The Diagonal Method and Hypercomputation

TL;DR

This paper critically examines the diagonal method's application to hypercomputation, arguing that claims of contradiction or arbitrariness are flawed because such contradictions only arise if a machine can compute its own diagonal function, which is unlikely in hypercomputation.

Contribution

The paper clarifies the limitations of the diagonal method in hypercomputation and refutes recent claims of contradictions in models claiming to solve the halting problem.

Findings

Contradictions only occur if a machine computes its own diagonal function

Hypercomputation methods do not compute their own diagonal functions

Claims of contradiction in hypercomputation models are flawed

Abstract

The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical models of computation that claim to be able to solve the halting problem for Turing machines. We show that such arguments are flawed -- a contradiction only occurs if a type of machine can compute its own diagonal function. We then demonstrate why such a situation does not occur for the methods of hypercomputation under attack and why it is unlikely to occur in any other serious methods.