Constructivists assert that one has to construct a mathematical object in order to show that it exists. Thus they reject hypercomputation. However, 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. 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 this is the largest number! To me such ideas are absurd. So 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.
Δευτέρα, 25 Αυγούστου 2014
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:
Σάββατο, 12 Ιουλίου 2014
In the preface of my book on hypercomputation I have stated that all models of computation described in the book assume the axiom of choice. Instead of explaining explicitly why it is needed. I give an excerpt from Gregory H. Moore's prologue to Zermelo's Axiom of Choice in the hope that readers will understand why it is needed.
Yet without the Axiom, mathematics today would be quite different. The very nature of modern mathematics would be altered and, if the Axiom's most severe constructivist critics prevailed, mathematics would be reduced to a collection of algorithms. Indeed, the Axiom epitomizes the fundamental changes—mathematical, philosophical, and psycological—that took place when mathematicians seriously began to study infinite collections of sets.
Πέμπτη, 20 Φεβρουαρίου 2014
The 7th AISB Symposium on Computing and Philosophy will examine whether computation is observer-related. In different words, participants will discuss whether computation is a sponteneous natural phenomenon or not. I think that computation, like art, is not a natural phenomenon. Nature is not a sculpturer or a painter, and for that matter not a programmer. To animals, a sculpture is just a stone or a piece of metal and that's all. Flowers are not beautiful or ugly: they just attract bees and other insects. Mountains are not fearsome and lakes are not picturesque. Only humans give this attributes to these physical entities. Similarly, no chair and no desk computes anything. In fact, even a computer does not compute anything unless someone would be able to interpret the result of the computation. I am sure that if one could present a computer to Aristotle he could not easily realize what kind of machine it is.
Quite recently, Takaaki Musha published a book entitled Superluminal Particles and Hypercomputation. I have not read the book but from the description I see that he proposes a new model of computation that is based on the existence of tachyons, that is, particles that travel faster than the light. I suppose that in Possibility of Hypercomputation from the Standpoint ofSuperluminal Particles he presents an earlier version of his idea.
Σάββατο, 18 Ιανουαρίου 2014
Today I read critical blog-post on the Mathematical Universe, that is the idea that "physical reality is a mathematical structure". Of course, this idea is surprisingly similar to the idea that the universe is a computer. In a sense, one could argue that the two ideas are identical. What puzzles me the most is that both physics and computer science are following very dangerous paths…