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A first-order structure whose domain is the Herbrand universe, with constants denoting themselves and function symbols acting by syntactic term formation. Its predicates are specified by assigning truth values to ground atoms. With first-order equality, ordinary equality of syntactic terms imposes additional restrictions; more general term models may require quotienting by a congruence.

Equivalently, the predicate meanings of a Herbrand interpretation correspond to a subset of its Herbrand base.

Fix a signature, adding a fresh constant if it otherwise has no ground terms. Its Herbrand universe $U$ is the set of all ground terms, and its Herbrand base $B$ is the set of all ground atoms over $U$. A Herbrand interpretation is identified with a subset $I\subseteq B$: precisely its members are true. Function symbols and constants have their term interpretations.

A least Herbrand model of a knowledge base $\mathcal{K}$ is a Herbrand model $M$ satisfyingIf it exists, it is unique. This is stronger than being minimal: a minimal model merely has no strictly smaller model contained in it.

For a positive definite Datalog program with facts, let $T$ add every fact and the head of every ground rule whose body is already true. Starting from the empty set and repeatedly adding consequences yieldswhere $T$ is taken inflationary, so it also retains its input. Every added ground atom belongs to every model, and the union is closed under every finite rule body. These two facts establish that it is the least Herbrand model. The intersection assertion is guaranteed here by the definite Horn form, not for arbitrary knowledge bases.