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sav08:ground_terms [2008/03/20 12:15] vkuncak |
sav08:ground_terms [2013/05/10 11:02] vkuncak |
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| i.e. built from constants using function symbols. | i.e. built from constants using function symbols. | ||
| - | **Example** | + | **Example**\\ |
| + | ${\cal L}=\{a, f\}$ \\ | ||
| + | $GT=\{a, f(a), f(f(a)), f(f(f(a))), ...\}$ | ||
| 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. | 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 $I_H = (HU,\alpha_H)$. | + | 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 $\alpha_H$? | How to define $\alpha_H$? | ||
| Line 21: | Line 23: | ||
| Let $ar(f)=n$. Then $f : GT^n \to GT$ | Let $ar(f)=n$. Then $f : GT^n \to GT$ | ||
| - | $\alpha_H(f)(t_1,\ldots,t_n) =$ ++| $f(t_1,\ldots,t_n)$++ | + | $\alpha_H(f)(t_1,\ldots,t_n) = f(t_1,\ldots,t_n)$ |
| - | This defines $\alpha_H(f)$. How to define $\alpha_H(R)$ to ensure that elements of a set are true, i.e. that $e_S(S)(I_H) = {\it true}$? | + | This defines $\alpha_H(f)$. How to define $\alpha_H(R)$ to ensure that elements of a set are true, i.e. that $e_S(S)(I_H) = {\it true}$?\\ |
| - | * is this possible for arbitrary set? | + | 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** | **Example** | ||
| + | Consider a set that is not satisfiable : $\{P(a),\ \neg P(a)\}$ | ||
| ===== Ground Atoms ===== | ===== Ground Atoms ===== | ||
| Line 41: | Line 45: | ||
| We will write $p(A)$. | We will write $p(A)$. | ||
| - | **Example** | + | **Example**\\ |
| + | ${\cal L}=\{a, f_1, P_1, R_2\}$ \\ | ||
| + | $GT=\{a, f(a), f(f(a)), f(f(f(a))), ...\}$\\ | ||
| + | $HA=\{P(a), R(a,a), P(f(a)), R(a, f(a)), ...\}$\\ | ||
| + | $V=\{p_1, p_2, p_3, p_4, ...\}$\\ | ||
| + | We define p such that :\\ | ||
| + | $p(P(a))=p_1,\ p(R(a,a))=p_2,\ p(P(f(a)))=p_3,\ p(R(a, f(a)))=p_4,...$ | ||