An Alternative Representation of Turing Machines by Means of the Iota-Delta Function

The evolution of universal systems has been of great interest to computer scientists. In particular, the role of Turing machines in the study of computational universality is widely recognized. Even though the patterns emerging from the evolution of this kind of dynamical system have been studied in much detail, the transition functions themselves have received less attention. In the present paper, the iota-delta function is used to encode the transition function of one-head Turing machines. In order to illustrate the methodology, we describe the transition functions of two universal Turing machines in terms of the latter function. By using the iota-delta function in this setting, Turing machines can be represented as a system of transition functions. This new representation allows us to write the transition functions as a linear combination of evolution variables wrapped by the iota-delta function. Thus, the nonlinear part of the evolution is totally described by the iota-delta function.