Consecutive Repeating State Cycles Determine Periodic Points in a Turing Machine

Abstract

The Turing machine is studied with new methods motivated by the notion of recurrence in classical dynamical systems theory. The state cycle of a Turing machine is introduced. It is proven that each consecutive repeating state cycle in a Turing machine determines a unique periodic configuration (point) and vice versa. This characterization is a periodic point theorem for Turing machines. A Turing machine is defined to be periodic if it has at least one periodic configuration or it only has halting configurations. Using the notion of a prime directed edge and a mathematical operation called edge pattern substitution, a search procedure finds consecutive repeating state cycles. If the Turing machine is periodic, then this procedure eventually finds each periodic point or this procedure determines that the machine has only halting configurations.Newmathematical techniques are demonstrated such as edge pattern substitution and prime directed edge sequences that could be quite useful in the further study of the aperiodic Turing machines. The aperiodicity appears to play an integral role in the undecidability of the Halting problem.