n-Skip Turing Machines

The Turing Machine was designed by Alan Turing to serve as a general computational model. The machine has a tape to which data can be written by a head. A tape consists of cells where each cell has a value (similar to a cell on a hard-drive, albeit the cells of a Turing Machine’s tape can have more than two states) and that value can be changed by the head. The head can also be in multiple states and it may move. What the head writes to the tape for a given iteration is determined by the value (color) of the cell and the state of the head. For a given case, the cell’s color, the head’s state, as well as the head’s position may change. Of particular note, however, is that the head may only move by one cell in either direction [1]. A Turing Machine can be made to exhibit complex behavior for simple initial conditions by adding states, colors, or both. In [2] Wolfram shows that the simplest Turing Machine with complex behavior with an empty tape as input has four states and two colors. What we consider is whether one can create a Turing Machine which exhibits complex behavior by letting the head move by more than one cell in either direction (we assume a one-dimensional tape). We define such a machine to be a n-Skip Turing Machine, where the n refers to the maximum number of cells a head can skip for a given iteration. Notice that we do not impose the limitation that the head must move n cells. We define the rule of a n-Skip Turing Machine as follows (with two states and two colors):