Ground expression

In mathematical logic, a ground term of a formal system is a term that does not contain any free variables.

Similarly, a ground formula is a formula that does not contain any free variables. In first-order logic with identity, the sentence  x (x=x) is a ground formula.

A ground expression is a ground term or ground formula.

Examples

Consider the following expressions from first order logic over a signature containing a constant symbol 0 for the number 0, a unary function symbol s for the successor function and a binary function symbol + for addition.

• s(0), s(s(0)), s(s(s(0))) ... are ground terms;
• 0+1, 0+1+1, ... are ground terms.
• x+s(1) and s(x) are terms, but not ground terms;
• s(0)=1 and 0+0=0 are ground formulae;
• s(1) and ∀x: (s(x)+1=s(s(x))) are ground expressions.

Formal definition

What follows is a formal definition for first-order languages. Let a first-order language be given, with the set of constant symbols, the set of (individual) variables, the set of functional operators, and the set of predicate symbols.

Ground terms

Ground terms are terms that contain no variables. They may be defined by logical recursion (formula-recursion):

1. elements of C are ground terms;
2. If f∈F is an n-ary function symbol and α₁, α₂, ..., α_n are ground terms, then f(α₁, α₂, ..., α_n) is a ground term.
3. Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms).

Roughly speaking, the Herbrand universe is the set of all ground terms.

Ground atom

A ground predicate or ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.

If p∈P is an n-ary predicate symbol and α₁, α₂, ..., α_n are ground terms, then p(α₁, α₂, ..., α_n) is a ground predicate or ground atom.

Roughly speaking, the Herbrand base is the set of all ground atoms, while a Herbrand interpretation assigns a truth value to each ground atom in the base.

Ground formula

A ground formula or ground clause is a formula without free variables.

Formulas with free variables may be defined by syntactic recursion as follows:

1. The free variables of an unground atom are all variables occurring in it.
2. The free variables of ¬p are the same as those of p. The free variables of p∨q, p∧q, p→q are those free variables of p or free variables of q.
3. The free variables of  x p and  x p are the free variables of p except x.

References

• Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G., Handbook of discrete and combinatorial mathematics, p. 68 
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6 
• First-Order Logic: Syntax and Semantics