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Non-Ground Instantiation and Resolution
Non-Ground Resolution
Why apply resolution only on ground terms? Consider arbitrary clauses and any atom . \[ \frac{C \cup \{\lnot A\}\ \ \ D \cup \{A\}}
{C \cup D}
\]
Non-Ground Instantiation
For arbitrary substitution: \[ \frac{C}{subst(\sigma)(C)} \]
Is this non-ground instantiation rule sound?
Illustration of rules.
Reducing the Search Space
More powerful rules.
But more choices at each step.
If we try to do the proof, how do we know what to instantiate with?
Resolution with instantiation \[ \frac{C \cup \{\lnot A_1\}\ \ \ D \cup \{A_2\}}
{subst(\sigma_1)(C) \cup subst(\sigma_2)(D)}
\] such that .
Resolution with instantiation generalizes resolution and ground resolution.
One complete proof system contains:
- instantiation
- resolution with instantiation
Note: if we apply instantiation that renames variables in each clause, then and can have disjoint domains and we let , obtaining
Note: such that is called a unifier for .
Further step: do we need to consider all unifiers?
Most general unifier. To compute it we can use the standard Unification algorithm.
Factoring with instantiation \[ \frac{C \cup \{A_1, A_2\}}
{subst(\sigma)(C)}
\] where .