Ground expression


In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables.
In first-order logic with identity with constant symbols and, the sentence is a ground formula. A ground expression is a ground term or ground formula.

Examples

Consider the following expressions in first order logic over a signature containing the constant symbols and for the numbers 0 and 1, respectively, a unary function symbol for the successor function and a binary function symbol for addition.
  • are ground terms;
  • are ground terms;
  • are ground terms;
  • and are terms, but not ground terms;
  • and are ground formulae.

    Formal definitions

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 functional operators, and the set of predicate symbols.

Ground term

A is a term that contains no variables. Ground terms may be defined by logical recursion :
  1. Elements of are ground terms;
  2. If is an -ary function symbol and are ground terms, then is a ground term.
  3. Every ground term can be given by a finite application of the above two rules.
Roughly speaking, the Herbrand universe is the set of all ground terms.

Ground atom

A ', ' or is an atomic formula all of whose argument terms are ground terms.
If is an -ary predicate symbol and are ground terms, then 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 ' or ' is a formula without variables.
Ground formulas may be defined by syntactic recursion as follows:
  1. A ground atom is a ground formula.
  2. If and are ground formulas, then,, and are ground formulas.
Ground formulas are a particular kind of closed formulas.