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sav08:interpolation_for_propositional_logic [2008/03/11 16:18] vkuncak |
sav08:interpolation_for_propositional_logic [2015/04/21 17:30] (current) |
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**Definition of Interpolant:** | **Definition of Interpolant:** | ||
- | Given two propositional formulas $F$ and $G$, an interpolant for $(F,G)$ is a propositional formula $H$ such that: \\ | + | 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) $\models H \rightarrow G$ \\ | ||
Line 26: | Line 26: | ||
**Definition:** ${\cal I}(F,G)$ 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, | ||
- | \[ | + | \begin{equation*} |
{\cal I}(F,G) = \{ H \mid H \mbox{ is interpolant for $(F,G)$ \} | {\cal I}(F,G) = \{ H \mid H \mbox{ is interpolant for $(F,G)$ \} | ||
- | \] | + | \end{equation*} |
**Theorem:** The following properties hold for $H_{min}$, $H_{max}$, ${\cal I}(F,G)$ defined above: | **Theorem:** The following properties hold for $H_{min}$, $H_{max}$, ${\cal I}(F,G)$ defined above: |