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sav08:interpolation_for_propositional_logic [2008/03/10 11:30] vkuncak |
sav08:interpolation_for_propositional_logic [2008/03/17 20:24] piskac |
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| ====== Interpolation for Propositional Logic ====== | ====== Interpolation for Propositional Logic ====== | ||
| - | Given two propositional formulas $F$ and $G$, an interpolant for $(F,G)$ is a propositional formula $H$ such that: \\ | + | **Definition of Interpolant:** |
| + | Given two propositional formulas $F$ and $G$, such that $\models F \rightarrow G$, an interpolant for $(F,G)$ is a propositional formula $H$ such that: \\ | ||
| 1) $\models F \rightarrow H$ \\ | 1) $\models F \rightarrow H$ \\ | ||
| 2) $\models H \rightarrow G$ \\ | 2) $\models H \rightarrow G$ \\ | ||
| 3) $FV(H)\subseteq FV(F) \cap FV(G)$ | 3) $FV(H)\subseteq FV(F) \cap FV(G)$ | ||
| - | $FV$ denotes the set of free variables in the given propositional formula (see [[Propositional Logic Syntax]]). | + | ++++What is FV?| |
| + | $FV$ denotes the set of free variables in the given propositional formula, see [[Propositional Logic Syntax]]. | ||
| + | ++++ | ||
| - | Note that $F \rightarrow G$ by transitivity of $\rightarrow$. The interpolant extracts what makes $F$ imply $G$. | ||
| - | Here are two easy ways to create an interpolant. Let $elim$ denote the operator that eliminates propositional quantifiers (see [[QBF and Quantifier Elimination]]): | + | Note that $F \rightarrow G$ by transitivity of $\rightarrow$. The interpolant extracts the essential part of formula $F$ which makes $F$ imply $G$. |
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| + | Let $elim$ denote the operator that eliminates propositional quantifiers (see [[QBF and Quantifier Elimination]]). | ||
| + | |||
| + | Here are two **simple ways to construct an interpolant:** | ||
| * We can quantify existentially all variables in $F$ that are not in $G$. | * We can quantify existentially all variables in $F$ that are not in $G$. | ||
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| * We can quantify universally all variables in $G$ that are not in $F$. | * We can quantify universally all variables in $G$ that are not in $F$. | ||
| - | $H_{max} \equiv elim(\forall p_1, p_2, ..., p_n.\ G)$ where $\{p_1, p_2, ..., p_n \}\ = FV(G)\backslash FV(F)$ \\ | + | $H_{max} \equiv elim(\forall q_1, q_2, ..., q_m.\ G)$ where $\{q_1, q_2, ..., q_m \}\ = FV(G)\backslash FV(F)$ \\ |
| - | More precisely, let ${\cal I}(F,G)$ denote the set of all interpolants for $(F,G)$, that is, | + | **Definition:** ${\cal I}(F,G)$ denote the set of all interpolants for $(F,G)$, that is, |
| \[ | \[ | ||
| {\cal I}(F,G) = \{ H \mid H \mbox{ is interpolant for $(F,G)$ \} | {\cal I}(F,G) = \{ H \mid H \mbox{ is interpolant for $(F,G)$ \} | ||
| \] | \] | ||
| - | Then the following properties hold: | + | |
| + | **Theorem:** The following properties hold for $H_{min}$, $H_{max}$, ${\cal I}(F,G)$ defined above: | ||
| - $H_{min} \in {\cal I}(F,G)$ | - $H_{min} \in {\cal I}(F,G)$ | ||
| - $\forall H \in {\cal I}(F,G).\ \models (H_{min} \rightarrow H)$ | - $\forall H \in {\cal I}(F,G).\ \models (H_{min} \rightarrow H)$ | ||