On Mon, Jul 7, 2008 at 5:44 PM, Linas Vepstas <[EMAIL PROTECTED]> wrote: > 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.
If a sphere has infinite number of points then it is an infinite bounded space and you can theoretically build a hypercomputer on it. Any level in the arithmetical hierarchy can correspond to a level in the physical world. It depends on the cardinality of the physical space, from the continuum to any power of it, and whether its dense and how dense. > 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. The big difference is that in principle you can verify the output of any digital computer by hand in finite time, however you cannot verify any non-Turing output from a hypercomputer by hand, not even in principle unless having infinite time. > > --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/?& > Powered by Listbox: http://www.listbox.com > -- Hector Zenil http://zenil.mathrix.org ------------------------------------------- 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