Ground Terms as Domain of Interpretation

Recall syntax of first-order logic terms in First-Order Logic Syntax.

Ground term is a term $t$ without variables, i.e. $FV(t)=\emptyset$, i.e. given by grammar:

    GT ::= f(GT,\ldots,GT)

i.e. built from constants using function symbols.

${\cal L}=\{a, f\}$
$GT=\{a, f(a), f(f(a)), f(f(f(a))), ...\}$

If ${\cal L}$ has no constants then $GT$ is empty. In that case, we add a fresh constant $a_0$ into the language and consider ${\cal L} \cup \{a_0\}$ that has a non-empty $GT$. We call the set $GT$ 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 $I_H = (GT,\alpha_H)$.

How to define $\alpha_H$?

Term Algebra Interpretation for Function Symbols

Let $ar(f)=n$. Then $f : GT^n \to GT$

$\alpha_H(f)(t_1,\ldots,t_n) = f(t_1,\ldots,t_n)$

This defines $\alpha_H(f)$. How to define $\alpha_H(R)$ to ensure that elements of a set are true, i.e. that $e_S(S)(I_H) = {\it true}$?
Partition $GT^n$ in two sets, one over which $\alpha_H(R)(t_1,...,t_n)$ 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 : $\{P(a),\ \neg P(a)\}$

Ground Atoms

If $R \in {\cal L}$, $ar(R)=n$ and $t_1,\ldots,t_n \in GT$, we call $R(t_1,\ldots,t_n)$ 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 $HA$ in sequence (e.g. sorted by length) and establish a bijection $p$ with propositional variables

   p : HA \to V

We will write $p(A)$.

${\cal L}=\{a, f_1, P_1, R_2\}$
$GT=\{a, f(a), f(f(a)), f(f(f(a))), ...\}$
$HA=\{P(a), R(a,a), P(f(a)), R(a, f(a)), ...\}$
$V=\{p_1, p_2, p_3, p_4, ...\}$
We define p such that :
$p(P(a))=p_1,\ p(R(a,a))=p_2,\ p(P(f(a)))=p_3,\ p(R(a, f(a)))=p_4,...$