▁and ▁complete ▁algorithm ▁for ▁dec iding ▁the ▁satisf i ability ▁of ▁a ▁propos itional ▁formula , ▁and , ▁by ▁extension , ▁the ▁valid ity ▁of ▁a ▁sentence ▁under ▁a ▁set ▁of ▁ax i oms . ▁ ▁This ▁resolution ▁technique ▁uses ▁proof ▁by ▁contradiction ▁and ▁is ▁based ▁on ▁the ▁fact ▁that ▁any ▁sentence ▁in ▁propos itional ▁logic ▁can ▁be ▁transformed ▁into ▁an ▁equivalent ▁sentence ▁in ▁conj unct ive ▁normal ▁form . ▁The ▁steps ▁are ▁as ▁follows . ▁▁ ▁All ▁sentences ▁in ▁the ▁knowledge ▁base ▁and ▁the ▁neg ation ▁of ▁the ▁sentence ▁to ▁be ▁proved ▁( the ▁conject ure ) ▁are ▁conj unct ively ▁connected . ▁ ▁The ▁resulting ▁sentence ▁is ▁transformed ▁into ▁a ▁conj unct ive ▁normal ▁form ▁with ▁the ▁conj unct s ▁viewed ▁as ▁elements ▁in ▁a ▁set , ▁S , ▁of ▁cla uses . ▁For ▁example , ▁▁▁ ▁gives ▁rise ▁to ▁the ▁set ▁. ▁ ▁The ▁resolution ▁rule ▁is ▁applied ▁to ▁all ▁possible ▁pairs ▁of ▁cla uses ▁that ▁contain ▁complement ary ▁liter als . ▁After ▁each ▁application ▁of ▁the ▁resolution ▁rule , ▁the ▁resulting ▁sentence ▁is ▁simplified ▁by ▁removing ▁repeated ▁liter als . ▁If ▁the ▁clause ▁contains ▁complement ary ▁liter als , ▁it ▁is ▁disc arded ▁( as ▁a ▁t aut ology ). ▁If ▁not , ▁and ▁if ▁it ▁is ▁not ▁yet ▁present ▁in ▁the ▁clause ▁set ▁S , ▁it ▁is ▁added ▁to ▁S , ▁and ▁is ▁considered ▁for ▁further ▁resolution ▁infer ences . ▁ ▁If ▁after ▁applying ▁a ▁resolution ▁rule ▁the ▁empty ▁clause ▁is ▁derived , ▁the ▁original ▁formula ▁is ▁uns