Δευτέρα, 25 Αυγούστου 2014

Constructive Hypercomputation

Constructivists assert that one has to construct a mathematical object in order  to show that it exists. And for some reasons they reject hypercomputation. In particular, Rasoul Ramezanian notes correctly in A Hypercomputation in Brouwer's Constructivism that for Brouwer, who was the founder of the mathematical philosophy of intuitionism, something exists as long there is a mental construction for it and this is exactly the reason for the rejection. Some constructivists do not accept that there are infinite objects at all. In fact, some assert that there are 21000 elementary particles in the Universe and so they believe this is the largest number! To me such ideas are absurd. But Ramezanian concludes that intuitionism can co-exist with hypercomputation. Moreover, he presents his Persistent Evolutionary Turing Machines, which is a couple N = (⟨z0, z1,…, zi⟩, f) where z0, z1,…, zi is a growing sequence of codes of deterministic Turing machines, and f (called the persistently evolutionary function) is a computable partial function from Σ× Σ to Σ. Ramezanian demands that f has certain properties and from there he goes on to explore the hypercomputational capabilities of this machine.

Implementing an Analog Recurrent Neural Network

A. Steven Younger, Emmett Redd, and Hava Siegelmann published a paper entitled  Development of Physical Super-Turing Analog Hardware, where they report their efforts to build a real hypercomputer. In particular, they present their work on the realization of Analog Recurrent Neural Networks (ARNN, for short). The theory of ARNNs is presented in Neural Networks and Analog Computation.In a nutshel, the ARNNs are generally more powerful than Turing machines and so they are classified as hypecomputers. Younger  et al. have designed and developed an OpticARNN which is depicted in the figure that follows:
Also, they have developed an electronic ARNN whose functional schematic follows:
These system have not been tested thoroughly and so one cannot draw definitive conclusions. The authors plan to build larger devices and continue their studies.

A "Solution" to Riemann Hypothesis

Riemann hypothesi s is "is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex n...