Zeno machines and hypercomputation

Abstract

This paper reviews the Church-Turing Thesis (or rather, theses) with reference to their origin and application and considers some models of "hypercomputation", concentrating on perhaps the most straight-forward option: Zeno machines (Turing machines with accelerating clock). The halting problem is briefly discussed in a general context and the suggestion that it is an inevitable companion of any reasonable computational model is emphasised. It is suggested that claims to have "broken the Turing barrier" could be toned down and that the important and well-founded rôle of Turing computability in the mathematical sciences stands unchallenged.

Scale-invariant cellular automata and self-similar Petri nets

“…The construction of a hypercomputer was a first demonstration of the extraordinary computational capabilities of these models. Further investigations are necessary to determine their limits, and to relate them with the emerging field of hypercomputa-tion [21,22,23,27,31,42,43]. Another line of research would be the investigation of their phenomenological properties, analogous to the statistical mechanics of cellular automata [8,44].…”

Computability and Complexity of Unconventional Computing Devices

“…An example of a theoretical hypercomputer is the Zeno Machine (Potgieter;2006). A Zeno Machine is an Accelerated Turing Machine that takes 1/2 n units of time to perform its n-th step; thus, the first step takes 1/2 units of time, the second takes 1/4, the third 1/8, and so on, so that after one unit of time, a countably infinite number of steps will have been performed.…”

A note on accelerated Turing machines

“…Copeland (2002) and Stewart (1991) applied this idea to Turing computations. Accelerated Turing machines have been studied by various authors including Barrow (Barrow 2005), Boolos and Jeffrey (Boolos and Jeffrey 1980), Calude and Pȃun (Calude and Pȃun 2004), Ord (Ord 2002), Potgieter (Potgieter 2006), Shagrir (Shagrir 2005; and Svozil (Svozil 1998). Such a machine can run an infinite number of steps in one unit of time.…”