Interpolation for Propositional Logic

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) $\models F \rightarrow H$
2) $\models H \rightarrow G$
3) $FV(H)\subseteq FV(F) \cap FV(G)$

What is FV?

Note that $F \rightarrow G$ by transitivity of $\rightarrow$. The interpolant extracts the essential part of formula $F$ which makes $F$ imply $G$.

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 $G$.

$H_{min} \equiv elim(\exists p_1, p_2, ..., p_n.\  F)$ where $\{p_1, p_2, ..., p_n \}\ = FV(F)\backslash FV(G)$

  • We can quantify universally all variables in $G$ that are not in $F$.

$H_{max} \equiv elim(\forall q_1, q_2, ..., q_m.\ G)$ where $\{q_1, q_2, ..., q_m \}\ = FV(G)\backslash FV(F)$

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 $(F,G)$ \}

Theorem: The following properties hold for $H_{min}$, $H_{max}$, ${\cal I}(F,G)$ defined above:

  1. $H_{min} \in {\cal I}(F,G)$
  2. $\forall H \in {\cal I}(F,G).\ \models (H_{min} \rightarrow H)$
  3. $H_{max} \in {\cal I}(F,G)$
  4. $\forall H \in {\cal I}(F,G).\ \models (H \rightarrow H_{max})$

We can also derive interpolants from resolution proofs, as we will see in lecture07.