On Mon, Jan 20, 2014 at 8:50 PM, Sergio Donal <[email protected]> wrote:
> Perhaps the trick is to move the logic to some space where you can > approximate Cn by a simple matrix multiplication, very much like a kernel > method (perhaps you could apply the kernel trick for some form of Kernel > Regression that gives the projection :-?). > I guess Cn is too complicated to approximate at once, but maybe the single step St is easier... St: facts X rules --> facts but it seems to be a set-valued function. Operationally, St(F) performs the matching of the set of rules (R) with F, via the unification algorithm. The substitution gotten from unification is then applied to the "consequence" or "tail" part of the rule (in Prolog it's called the "head" but that seems to be a misnomer). The substituted tail is then added to the facts space. The question is how to approximate this via matrices. But a matrix multiplication from and to the same space merely moves the points; it does not add points to the space that are not originally there. Perhaps we'd want the matrix multiplication to map to a set of consequence points (for a single step), which could then be added to the facts space. That means the matrix multiplication would condense the operations of: 1) unification of heads with facts and then 2) substitution to the tails. This still seems to be too complex... This "operator" St_R() maps a set of facts to new consequences according to a set of rules R. So St_R() takes R as parameter. For each fixed R, St_R(F) returns a new set of facts. It seems to ignore the facts in F that are irrelevant to the conclusion. It's starting to feel like I'm merely re-stating the deduction algorithm, with unification etc... Another (perhaps not so) related idea could be to apply some form of > compressed sensing to the space of the arguments, which basically means > that if the arguments involve a dense region of some (continuous) space, > you may find a basis such that the projection is very sparse without loss > of information. > We cannot represent the space of arguments (or "objects" in logic) flatly, because the size of such a space would be the same as the real world... The space of logic formulas is a condensation (ie compression) of the real world. Logic is the compression scheme. It is highly non-linear and so far seems to be the only scheme capable of compressing the real world to a manageable size. This is the so-called "expressive power" of logic. Just throwing in more points, no solution yet... =) ------------------------------------------- AGI Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/21088071-f452e424 Modify Your Subscription: https://www.listbox.com/member/?member_id=21088071&id_secret=21088071-58d57657 Powered by Listbox: http://www.listbox.com
