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sav08:deriving_propositional_resolution [2008/03/19 17:12] tatjana |
sav08:deriving_propositional_resolution [2012/05/01 12:26] vkuncak |
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==== Projection Proof Rules ==== | ==== Projection Proof Rules ==== | ||
- | Th justified the use of $ProjectForm(F_1,F_2,p)$ as an inference rule. We write such rule: | + | Above we justified the use of $ProjectForm(F_1,F_2,p)$ as an inference rule. We write such rule: |
\[ | \[ | ||
\frac{F_1 \; \ F_2} | \frac{F_1 \; \ F_2} | ||
Line 54: | Line 54: | ||
\] | \] | ||
The soundness of projection rule follows from the fact that | The soundness of projection rule follows from the fact that | ||
- | for every interpretation $I$, if $I \models S$, then also $I \models Proj(S,p)$. | + | for every interpretation $I$, if $I \models S$, then also $I \models ProjectSet(S,p)$. |
Applying the projection rule we obtain formulas with fewer and fewer variables. We therefore also add the "ground contradiction rule" | Applying the projection rule we obtain formulas with fewer and fewer variables. We therefore also add the "ground contradiction rule" | ||
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{{\it false}} | {{\it false}} | ||
\] | \] | ||
- | where $F$ is formula that has no variables and that evaluates to //false// (ground contradictory formula). This rule is trivially sound. | + | where $F$ is formula that has no variables and that evaluates to //false// (ground contradictory formula). This rule is trivially sound: we can never |
+ | have a model of a ground formula that evaluates to false. | ||
==== Iterating Rule Application ==== | ==== Iterating Rule Application ==== | ||
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\[\begin{array}{l} | \[\begin{array}{l} | ||
P_0(S) = S \\ | P_0(S) = S \\ | ||
- | P_{n+1}(S) = Proj(P_n(S),p_{n+1}) \\ | + | P_{n+1}(S) = ProjectSet(P_n(S),p_{n+1}) \\ |
P^*(S) = \bigcup_{n \geq 0} P_n(S) | P^*(S) = \bigcup_{n \geq 0} P_n(S) | ||
\end{array}\] | \end{array}\] | ||
- | |||
==== Completeness of Projection Rules ==== | ==== Completeness of Projection Rules ==== | ||
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The first statement follows from soundness of projection rules. We next prove the second statement. | The first statement follows from soundness of projection rules. We next prove the second statement. | ||
- | Suppose that it ${\it false} \notin P^*(S)$. We claim that $S$ is satisfiable. We show that every finite set is satisfiable, so the property follows from the [[Compactness Theorem]]. | + | Suppose that it ${\it false} \notin P^*(S)$. We claim that $S$ is satisfiable. We show that every finite set is satisfiable, so the property will follow from the [[Compactness Theorem]]. |
Consider any finite $T \subseteq S$. We show that $T$ it is satisfiable. Let $T = \{F_1,\ldots,F_n\}$ and let $FV(F_1) \cup \ldots \cup FV(F_n) \subseteq \{p_1,\ldots,p_M\}$. Consider the set | Consider any finite $T \subseteq S$. We show that $T$ it is satisfiable. Let $T = \{F_1,\ldots,F_n\}$ and let $FV(F_1) \cup \ldots \cup FV(F_n) \subseteq \{p_1,\ldots,p_M\}$. Consider the set | ||
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Therefore, for clauses, projection (with some elimination of redundant conclusions) is exactly the resolution proof rule. | Therefore, for clauses, projection (with some elimination of redundant conclusions) is exactly the resolution proof rule. | ||
+ |