Hypercomputation.info

Konstantine Arkoudas argues in " Computation, hypercomputation, and physical science " why in his opinion [T]he idea that physical science will be able to discover fundamental computability limits is untenable. A computation is carried out by concrete computational devices whose operation and capabilities are delimited by the laws of physics. It is one thing to argue that we have no idea what are the limits of computation and another to simple say that the limits of computation have nothing to do with physical reality. On the other hand, it is more than sure that there is a limit to what we can achieve with computing devices , but for the time being we simply do not know this limit. And this is exactly the essence of hypercomputation.

German Sierra and Paul K. Townsend have recently presented an idea that may lead to the solution of the Riemann hypothesis. The Riemann hypothesis is a co-recursively enumerable problem (roughly, there is an algorithm that, when given an input number, eventually halts if and only if the input satisfies the problem, but no algorithm can decide if an arbitrary input satisfies the problem or not). The solution of this and other similar problems would falsify Church's Thesis. Interestingly, Fermat's last theorem and Poincaré's conjecture are co-recursively enumerable problems, nonetheless, these problems have been decided! A proof that Church's thesis is false...?

I am really happy to announce that my book on Hypercomputation has been published by Springer. Hypercomputation is a new discipline that emerged from the ascertainment that the Church-Turing Thesis (CTT) cannot be possible true. Roughly, the CTT is about what can be computed with any real or conceptual computing device. In particular, if some function is computable, then it is definitely computable by a Turing mchine, that is, Turing's archetypal conceptual computing device. Hypercomputation, asserts that functions and numbers that cannot be computed by a Turing machine, can be computed by other more powerful computing devices. The description of such machines and their computational power is the subject of this book. In particular, the book describes the various approaches to hypercomputation in nine chapters Introduction On the Church–Turing Thesis Early Hypercomputers Infinite-Time Turing Machines Interactive Computing Hyperminds Computing Real Numbers Relativistic and Quant...