Zeno machine

Abstract: The accelerated Turing machine (ATM) is the work-horse of hypercomputation. In certain cases, a machine having run through a countably infinite number of steps is supposed to have decided some interesting question such as the Twin Prime conjecture. One is, however, careful to avoid unnecessary discussion of either the possible actual use by such a machine of an infinite amount of space, or the difficulty (even if only a finite amount of space is used) of defining an outcome for machines acting like Thomson's lamp. It is the authors' impression that insufficient attention has been paid to introducing a clearly defined counterpart for ATMs of the halting/non-halting dichotomy for classical Turing computation. This paper tackles the problem of defining the output, or final message, of a machine which has run for a countably infinite number of steps. Non-standard integers appear quite useful in this regard and we describe several models of computation using filters. Die versnelde Turing-masjien (VTM) is die trekperd van hiperberekening. In sekere gevalle word veronderstel dat 'n masjien wat aftelbaar oneindig aantal stappe uitgevoer het, 'n interessante probleem soos die Tweelingpriemvermoede sou beslis het. 'n Mens le egter sorg aan die dag om 'n uiteensetting van of die potensiele benutting van oneindig veel ruimte deur 'n dergelike masjien of die probleem (indien slegs eindig veel ruimte gebruik is) om 'n eindtoestand te definieer vir masjiene wat optree soos Thomson se lamp. Die outeurs is onder die indruk dat te min aandag gegee word aan die invoer van 'n deeglik gedefinieerde eweknie van die halt/niehalt-tweespalt in klassieke Turing-berekening. Hierdie artikel takel die probleem van definisie van die uitvoer, of finale boodskap, van 'n masjien wat 'n aftelbaar oneindige aantal stappe uitgevoer het. Die niestandaard-heelgetalle skyn nuttig te wees in hierdie verband en ons beskryf 'n aantal berekeningsmodelle met filters.