OK, I see what you're asking now I think some bounds on the number you're looking for, are given by some classical combinatorial theorems, such as you may find in
http://www.math.ucla.edu/~bsudakov/cross-*intersections*.pdf (take their set L to consist of {0,...,O} ... and set A_1 = A_2), and the references given therein. Anyway that paper should clue you in as to the right keyphrases to use in hunting down related theorems if you want to. You are right that it's a nontrivial combinatorial problem -- Ben On Thu, Oct 16, 2008 at 11:08 AM, Ed Porter <[EMAIL PROTECTED]> wrote: > > Eric, > > Actually I am looking for a function A =f(N,S,O). > > If one leaves out the O, and merely wants to find the number of > subcombinations of size S that can be formed from a population of size N, > just apply the standard formula for combinations. But adding the > limitation > that none of the combinations in A is allowed to overlap by more than O > with > any other combination in A makes things much more complex, and way beyond > my > understanding. > > Ed Porter > > -----Original Message----- > From: Eric Burton [mailto:[EMAIL PROTECTED] > Sent: Wednesday, October 15, 2008 8:05 PM > To: [email protected] > Subject: Re: [agi] Who is smart enough to answer this question? > > >Is anybody on this list smart and/or knowledgeable enough to come up with > a > >formula for the following (I am not): > > I don't think I'm the person to answer this for you. But I do have > some insights. > > > >Given N neural net nodes, what is the number A of unique node assemblies > >(i.e., separate subsets of N) of size S that can have less than O > >overlapping nodes, with the population of any other such node assembly > >similarly selected from the N nodes to have the same size S and less than > >the same O overlapping nodes with any other such node assembly. > > Good question. Let's call the function that returns A for N, "f(N)", > and continue. > > > >For example, if you have 1 billion nodes (N = 1G), how many cell > assemblies > >(A) of size 10,000 (S=10K) will have less than 5,000 nodes (0 = 5K) in > >common with the population of any other node assembly. > > So at this point we are seeking f(1000000000), which I'm going to go > ahead and call P. > > > >Its easy to figure out how many unique cell assemblies drawn from a > >population of N nodes that can have a size S, but I haven't a clue, other > >than by computational exploration to figure out how many will each have > less > >than a given level of overlap with any other unique cell assemblies. > > This is why we have f. > > > >And for anyone who knows how to solve the above, if possible, could you > also > >please also tell me, once you have close to A node assemblies selected > that > >have less than O overlap, how can you rapidly determine the population of > a > >new node assembly that has less than O overlap? > > For my purposes, f is a black box. You'd have to delve into its > internals to answer this yourself. > > > >This is not just an meaningless math problem. > >A lot of people believe the human brain uses cell assemblies to represent > >nodes in a representation of semantic knowledge. Such cell assemblies > >create problems with current computer hardware because they tend to > require > >very high internal bandwidth, but in future architectures this problem may > >not exist, and if the number of cell assemblies that can be created with a > >sufficiently low cross-talk is large relative to the number of nodes, the > >use of cell assemblies can allow for redundancy, high representational > >capacity, and gradual degrading of memories over time to make room for > more > >memories. > > Hence, I suppose, the value of your inquiry. If I'm visualizing the > issue correctly you'd like a function that returns the number of > unique neural nets with a number of nodes either less or greater than > n and interconnects either less or greater than i... I'll reiterate > that I'm not a student of mathematics and aren't qualified to address > the details of the problem. But surely this function is what we've > been calling f. I'd welcome corrections and clarifications. I may not > understand the question. > > Eric B > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] "Nothing will ever be attempted if all possible objections must be first overcome " - Dr Samuel Johnson ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
