▁different ▁experiments ▁with ▁back prop ag ation ▁neural ▁network ▁mod elling . ▁▁ ▁Ex periment ▁ 1 : ▁Learning ▁the ▁ones ▁and ▁tw os ▁addition ▁facts ▁In ▁their ▁first ▁experiment ▁they ▁trained ▁a ▁standard ▁back prop ag ation ▁neural ▁network ▁on ▁a ▁single ▁training ▁set ▁consisting ▁of ▁ 1 7 ▁single - digit ▁ones ▁problems ▁( i . e ., ▁ 1 ▁+ ▁ 1 ▁through ▁ 9 ▁+ ▁ 1 , ▁and ▁ 1 ▁+ ▁ 2 ▁through ▁ 1 ▁+ ▁ 9 ) ▁until ▁the ▁network ▁could ▁represent ▁and ▁respond ▁properly ▁to ▁all ▁of ▁them . ▁The ▁error ▁between ▁the ▁actual ▁output ▁and ▁the ▁desired ▁output ▁stead ily ▁decl ined ▁across ▁training ▁sessions , ▁which ▁reflected ▁that ▁the ▁network ▁learned ▁to ▁represent ▁the ▁target ▁outputs ▁better ▁across ▁tri als . ▁ ▁Next ▁they ▁trained ▁the ▁network ▁on ▁a ▁single ▁training ▁set ▁consisting ▁of ▁ 1 7 ▁single - digit ▁tw os ▁problems ▁( i . e ., ▁ 2 ▁+ ▁ 1 ▁through ▁ 2 ▁+ ▁ 9 , ▁and ▁ 1 ▁+ ▁ 2 ▁through ▁ 9 ▁+ ▁ 2 ) ▁until ▁the ▁network ▁could ▁represent , ▁respond ▁properly ▁to ▁all ▁of ▁them . ▁They ▁noted ▁that ▁their ▁procedure ▁was ▁similar ▁to ▁how ▁a ▁child ▁would ▁learn ▁their ▁addition ▁facts . ▁Following ▁each ▁learning ▁trial ▁on ▁the ▁tw os ▁facts , ▁the ▁network ▁was ▁tested ▁for ▁its ▁knowledge ▁on ▁both ▁the ▁ones ▁and ▁tw os ▁addition ▁facts . ▁Like ▁the ▁ones ▁facts , ▁the ▁tw os ▁facts ▁were ▁readily ▁learned