2008/7/2 Hector Zenil <[EMAIL PROTECTED]>: > Hypercomputational models basically pretend to take advantage from > either infinite time or infinite space (including models such as > infinite resources, Zeno machines or the Omega-rule, real computation, > etc.), from the continuum. Depending of the density of that space/time > continuum one can think of several models taking advantage at several > levels of the arithmetical hierarchy.
Lest various readers of this thread be confused, lets be careful to make a distinction between physical infinity and the multitude of mathematical infinities. Whether or not one beleives that some model of of physical quantum computer depends on what one might believe about physics at the plank length -- which would be speculative. Next, using words like "infinite space" is misleading: the surface of a sphere has an infinite number of points; yet, upon hearing the words "infinite space", its unusual to think of a sphere as an example. Note that a single qubit is a 2-sphere, and so, in the same way that there are an infinite number of points in the surface of a sphere, so to are there an infinite number of states in a qubit. Does any given quantum computer make use of the infinite number of states in one qubit? That depends ... but the point is that one can ponder hypercomputation in a finite volume; this is crucial; don't let words like "infinite space" accidentally imply "infinite volume" - it does not. > But even if there is infinite > space or time another issue is how to verify a hypercomputation. One > would need another hypercomputer to verify the first and then trust in > one. Sure ... but we can express many theorems about Turing machines, and know them to be true, without "verifying" them on some other computer which we need to magically trust. Similarly, one can make plenty of true statements about hypercomputation without actually having to build one and verifying its operation. But perhaps I missed some subtle point. --linas ------------------------------------------- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244&id_secret=106510220-47b225 Powered by Listbox: http://www.listbox.com
