Herbrand universe

In first-order logic, a Herbrand structure S is a structure over a vocabulary σ, that is defined solely by the syntactical properties of σ. The idea is to take the symbols of terms as their values, e.g. the denotation of a constant symbol c is just 'c' (the symbol).

Herbrand structures play an important role in the foundations of logic programming.

Herbrand universe will serve as the universe in 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.

(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.

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

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.

A Herbrand structure interprets terms on top of 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

For a constant symbol c and a 1-ary function symbol f(.) we have the following interpretation:

In addition to the universe, defined in Herbrand universe, and the term denotations, defined in Herbrand structure, the Herbrand base completes the interpretation by denoting the relation symbols.

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

For a 2-ary 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), ...}

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