Ground expression

In mathematical logic, a ground term of a formal system is a term that does not contain any variables at all, and a closed term is a term that has no free variables. In first-order logic all closed terms are ground terms, but in lambda calculus the closed term λ x. x (λ y. y) is not a ground term.

Similarly, a ground formula is a formula that does not contain any variables, and a closed formula or sentence is a formula that has no free variables. In first-order logic with identity, the sentence $\forall$ x (x=x) is not 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(z)=1 and ∀x: (s(x)+1=s(s(x))) are expressions, but are not ground expressions.

Ground expressions are necessarily closed. The last example, ∀x: (s(x)+1=s(s(x))), shows that a closed expression is not necessarily a ground expression. So, this formula is a closed formula, but not a ground formula, because it contains a logical variable, even though that variable is not free.

Formal definition

What follows is a formal definition for first-order languages. Let a first-order language be given, with the $C$ the set of constant symbols, $V$ the set of (individual) variables, $F$ the set of functional operators, and $P$ 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 is an atomic formula all of whose terms are ground terms. That is,

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 all of whose arguments are ground atoms.

Ground formulae may be defined by syntactic recursion as follows:

1. A ground atom is a ground formula; that is, 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 formula (and is a ground atom);
2. If p and q are ground formulae, then ¬(p), (p)∨(q), (p)∧(q), (p)→(q), formulas composed with logical connectives, are ground formulae, too.
3. If p is a ground formula and we can get q from it that way some ( or ) we delete or insert in the p formula, and then the result, q is well-formed and equivalent with p, then q is a ground formula.
4. We can get all ground formulae applying these three rules.

References

• Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.), 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