On 1/29/21, Matt Mahoney <[email protected]> wrote: > Well I disagree that set D^D can be isomorphic to D violating Cantor's > theorem by restricting to continuous functions. Each point in the > continuous subset of R^R real valued functions still can encode infinite > information while meeting the definition of continuity.
Continuity here should mean "Scott-continuous" and I'm sure the math works because it is well-established. But I did not look into the details and my description may be slightly inaccurate. Perhaps this is a more accurate description: “Scott's models are directed-complete partial orders (DCPOs), which form a cartesian closed category, and the exponential A^B of two DCPOs does not consist of all functions from B to A, but only those which preserve directed joins." > More importantly, how does logic in Hilbert space solve any existing > problems? How does it get us closer to AGI? My proposed model has some important properties: 1. uses deep learning to learn logic formulas 2. allows logic formulas to act on other logic formulas 3. the map from logic formulas to logic formulas is "continuous" in a precise sense None other AGI systems as far as I know, have all these properties. * If they are based on symbolic logic, they usually don't use deep learning to learn logic rules. * Even if they do, the domain of symbolic logic formulas is discrete, causing the mapping to be discontinuous, which may have adverse effects on machine learning. Also, such mappings are purely based on the "syntax" of logic formulas, but my mapping actually acts on semantic models. The continuity notion means that we can endow our representation space with a topology structure and even metric structure (ie, distances). Traditional machine learning theory relies on these ideas. Like I said in the conclusion of the paper, I'm not strongly saying that this route must be taken. I'm just saying: "If you want these nice properties, then here is a theory that provides one such model".... YKY ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T54594b98b5b98f83-M5e3a0b0c80d0b33d18f83050 Delivery options: https://agi.topicbox.com/groups/agi/subscription
