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 symbol strings of terms as their values, e.g. the denotation of a constant symbol c is just "c" (the symbol). It is named after Jacques Herbrand.

Herbrand structures play an important role in the foundations of logic programming.[1]

Herbrand universe

Definition

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.

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)), ...}.

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 W be the set of all terms over σ and W₀ be the subset of all variable-free terms. S is said to be a Herbrand structure iff

1. U = W₀
2. f^S(t₁, ..., t_n) = f(t₁, ..., t_n) for every n-ary function symbol f ∈ σ and t₁, ..., t_n ∈ W₀
3. c^S = c for every constant c in σ

Remarks

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

Examples

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

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 whose argument terms are elements of the Herbrand universe.

Examples

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), ...}

See also

• Herbrand's theorem
• Herbrandization
• Herbrand interpretation

Notes

1. "Herbrand Semantics".

References

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