Say a Turing machine is simulated by the following rules: A number of states can be defined. Each state is defined by : specifying what the head writes in the current cell in case of each symbol of the alphabet specifying where the head moves (left or right) in case of each symbol of the alphabet of the current cell specifying which state the current state transitions to in case of each symbol of the alphabet Is this a Turing-complete (ignoring memory limitations) machine? I have seen very different Turing machines on the Internet by some people. For example palindrome checking-capable machine which, at some point during its procedure moves the head by more than 1 cell. Or at some other point it searches for a specific value on the tape. Both of these tasks seem impossible on the approach I outlined. Does it make the approach I described result in a non-Turing-complete machine?