# In SAT, do we require an assignment for arbitrary variables?

I am reading about the Satisfiability Problem, in page (5) the author gives the following example :

$$(P \lor Q \lor R) \wedge (\bar{P} \lor Q \lor \bar{R}) \wedge (P \lor \bar{Q} \lor S) \wedge (\bar{P} \lor \bar{R} \lor \bar{S})$$

A satisfying assignement is $$: P,Q,\bar{R},\bar{S}$$.

A different assignments is $$P,\bar{R}$$ to satisfy the four caluses. This means that $$Q,S$$ can be set arbitrarily to $$True$$ or $$False$$.

Do we require an assignment for all variables or we only reqiure to set variables that satisfy the input (i.e. $$P,\bar{R}$$ discluding $$Q,S$$) ?

Does that mean that the satisfying assignment of $$P,\bar{R}$$ is more efficient than $$P,Q,\bar{R},\bar{S}$$ given that it uses less variables ? Is there any resources to read about it ?

Also, given that $$P,\bar{R}$$ satisfy the input, does that mean that we can disclude $$Q,S$$ from the original Boolean Formula ?

• A truth assignment assigns values to all variables. – Yuval Filmus Nov 24 '18 at 19:07