Ed, After a little more thought, it occurred to me that this problem was already solved in coding theory ... just take the bound given here, with q=2:
http://en.wikipedia.org/wiki/Hamming_bound The bound is achievable using Hamming codes (linked to from that page), so it's realizable. What they do there is, look at q-ary code strings of length n (in the binary case, q=2). Then, they say that d errors in transmission will occur while the code is being sent, so they want any two code strings to differ in at least d places. The bound tells how many different messages can be encoded in this way. I think this is the same as your problem. Assume you have n neurons. Then each potential cell assembly may be represented as an length-n bit string, where the i'th neuron in the network corresponds to the i'th bit in the bit string, and an assembly A has a 1 in position i iff it contains the i'th neuron. Then, what we want is for no two assemblies to have more than d neurons overlap -- because there may be some noise in memory retrieval. So, the optimal way to do this would be to assign memories to cell assemblies according to a perfect code, such as a Hamming code. Of course this is not how the brain works, though. But anyway, that seems to resolve your combinatorics problem... -- Ben On Thu, Oct 16, 2008 at 3:51 PM, Ed Porter <[EMAIL PROTECTED]> wrote: > Thanks, for offering to look into this.. The bounds I saw were upper > bounds (such as just the number of possible combinations), and I was more > interested in lower bounds. > > > > -----Original Message----- > *From:* Ben Goertzel [mailto:[EMAIL PROTECTED] > *Sent:* Thursday, October 16, 2008 2:45 PM > *To:* [email protected] > *Subject:* Re: [agi] Who is smart enough to answer this question? > > > > > I am pretty sure their formulas give bounds on the number you want, but not > an exact calculation... > > Sorry the terminology is a pain! At some later time I can dig into this > for you but this week I'm swamped w/ practical stuff... > > On Thu, Oct 16, 2008 at 2:35 PM, Ed Porter <[EMAIL PROTECTED]> wrote: > > Ben, > > > > Thanks. I spent about an hour trying to understand this paper, and, from > my limited reading and understanding, it was not clear it would answer my > question, even if I took the time that would be necessary to understand it, > although it clearly was in the same field of inquiry. > > > > After about an hour I gave up, since it uses a lot of terminology I do not > know and since it is sufficiently deep that without the help of someone to > guide me, I am not even clear that I would be able to understand it enough > to extract from it guidance as to how to solve the problem I posed. > > > > Ed Porter > > > > -----Original Message----- > *From:* Ben Goertzel [mailto:[EMAIL PROTECTED] > *Sent:* Thursday, October 16, 2008 11:32 AM > *To:* [email protected] > *Subject:* Re: [agi] Who is smart enough to answer this question? > > > > > 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-<http://www.math.ucla.edu/%7Ebsudakov/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> > <https://www.listbox.com/member/archive/rss/303/>| > Modify<https://www.listbox.com/member/?&;>Your Subscription > > <http://www.listbox.com> > > > ------------------------------ > > *agi* | Archives <https://www.listbox.com/member/archive/303/=now> > <https://www.listbox.com/member/archive/rss/303/>| > Modify<https://www.listbox.com/member/?&;>Your Subscription > > <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> > <https://www.listbox.com/member/archive/rss/303/>| > Modify<https://www.listbox.com/member/?&;>Your Subscription > > <http://www.listbox.com> > > > ------------------------------ > *agi* | Archives <https://www.listbox.com/member/archive/303/=now> > <https://www.listbox.com/member/archive/rss/303/> | > Modify<https://www.listbox.com/member/?&;>Your Subscription > <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
