Approximate Outputs of Accelerated Turing Machines Closest to Their Halting Point

Abstract

The accelerated Turing machine (ATM) which can compute super-tasks are devices with the same computational structure as Turing machines (TM) and they are also defined as the work-horse of hypercomputation. Is the final output of the ATM can be produced at the halting state? We supported our analysis by reasoning on Thomson’s paradox and by looking closely the result of the Twin Prime conjecture. We make sure to avoid unnecessary discussion on the infinite amount of space used by the machine or considering Thomson’s lamp machine, on the difficulty of specifying a machine’s outcome. Furthermore, it’s important for us that a clear definition counterpart for ATMs of the non-halting/halting dichotomy for classical Turing must be introduced. Considering a machine which has run for a countably infinite number of steps, this paper addresses the issue of defining the output of a machine close or at the halting point.