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sav08:deriving_propositional_resolution [2008/03/19 17:12] tatjana |
sav08:deriving_propositional_resolution [2008/03/19 17:15] 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 ==== | ||
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\] | \] | ||
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}\] |