I wrestled with similar issues regarding associativity and cognitive operations when writing this paper
https://arxiv.org/abs/2004.05269 -- see Section 6.2.1 where I show that if you have a set of mutually associative combinational operators, then you get a very nice subpattern hierarchy ... which is much messier to get without mutual associativity... An interesting question is whether: IF you have a set of mutually associative combinational operators, can you then isomorphically map the functions being combined into Hilbert space vectors to that the (associative) combinations map into vector operations? I have not thought about this before and am not sure if it can be made to work Quantum mechanics uses complex Hilbert spaces, and i did go thru some work to make an approximate isomorphism btw paraconsistent logic truth values and complex amplitudes, https://arxiv.org/abs/2101.07498 but that addresses a different part of the same problem, it seems -- Ben On Sun, Jan 31, 2021 at 2:45 AM YKY (Yan King Yin, ηζ―θ΄€) < [email protected]> wrote: > On 1/30/21, Ben Goertzel <[email protected]> wrote: > > Unless I remember wrong (which is possible), function application in a > > Scott domain is not associative, e.g. > > > > (f(g) ) (h) > > > > is not in general equal to > > > > f( g(h) ) > > > > However function composition is associative, and the standard products > > on vectors in Hilbert space are associative > > > > So it seems what you're doing may not be quite right, and you need to > > be looking at some sort of fairly general nonassociative algebras over > > Hilbert space instead ... or something... > > > > Or am I misunderstanding something? > > > Damn... you're right π > > I have no idea how to make one function "apply" to another, except by > function composition, so that was what I did. But I forgot about > associativity... > > I don't know if I should pursue along this any further.... it seems > computer-implementable but it's very complicated... and I currently > don't know how to make non-associative algebras.... > > Thanks for pointing out my mistake, it's helpful π > YKY -- Ben Goertzel, PhD http://goertzel.org βHe not busy being born is busy dying" -- Bob Dylan ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T54594b98b5b98f83-M5d4d8e1d1e09d9b49c528cc7 Delivery options: https://agi.topicbox.com/groups/agi/subscription
