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sav08:non-ground_instantiation_and_resolution [2008/04/01 17:39] vkuncak |
sav08:non-ground_instantiation_and_resolution [2008/04/02 10:50] vkuncak |
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| Illustration of rules. | 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.++ | 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\}} | \frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}} | ||
| Line 46: | Line 46: | ||
| such that $subst(\sigma_1)(A_1) = subst(\sigma_2)(A_2)$. | such that $subst(\sigma_1)(A_1) = subst(\sigma_2)(A_2)$. | ||
| - | Resolution with instantiation generalizes resolution and ground resolution. | + | 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: | One complete proof system contains: | ||
| + | * instantiation followed by resolution | ||
| * instantiation | * instantiation | ||
| - | * resolution with instantiation | ||
| - | |||
| - | 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 | ||
| 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 $\{A_1,A_2\}$. | ||
| - | Further step: do we need to consider all unifiers? | + | Note: a //renaming// is substitution whose domain and codomain are variables and which is injective. That, is, it renames variables without clashing variable names. |
| + | |||
| + | Further step: do we need to consider all possible 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. |