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sav07_lecture_6_skeleton [2007/03/30 16:08] vkuncak |
sav07_lecture_6_skeleton [2007/03/30 19:57] vkuncak |
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| We typically do not write universal quantifiers either, because we know that all variables in a clause are universally quantified. | We typically do not write universal quantifiers either, because we know that all variables in a clause are universally quantified. | ||
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| **Herbrand's theorem**: If a set of clauses in language L is true in some interpretation with domain D, then it is true in an interpretation with domain Term(L). | **Herbrand's theorem**: If a set of clauses in language L is true in some interpretation with domain D, then it is true in an interpretation with domain Term(L). | ||
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| + | **Proof sketch**: If set $S$ of clauses is true in interpretation $e$, then define interpretation $e_T$ with domain Term(L) by evaluating functions as in the term model $[[f]](t_1,t_2) = f(t_1,t_2)$, and defining the value of relations by $[[R]]e_T = \{(t_1,t_2)\mid [[R(t_1,t_2)]]e = \mbox{true} \}$ (end of sketch). | ||
| Note: this is in language without equality. With equality, we need to take equivalence classes of equal terms. So, in the presence of equality the term model can be finite. | Note: this is in language without equality. With equality, we need to take equivalence classes of equal terms. So, in the presence of equality the term model can be finite. | ||