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sav08:non-ground_instantiation_and_resolution [2008/04/01 16:00] vkuncak created |
sav08:non-ground_instantiation_and_resolution [2008/04/01 16:34] vkuncak |
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If we try to do the proof, how do we know what to instantiate with? ++|we instantiate to enable subsequent resolution rule.++ | 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: | + | **Resolution with instantiation** |
\[ | \[ | ||
\frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}} | \frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}} | ||
- | {subst(\sigma,C \cup D)} | + | {subst(\sigma_1)(C) \cup subst(\sigma_2)(D)} |
\] | \] | ||
- | such that $subst(\sigma,A_1) = subst(\sigma,A_2)$. | + | such that $subst(\sigma_1)(A_1) = subst(\sigma_2)(A_2)$. |
- | Note: $\sigma$ such that $subst(\sigma,A_1) = subst(\sigma,A_2)$ is called a **unifier** for $\{A_1,A_2\}$. | + | Note: $\sigma$ such that $subst(\sigma)(A_1) = subst(\sigma)(A_2)$ is called a **unifier** for $\{A_1,A_2\}$. |
- | Does resolution with instantiation subsume: | + | **Factoring with instantiation** |
- | * ++non-ground resolution rule?|yes, take empty substitution++ | + | \[ |
- | * ++instantiation rule?|not by itself++ | + | \frac{C \cup \{A_1, A_2\}} |
+ | {subst(\sigma)(C)} | ||
+ | \] | ||
+ | where $subst(\sigma)(A_1)=subst(\sigma)(A_2)$. | ||
Further step: do we need to consider all unifiers? | Further step: do we need to consider all unifiers? | ||
- | Most general unifier. | + | Most general unifier. To compute it we can use the standard [[Unification]] algorithm. |