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sav08:ground_terms [2009/05/13 10:29] vkuncak |
sav08:ground_terms [2015/04/21 17:30] |
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- | ====== Ground Terms as Domain of Interpretation ====== | ||
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- | Recall syntax of first-order logic terms in [[First-Order Logic Syntax]]. | ||
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- | //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. | ||
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- | **Example**\\ | ||
- | ${\cal L}=\{a, f\}$ \\ | ||
- | $GT=\{a, f(a), f(f(a)), f(f(f(a))), ...\}$ | ||
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- | 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. | ||
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- | 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 $I_H = (HU,\alpha_H)$. | ||
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- | How to define $\alpha_H$? | ||
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- | ===== Term Algebra Interpretation for Function Symbols ===== | ||
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- | Let $ar(f)=n$. Then $f : GT^n \to GT$ | ||
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- | $\alpha_H(f)(t_1,\ldots,t_n) =$ ++| $f(t_1,\ldots,t_n)$++ | ||
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- | 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 ++ | ||
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- | **Example** | ||
- | Consider a set that is not satisfiable : $\{P(a),\ \neg P(a)\}$ | ||
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- | ===== Ground Atoms ===== | ||
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- | 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: | ||
- | \[ | ||
- | HA = \{ R(t_1,\ldots,t_n) \mid R \in {\cal L}\ \land \ t_1,\ldots,t_n \in GT \} | ||
- | \] | ||
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- | 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 $p(A)$. | ||
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- | **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 :\\ | ||
- | $p(P(a))=p_1,\ p(R(a,a))=p_2,\ p(P(f(a)))=p_3,\ p(R(a, f(a)))=p_4,...$ | ||