidable under this semantic promise. Moreover, it belongs to 2EXPTIME, and to EXPTIME for every fixed bound on the arity of the input signature. Thus the semantic Horn fragment admits an unconditional decision procedure for signatures of unbounded relational arity. Our proof starts from the inside-out correspondence, which we use as a black box for the semantic reduction to a finite completion problem. We encode finite completions as homomorphisms to a finite relational template and introduce a finite set-valued local-consistency certificate for completion problems whose template has bounded width. For semantic Horn inputs, the local completions over each fixed source chart are closed under relationwise intersection of the added relations, and these intersections are compatible with restriction maps. This yields a semilattice polymorphism of the completion template. Since the template is binary, the semilattice operation gives width $2$, making the local-consistency certificate complete. The same construction gives a decision procedure whenever the associated completion template has bounded width.

Problem

• amalgamation
• semantic Horn
• finite models
• universal first-order sentence
• decision problem

Innovation

• amalgamation
• semantic Horn
• bounded width
• semilattice polymorphism
• local consistency