▁functions ▁which ▁can ▁be ▁computed ▁by ▁a ▁T uring ▁machine ▁that ▁always ▁hal ts . ▁This ▁is ▁because ▁existence ▁of ▁such ▁a ▁programming ▁language ▁would ▁be ▁a ▁contradiction ▁to ▁the ▁non - sem i - dec id ability ▁of ▁the ▁problem ▁whether ▁a ▁T uring ▁machine ▁hal ts ▁on ▁every ▁input . ▁ ▁Rel ations hip ▁to ▁partial ▁T uring ▁machines ▁▁ ▁A ▁general ▁T uring ▁machine ▁will ▁compute ▁a ▁partial ▁function . ▁ ▁Two ▁questions ▁can ▁be ▁asked ▁about ▁the ▁relationship ▁between ▁partial ▁T uring ▁machines ▁and ▁total ▁T uring ▁machines : ▁ ▁Can ▁every ▁partial ▁function ▁comput able ▁by ▁a ▁partial ▁T uring ▁machine ▁be ▁extended ▁( that ▁is , ▁have ▁its ▁domain ▁en larg ed ) ▁to ▁become ▁a ▁total ▁comput able ▁function ? ▁ ▁Is ▁it ▁possible ▁to ▁change ▁the ▁definition ▁of ▁a ▁T uring ▁machine ▁so ▁that ▁a ▁particular ▁class ▁of ▁total ▁T uring ▁machines , ▁computing ▁all ▁the ▁total ▁comput able ▁functions , ▁can ▁be ▁found ? ▁The ▁answer ▁to ▁each ▁of ▁these ▁questions ▁is ▁no . ▁ ▁The ▁following ▁theorem ▁shows ▁that ▁the ▁functions ▁comput able ▁by ▁machines ▁that ▁always ▁halt ▁do ▁not ▁include ▁extensions ▁of ▁all ▁partial ▁comput able ▁functions , ▁which ▁implies ▁the ▁first ▁question ▁above ▁has ▁a ▁negative ▁answer . ▁▁ ▁This ▁fact ▁is ▁closely ▁related ▁to ▁the ▁algorithm ic ▁un sol v ability ▁of ▁the ▁Hal ting ▁problem . ▁ ▁Theorem . ▁There ▁are ▁T uring ▁comput able ▁partial ▁functions ▁that ▁have ▁no ▁extension ▁to ▁a ▁total ▁T uring ▁comput able ▁function