. ▁ ▁In ▁particular , ▁the ▁partial ▁function ▁f ▁defined ▁so ▁that ▁f ( n ) ▁= ▁m ▁if ▁and ▁only ▁if ▁the ▁T uring ▁machine ▁with ▁index ▁n ▁hal ts ▁on ▁input ▁ 0 ▁with ▁output ▁m ▁has ▁no ▁extension ▁to ▁a ▁total ▁comput able ▁function . ▁ ▁Indeed , ▁if ▁g ▁were ▁a ▁total ▁comput able ▁function ▁extending ▁f ▁then ▁g ▁would ▁be ▁comput able ▁by ▁some ▁T uring ▁machine ; ▁fix ▁e ▁as ▁the ▁index ▁of ▁such ▁a ▁machine . ▁▁ ▁Build ▁a ▁T uring ▁machine ▁M , ▁using ▁Kle ene ' s ▁recursion ▁theorem , ▁which ▁on ▁input ▁ 0 ▁sim ul ates ▁the ▁machine ▁with ▁index ▁e ▁running ▁on ▁an ▁index ▁n M ▁for ▁M ▁( th us ▁the ▁machine ▁M ▁can ▁produce ▁an ▁index ▁of ▁itself ; ▁this ▁is ▁the ▁role ▁of ▁the ▁recursion ▁theorem ). ▁▁ ▁By ▁assumption , ▁this ▁simulation ▁will ▁eventually ▁return ▁an ▁answer . ▁Define ▁M ▁so ▁that ▁if ▁g ( n M ) ▁= ▁m ▁then ▁the ▁return ▁value ▁of ▁M ▁is ▁m ▁+ ▁ 1 . ▁ ▁Thus ▁f ( n M ), ▁the ▁true ▁return ▁value ▁of ▁M ▁on ▁input ▁ 0 , ▁will ▁not ▁equal ▁g ( n M ). ▁ ▁Hence ▁g ▁does ▁not ▁extend ▁f . ▁ ▁The ▁second ▁question ▁asks , ▁in ▁ess ence , ▁whether ▁there ▁is ▁another ▁reasonable ▁model ▁of ▁computation ▁which ▁comput es ▁only ▁total ▁functions ▁and ▁comput es ▁all ▁the ▁total ▁comput able ▁functions . ▁ ▁Inform ally , ▁if ▁such ▁a