atisf i able ▁( or ▁contradict ory ), ▁and ▁hence ▁it ▁can ▁be ▁concluded ▁that ▁the ▁initial ▁conject ure ▁follows ▁from ▁the ▁ax i oms . ▁ ▁If , ▁on ▁the ▁other ▁hand , ▁the ▁empty ▁clause ▁cannot ▁be ▁derived , ▁and ▁the ▁resolution ▁rule ▁cannot ▁be ▁applied ▁to ▁derive ▁any ▁more ▁new ▁cla uses , ▁the ▁conject ure ▁is ▁not ▁a ▁theorem ▁of ▁the ▁original ▁knowledge ▁base . ▁ ▁One ▁instance ▁of ▁this ▁algorithm ▁is ▁the ▁original ▁Davis – Put nam ▁algorithm ▁that ▁was ▁later ▁ref ined ▁into ▁the ▁D P LL ▁algorithm ▁that ▁removed ▁the ▁need ▁for ▁explicit ▁representation ▁of ▁the ▁resol vent s . ▁ ▁This ▁description ▁of ▁the ▁resolution ▁technique ▁uses ▁a ▁set ▁S ▁as ▁the ▁underlying ▁data - structure ▁to ▁represent ▁resolution ▁deriv ations . ▁List s , ▁Tre es ▁and ▁Direct ed ▁A cyc lic ▁Graph s ▁are ▁other ▁possible ▁and ▁common ▁alternatives . ▁Tree ▁representations ▁are ▁more ▁faithful ▁to ▁the ▁fact ▁that ▁the ▁resolution ▁rule ▁is ▁binary . ▁T ogether ▁with ▁a ▁sequ ent ▁notation ▁for ▁cla uses , ▁a ▁tree ▁representation ▁also ▁makes ▁it ▁clear ▁to ▁see ▁how ▁the ▁resolution ▁rule ▁is ▁related ▁to ▁a ▁special ▁case ▁of ▁the ▁cut - rule , ▁restricted ▁to ▁atomic ▁cut - form ulas . ▁However , ▁tree ▁representations ▁are ▁not ▁as ▁compact ▁as ▁set ▁or ▁list ▁representations , ▁because ▁they ▁explicitly ▁show ▁redundant ▁sub der iv ations ▁of ▁cla uses ▁that ▁are ▁used ▁more ▁than ▁once ▁in ▁the ▁deriv ation ▁of ▁the ▁empty ▁clause . ▁Graph ▁representations ▁can