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sav08:non-ground_instantiation_and_resolution [2008/04/01 16:21] vkuncak |
sav08:non-ground_instantiation_and_resolution [2008/04/01 17:39] vkuncak |
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\[ | \[ | ||
\frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}} | \frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}} | ||
- | {subst(\sigma,C \cup D)} | + | {subst(\sigma_1)(C) \cup subst(\sigma_2)(D)} |
\] | \] | ||
- | 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 $\{A_1,A_2\}$. | + | Resolution with instantiation generalizes resolution and ground resolution. |
- | **Factoring with instantiation** | + | One complete proof system contains: |
- | \[ | + | * instantiation |
- | \frac{C \cup \{\lnot A_1, A_2\}} | + | * resolution with instantiation |
- | {subst(\sigma,C \cup D)} | + | |
- | \] | + | 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 |
- | where $subst(\sigma)(A_1)=subst(\sigma)(A_2)$. | + | |
+ | 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? | Further step: do we need to consider all unifiers? |