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--- sav08:non-ground_instantiation_and_resolution [2008/04/01 16:34]
vkuncak
+++ sav08:non-ground_instantiation_and_resolution [2008/04/02 10:49]
vkuncak
@@ Line 31: / Line 31: @@
 Illustration of rules.
-===== Reducing the Search Space =====
+===== Combining Instantiation and Resolution =====
-More powerful rules.
+These two are more powerful rules.
-But more choices at each step.
+But we therefore have more choices at each step.
 If we try to do the proof, how do we know what to instantiate with? ++|we instantiate to enable subsequent resolution rule.++
-**Resolution with instantiation**
+**Instantiation followed by resolution:**
 \[
 \frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}}
@@ Line 45: / Line 45: @@
 \]
 such that $subst(\sigma_1)(A_1) = subst(\sigma_2)(A_2)$.
+This rule generalizes the above non-ground resolution and ground resolution.
+Note: if we apply instantiation that renames variables in each clause, then $\sigma_1$ and $\sigma_2$ can have disjoint domains and we let $\sigma = \sigma_1 \cup \sigma_2$, obtaining
+One complete proof system contains:
+  * instantiation followed by resolution
+  * instantiation
 Note: $\sigma$ such that $subst(\sigma)(A_1) = subst(\sigma)(A_2)$ is called a **unifier** for $\{A_1,A_2\}$.
-**Factoring with instantiation**
+Note: a //renaming// is substitution whose domain and codomain are variables and which is injective.  That, is, it renames variables without clashing variable names.
-\[
-\frac{C \cup \{A_1, A_2\}}
+Further step: do we need to consider all possible unifiers?
-     {subst(\sigma)(C)}
-\]
-where $subst(\sigma)(A_1)=subst(\sigma)(A_2)$.
-Further step: do we need to consider all unifiers?
+Most general unifier for $\{A_1,A_2\}$, denoted $mgu(A_1,A_2)$
-Most general unifier.  To compute it we can use the standard [[Unification]] algorithm.
+To compute it we can use the standard [[Unification]] algorithm.