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Ground Terms as Domain of Interpretation
Recall syntax of first-order logic terms in First-Order Logic Syntax.
Ground term is a term without variables, i.e. , i.e. given by grammar: \[
GT ::= f(GT,\ldots,GT)
\] i.e. built from constants using function symbols.
Example
If has no constants then is empty. In that case, we add a fresh constant into the language and consider that has a non-empty . We call the set Herbrand Universe.
Goal: show that if a formula without equality (for now) has a model, then it has a model whose domain is Herbrand universe, that is, a model of the form .
How to define ?
Term Algebra Interpretation for Function Symbols
Let . Then
This defines . How to define to ensure that elements of a set are true, i.e. that ?
Partition in two sets, one over which is true and the other over which it is false.
- is this possible for arbitrary set? no
Example Consider a set that is not satisfiable :
Ground Atoms
If , and , we call an Herbrand Atom. HA is the set of all Herbrand atoms: \[
HA = \{ R(t_1,\ldots,t_n) \mid R \in {\cal L}\ \land \ t_1,\ldots,t_n \in GT \}
\]
We order elements of in sequence (e.g. sorted by length) and establish a bijection with propositional variables \[
p : HA \to V
\] We will write .
Example
We define p such that :