Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
sav08:instantiation_plus_ground_resolution [2009/05/14 11:57] vkuncak |
sav08:instantiation_plus_ground_resolution [2015/04/21 17:30] (current) |
||
|---|---|---|---|
| Line 11: | Line 11: | ||
| === Ground Instantiation Rule === | === Ground Instantiation Rule === | ||
| - | \[ | + | \begin{equation*} |
| \frac{C}{subst(\sigma)(C)} | \frac{C}{subst(\sigma)(C)} | ||
| - | \] | + | \end{equation*} |
| where $C$ is a clause and $\sigma$ is a ground substitution. | where $C$ is a clause and $\sigma$ is a ground substitution. | ||
| Line 21: | Line 21: | ||
| If $A$ is a ground atom and $C,D$ are ground claues, then | If $A$ is a ground atom and $C,D$ are ground claues, then | ||
| - | \[ | + | \begin{equation*} |
| \frac{C \cup \{\lnot A\}\ \ \ D \cup \{A\}} | \frac{C \cup \{\lnot A\}\ \ \ D \cup \{A\}} | ||
| {C \cup D} | {C \cup D} | ||
| - | \] | + | \end{equation*} |
| Note that this is propositional resolution where propositional variables have "long names" (they are ground atoms). | Note that this is propositional resolution where propositional variables have "long names" (they are ground atoms). | ||
| ==== Example ==== | ==== Example ==== | ||
| + | |||
| Line 35: | Line 36: | ||
| The proof is based on [[Herbrand's Expansion Theorem]] (see also the proof of [[Compactness for First-Order Logic]]). | The proof is based on [[Herbrand's Expansion Theorem]] (see also the proof of [[Compactness for First-Order Logic]]). | ||
| - | Suppose a set $S$ of clauses is contradictory. By [[Herbrand's Expansion Theorem]] and [[Compactness Theorem|Compactness Theorem for Propositional Formulas]], there is some finite subset $S_0 \subseteq expand(S)$ is contradictory. Then there exists a derivation of empty clause from $S_0$ viewed as set of propositional formulas, using propositional resolution. In other words, there exists a derivation of empty clause from $S_0$ using ground resolution rule. Each element of $S_0$ can be obtained from an element of $S$ using instantiation rule. This means that there exists a proof tree whose leaves are followed by a single application of instantiation rule, and inner nodes contain ground resolution steps. | + | Suppose a set $S$ of clauses is contradictory. By [[Herbrand's Expansion Theorem]] and [[Compactness Theorem|Compactness Theorem for Propositional Formulas]], some finite subset $S_0 \subseteq expand(S)$ is contradictory. Then there exists a derivation of empty clause from $S_0$ viewed as set of propositional formulas, using propositional resolution. In other words, there exists a derivation of empty clause from $S_0$ using ground resolution rule. Each element of $S_0$ can be obtained from an element of $S$ using instantiation rule. This means that there exists a proof tree whose leaves are followed by a single application of instantiation rule, and inner nodes contain ground resolution steps. |