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sav08:deriving_propositional_resolution [2008/03/19 17:12] tatjana |
sav08:deriving_propositional_resolution [2008/03/19 17:18] tatjana |
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| ProjectSet(S,p) = \{ ProjectForm(F_1,F_2,p) \mid F_1,F_2 \in S \} | ProjectSet(S,p) = \{ ProjectForm(F_1,F_2,p) \mid F_1,F_2 \in S \} | ||
| \] | \] | ||
| + | |||
| ==== Projection Proof Rules ==== | ==== Projection Proof Rules ==== | ||
| Line 54: | Line 55: | ||
| \] | \] | ||
| 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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| \] | \] | ||
| 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. | ||
| + | |||
| ==== 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}\] | ||
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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 | ||