Herbrand structure

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.^[1]

Herbrand universe

Definition

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.

• It 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.

Example

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.

Herbrand structure

A Herbrand structure interprets terms on top of a Herbrand universe.

Definition

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 ∈ σ

Remarks

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.

Examples

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

• U = { c, fc, ffc, fffc, ...}
• fc -> fc, ffc -> ffc, ...
• c -> c

Herbrand base

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.

Definition

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

Examples

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), ...}

See also

• Herbrand's theorem, Herbrand theory
• Herbrandization
• Herbrand interpretation
• Herbrand universe
• Herbrand base

References

• Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1996). Mathematical Logic. Springer. ISBN 978-0387942582. 

1. ↑ http://logic.stanford.edu/herbrand/herbrand.html