This is an old revision of the document!
Herbrand Universe for Equality
Recall example of formula : \[
P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. x=a \lor x=b)
\] Replace '=' with 'eq': \[
P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. eq(x,a) \lor eq(x,b))
\] call the resulting formula .
Consider Herbrand model for the set . The relation splits into two sets: the set of terms eq with , and the set of terms eq with .
Constructing Model for Formulas with Equality
Let be a set of formulas in first-order logic with equality and result of replacing '=' with 'eq' in . Suppose that is satisfiable. Let be Herbrand model for . We construct a new model using quotient construction, described as follows (recall notation in First-Order Logic Semantics).
For each element , define \[
[t] = \{ s \mid (s,t) \in I_H(eq) \}
\] Let \[
[GT] = \{ [t] \mid t \in GT \}
\] The constructed model will be where \[
I_Q(f) = \{ ([t_1],\ldots,[t_n], [f(t_1,\ldots,t_n)]) \mid t_1,\ldots,t_n \in GT \}
\] \[
I_Q(R) = \{ ([t_1],\ldots,[t_n]) \mid (t_1,\ldots,t_n) \in I_H(R) \}
\]
In particular, when is we have
Herbrand-Like Theorem for Equality
Theorem: For every set of formulas with equality the following are equivalent
- has a model
- has a model (where are Axioms for Equality and is result of replacing '=' with 'eq' in )
- has a model whose domain is the quotient of ground terms under some congruence