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. 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