On the Possibilities of Hypercomputing Supertasks

TL;DR

This paper argues that hypercomputing supertasks, such as Zeno-machines, are impossible in reality, supporting the continued validity of the Church-Turing thesis and challenging claims that hypercomputing refutes it.

Contribution

It critically examines proposals for digital hypercomputing with Zeno-machines and demonstrates their practical and logical impossibility, reaffirming the Church-Turing thesis.

Findings

Hypercomputing supertasks are impossible in the actual world.

Zeno-machines either lack outputs or involve contradictions.

No effective methods or rules can repair hypercomputing supertasks.

Abstract

This paper investigates the view that digital hypercomputing is a good reason for rejection or re-interpretation of the Church-Turing thesis. After suggestion that such re-interpretation is historically problematic and often involves attack on a straw man (the 'maximality thesis'), it discusses proposals for digital hypercomputing with Zeno-machines , i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zeno-machines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though non-effective methods or special rules for semi-decidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the Church-Turing thesis in its traditional interpretation.