Dear Bruno, Thank you for writing this remark! It is very helpful. I could see where there could be some debate on the constructability claim, as the set of all programs in L could be infinite and thus the lexicographic ordering would be a supertask in that case, but that can be ignored for now. My interest now is in the computational Word Problem. I have more homework to do.
On Sun, Jan 19, 2014 at 6:33 AM, Bruno Marchal <marc...@ulb.ac.be> wrote: > Stephen, Liz, > > On 18 Jan 2014, at 00:02, LizR wrote: > > On 18 January 2014 11:39, Stephen Paul King <stephe...@provensecure.com>wrote: > >> Dear John, >> >> I invite your comment on a statement and question: *There is not >> observable difference between "X is non-computable" and "there does not >> exist sufficient resources to complete the computation of Y".* >> >> Are X and Y effectively the same thing, everything else being equal? If >> there is a difference that makes a difference, what might it be? >> > > In other words, is anything non-computable because of some theoretical > reason, rather than "merely local geographical" ones (which might cease to > be restrictions if, say, our local even horizon expands, or we construct > wormholes to other universe) ? > > Surely the halting problem? > > > Of more easily the totality problem. It is more complex than halting, and > thus more easily shown to be non computable. > > Let me do it here, as I promise to Stephen. It is almost a direct > consequence of Church thesis. > > I recall that a function (from N to N) is said TOTAL computable if you can > explain in a finite set of words, in one formal language, with a decidable > simple grammar, how to compute it on each (cf TOTAL) natural numbers, in a > finite time, and this to a sufficiently dumb person. > > Church's thesis: there is a universal language L in which all TOTAL > functions can be described/coded, and my language "lambda calculus" is such > a language. > > Theorem: Church's thesis entails that L describes more than just all TOTAL > functions. > > Proof: suppose that L describes only the TOTAL functions. We can > lexicographically order all programs in L (due to the clear decidable > grammar) p0, p1, p2, p3, .... Let us call f_i the function computed by p_i > > Then we can define a function g such that g(n) = f_n(n) + 1. Indeed we can > even program it in L. (To compute it on n, generate the list up to p_n and > apply p_n on n, then add 1. As all f_n are total, this gives a total > function (everywhere defined on N). > > So that function g as some program p_k, and thus compute some function f_k > belonging to the list (L is universal). > > But then, let us apply g on its own code k, we have g(k) = f_k(k), given > that g = f_k. > But we have also that g(k) = f_k(k) + 1, by definition of g. So f_k(k) = > f_k(k) + 1. > And, by totality, f_k(k) is a number. So I can subtract it from both side > in the preceding equality, and we get > 0 = 1. Contradiction. > > So either L is not universal (and Church is wrong), or f_k(k) is not > defined, and f_k is not total. > > Then if you look in "lambda calculus", you can empirically observe that > f_k(k) is indeed not defined: p_k does not stop on k. Making Church thesis > consistent. > > So if Church thesis is true, and if some language L is truly universal, > the list of its programs p_i will go through all total computable functions > (by universality), but will also go through many NON TOTAL functions. > > And, the key point, you will not been able to use any program to > distinguish the code of the total function from the code of the non total > one. > > Why? Because if that was the case, you would be able to filter out the non > total function from the list of all programs p_i, and you would get a > computable enumeration (list) of all, and only the total functions, and the > diagonal above would again conclude that 0 = 1. > > So the attribute of a code "computing a total function" cannot be > computable. QED. > > OK? (you have to read this with pencil and paper!) > > This was a constructive proof, leading to a meaningful predicate ("TOTAL") > being not computable. > > Thi entails a strong form of incompleteness: no effective theory (with > checkable proofs) can ever decide if some arbitrary code is the code of a > total, or not, function. > > (Note that a far easier, but non constructive and less informative, proof > is given directly by Cantor theorem. The computable functions are codable, > making them enumerable, but N^N, the set of all functions from N to N is > NOT enumerable by Cantor theorem. > So almost all functions from N to N are not computable.) > > Bruno > > > > > > > > > > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to email@example.com. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > You received this message because you are subscribed to a topic in the > Google Groups "Everything List" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/everything-list/TBc_y2MZV5c/unsubscribe. > To unsubscribe from this group and all its topics, send an email to > everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to firstname.lastname@example.org. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > -- Kindest Regards, Stephen Paul King Senior Researcher Mobile: (864) 567-3099 stephe...@provensecure.com http://www.provensecure.us/ “This message (including any attachments) is intended only for the use of the individual or entity to which it is addressed, and may contain information that is non-public, proprietary, privileged, confidential and exempt from disclosure under applicable law or may be constituted as attorney work product. 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