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 [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], the sentence [math]\displaystyle{ Q(a) \lor P(b) }[/math] 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 [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1 }[/math] for the numbers 0 and 1, respectively, a unary function symbol [math]\displaystyle{ s }[/math] for the successor function and a binary function symbol [math]\displaystyle{ + }[/math] for addition.
- [math]\displaystyle{ s(0), s(s(0)), s(s(s(0))), \ldots }[/math] are ground terms;
- [math]\displaystyle{ 0 + 1, \; 0 + 1 + 1, \ldots }[/math] are ground terms;
- [math]\displaystyle{ 0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0 }[/math] are ground terms;
- [math]\displaystyle{ x + s(1) }[/math] and [math]\displaystyle{ s(x) }[/math] are terms, but not ground terms;
- [math]\displaystyle{ s(0) = 1 }[/math] and [math]\displaystyle{ 0 + 0 = 0 }[/math] are ground formulae.
Formal definitions
What follows is a formal definition for first-order languages. Let a first-order language be given, with [math]\displaystyle{ C }[/math] the set of constant symbols, [math]\displaystyle{ F }[/math] the set of functional operators, and [math]\displaystyle{ P }[/math] the set of predicate symbols.
Ground term
A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
- Elements of [math]\displaystyle{ C }[/math] are ground terms;
- If [math]\displaystyle{ f \in F }[/math] is an [math]\displaystyle{ n }[/math]-ary function symbol and [math]\displaystyle{ \alpha_1, \alpha_2, \ldots, \alpha_n }[/math] are ground terms, then [math]\displaystyle{ f\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right) }[/math] is a ground term.
- 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, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.
If [math]\displaystyle{ p \in P }[/math] is an [math]\displaystyle{ n }[/math]-ary predicate symbol and [math]\displaystyle{ \alpha_1, \alpha_2, \ldots, \alpha_n }[/math] are ground terms, then [math]\displaystyle{ p\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right) }[/math] is a ground predicate or ground atom.
Roughly speaking, the Herbrand base is the set of all ground atoms,[1] 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 variables.
Ground formulas may be defined by syntactic recursion as follows:
- A ground atom is a ground formula.
- If [math]\displaystyle{ \varphi }[/math] and [math]\displaystyle{ \psi }[/math] are ground formulas, then [math]\displaystyle{ \lnot \varphi }[/math], [math]\displaystyle{ \varphi \lor \psi }[/math], and [math]\displaystyle{ \varphi \land \psi }[/math] are ground formulas.
Ground formulas are a particular kind of closed formulas.
See also
References
- ↑ Alex Sakharov. "Ground Atom". http://mathworld.wolfram.com/GroundAtom.html.
- 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
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