Hypercomputation: computing more than the Turing machine

TL;DR

This paper surveys hypercomputation, models that surpass Turing machines, exploring their theoretical foundations, physical feasibility, and implications for classical computation, Godel's theorem, and mathematical randomness.

Contribution

It provides a comprehensive overview of hypercomputation models, their relation to classical theory, and discusses their physical realizability and philosophical implications.

Findings

Hypercomputation models can compute beyond Turing limits.

Physical realizability of hypercomputers remains uncertain.

Hypercomputation challenges the traditional understanding of mathematical incompleteness.

Abstract

Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation that can compute more than the Turing machine and addresses their implications. In this report, I survey much of the work that has been done on hypercomputation, explaining how such non-classical models fit into the classical theory of computation and comparing their relative powers. I also examine the physical requirements for such machines to be constructible and the kinds of hypercomputation that may be possible within the universe. Finally, I show how the possibility of hypercomputation weakens the impact of Godel's Incompleteness Theorem and Chaitin's discovery of 'randomness' within arithmetic.