They also note that according to their experiments, bounded-weight codes don't offer much improvement over constant-weight codes, for which analytical results *are* available... and for which lower bounds are given at
http://www.research.att.com/~njas/codes/Andw/ ben On Thu, Oct 16, 2008 at 6:40 PM, Ben Goertzel <[EMAIL PROTECTED]> wrote: > 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 > > > -- 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
