Full Resolution Rule - Artificial intelligence:

Now that we know about unification, we can correctly describe the complete edition of resolution:

p1 ∨... ∨pj ∨... ∨pm,     q1 ∨...∨qk  ∨... ∨qn

Subst(θ, p1 ∨... ∨pj-1 ∨pj+1 ∨...∨pm∨ q1 ∨... qk-1  ∨qk+1 ∨... ∨qn)

Unify(pj, ¬qk) = θ

This solves literals pj and qk. Note that we get to add ¬ to qk to make it join with pj, so it is in fact pj which is the negative truthful here. The principal is more ordinary than first-order binary resolution in that it allows an arbitrary number of literals in every clause.  Moreover,  θ  is  the  most  common  unifier,  rather  than  an  arbitrary unifying replacement.

To use the principal in practice, we first take a couple of statements and convey them in CNF using the actions defined above. Then we look for two literals, pj and qk for which may search a substitution mu to combine pj and ¬qk. Then we have a disjunction of all the literals (in both sentences) except pj and qk. Finally, we use the substitution θ to the new disjunction to agree on what we have just inferred using resolution.

In the next lecture, we will look at how resolution theorem verification is put into act, counting some example proofs, some heuristics for improving its appearance and some applications.