Herbrand structure

The Herbrand universe serves as the universe in the Herbrand structure.

(1) The Herbrand universe of a first-order language L^σ, is the set of all ground terms of L^σ. If the language has no constants, then the language is extended by adding an arbitrary new constant.

• The Herbrand universe is countably infinite if σ is countable and a function symbol of arity greater than 0 exists.
• In the context of first-order languages we also speak simply of the Herbrand universe of the vocabulary σ.

(2) The Herbrand universe of a closed formula in Skolem normal form F, is the set of all terms without variables, that can be constructed using the function symbols and constants of F. If F has no constants, then F is extended by adding an arbitrary new constant.

• This second definition is important in the context of computational resolution.

Let L^σ be a first-order language with the vocabulary

• constant symbols: c
• function symbols: f(.), g(.)

then the Herbrand universe of L^σ (or σ) is {c, f(c), g(c), f(f(c)), f(g(c)), g(f(c)), g(g(c)), ...}.

Notice that the relation symbols are not relevant for a Herbrand universe.

Let S be a structure, with vocabulary σ and universe U. Let T be the set of all terms over σ and T₀ be the subset of all variable-free terms. S is said to be a Herbrand structure iff

1. U = T₀
2. f^S(t₁, ..., t_n) = f(t₁, ..., t_n) for every n-ary function symbol f ∈ σ and t₁, ..., t_n ∈ T₀
3. c^S = c for every constant c in σ

1. U is the Herbrand universe of σ.
2. A Herbrand structure that is a model of a theory T, is called the Herbrand model of T.

For a constant symbol c and a unary function symbol f(.) we have the following interpretation:

• U = {c, fc, ffc, fffc, ...}
• fc → fc, ffc → ffc, ...
• c → c

A Herbrand base is the set of all ground atoms of whose argument terms are the Herbrand universe.

For a binary relation symbol R, we get with the terms from above:

{R(c, c), R(fc, c), R(c, fc), R(fc, fc), R(ffc, c), ...}