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sav08:deriving_propositional_resolution [2008/03/12 12:58] vkuncak |
sav08:deriving_propositional_resolution [2008/03/19 17:13] tatjana |
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We next consider proof rules for checking [[Satisfiability of Sets of Formulas]]. | We next consider proof rules for checking [[Satisfiability of Sets of Formulas]]. | ||
- | We extending the notion of [[Substitution Theorems for Propositional Logic|substitution on formulas]] to sets of formulas by | + | We are extending the notion of [[Substitution Theorems for Propositional Logic|substitution on formulas]] to sets of formulas by |
\[ | \[ | ||
subst(\sigma,S) = \{ subst(\sigma,F) \mid F \in S \} | subst(\sigma,S) = \{ subst(\sigma,F) \mid F \in S \} | ||
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We first derive a more abstract proof system and that show that resolution is a special case of it. | We first derive a more abstract proof system and that show that resolution is a special case of it. | ||
+ | |||
==== Key Idea ==== | ==== Key Idea ==== | ||
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Then we conclude that $\exists p. S$ is equivalent to $ProjectSet(S,p)$ defined by | Then we conclude that $\exists p. S$ is equivalent to $ProjectSet(S,p)$ defined by | ||
\[ | \[ | ||
- | Proj(S,p) = \{ ProjectForm(F_1,F_2,p) \mid F_1,F_2 \in S \} | + | ProjectSet(S,p) = \{ ProjectForm(F_1,F_2,p) \mid F_1,F_2 \in S \} |
\] | \] | ||
+ | |||
==== 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" |