Hypercomputation: computing more than the Turing machine

• DOI: 10.48550/arxiv.math/0209332
• Publication Date: 2002-01-01
• Authors: Ord, Toby
• Keywords: FOS: Computer and information sciences; Other Computer Science (cs.OH); FOS: Physical sciences; Mathematics - Logic; 06 humanities and the arts; Mathematical Physics (math-ph); 0603 philosophy, ethics and religion; Computer Science - Other Computer Science; 03D10 (Primary) 68Q10, 68Q10, 68Q30 (Secondary); FOS: Mathematics; Logic (math.LO); Mathematical Physics

57 pages, 9 figures<br/>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.<br/>