One more addition... Actually the Hamming-code problem is not exactly the same as your problem because it does not place an arbitrary limit on the size of the cell assembly... oops
But I'm not sure why this limit is relevant, since cell assemblies in the brain could be very large Anyway, it seems that to answer your precise question, one must look at **bounded weight codes**, as is done in http://www.jucs.org/jucs_5_12/a_note_on_bounded/Bent_R.html They do not give analytical formulas there, but they give some computational results they found using some nice graph theory software. The final paragraph of their paper is interestingly relevant: *** However, as a final comment, we mention that maximum-sized codes may have potential future uses in alternate models of computation/communication. Though this is currently hypothetical, it is not entirely implausible. Consider for example some future alternate model of information (storage or) transmission - perhaps biological, perhaps electrical, perhaps something else - in which each (stored or) transmitted "word" has *n* binary "bits" (which might be represented via genetic material, or via charged particles in a given location, or so on) but such that, due to constraints of the (storage or) transmission medium, if more than *w* of the bits are "on" there is the possibility that the information in the word will degrade, or that the computer or transmission lines will incur physical damage. Possible reasons might include power limitations, heat dissipation, or attraction between biological components. In this admittedly extremely hypothetical setting, bounded-weight codes might play a valuable role, as their limitation would be exactly suited to the physical constraints imposed by the (storage or) transmission medium. *** ;-) -- Ben On Thu, Oct 16, 2008 at 6:24 PM, Ben Goertzel <[EMAIL PROTECTED]> wrote: > > 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 > > > -- 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
