Pure Literal

logic

Definition

Pure Literal

A literal $L$ in a set of clauses $C$ is called pure if $S$ contains no clauses of the form $\bar{L}\vee C$.

Examples

Example

The following literals are pure in this set:

$$\begin{array}{r}\neg p_2\vee \neg p_3\\ p_1\vee \neg p_2\\ \neg p_1\vee p_2\vee \neg p_3\\ \neg p_1\vee \neg p_3\\ p_1\vee p_2\\ \neg p_1\vee \neg p_2\vee \neg p_3\end{array}$$

The literal $\neg p_3$ is pure in this set. We can remove all clauses containing this literal and record $p_3\mapsto 0$:

$$\begin{array}{r}{p}_1\vee \neg p_2\\ p_1\vee p_2\end{array}$$

Now, the literal $p_1$ is pure in this set. We can remove all clauses containing this literal and record $\{p_1\mapsto 1\}$.

We obtain the empty set of clauses, yielding two models:

$$\begin{array}{r}\{p_1\mapsto 1,p_2\mapsto 0,p_3\mapsto 0\}\\ \{p_1\mapsto 1,p_2\mapsto 1,p_3\mapsto 0\}\end{array}$$