> > > > Another possibility is to enforce "type theory" on the space of > operators. This > would just be a strongly-typed programming language, with all functions > and operators > embedded in Hilbert space ;) >
I think this is a promising approach in terms of consilience with the actual AGI design/dev work we're doing now w/ OpenCog Hyperon... We are designing a new gradually-typed higher order functional programming language, w/ dependent and probabilistic types. (Atomese 2) ... and it was in this context that I started looking at the relation btw paraconsistent, probabilistic and quantum logics recently... > > *But* we may be looking too far. Indeed, certain classes of non-linear > functions in > a Hilbert space can already constitute a Turing-universal system. Such > elements > would not have the ability of *self-application*, but this is not > absolutely required for > Turing-universality. > > Even if we want self-application, we can make a function act on another > function's > *Fourier spectrum*. But here the domain is somewhat different: if the > source function's > domain is *bounded and continuous*, then its Fourier domain would be > *unbounded > and* > *discrete*; and vice versa. We need a map between these 2 kinds of > domains, which > can be achieved by something like a stereographic projection or a periodic > map. > > > 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 > > If associative, then yes, it's just the matrix representation of the > algebra, and the composition operation would be matrix multiplication. > Hmm, well if we have a group G then if we set V equal to the space of complex-valued functions on G, we can make a homomorphism f from G into GL(V) via f(g) F(x) = F(xg) for F in V and x,g in G I guess that works quite generally .. AGI can be cast as a reinforcement learning problem, obeying the Hamilton- > Jacobi-Bellman equation. > Interesting... yes in fact the paper I'm in the middle of working on right now (slowly in evenings and weekends) represents all the OpenCog cognitive algorithms as either greedy algorithms or approximate stochastic dynamic programming (on appropriately defined abstract spaces) One then uses Galois connections together with this representation to map all the cognitive algorithms into metagraph chronomorphisms implemented continuation passing style... which is key to designing the guts of the Atomese 2 interpreter > The HJB equation can be transformed into the > Schrodinger equation via the ansatz Ψ = exp(-iℏS) where S is the utility > function in reinforcement learning. Which means that Ψ is actually just > the > utility function in disguise (exponentiated), and Ψ is the function that > tells > an AI how to act in the environment. The space X here would no longer be > 3D physical space, but the cognitive space of thoughts. > > If reinforcement learning is made stochastic, Ψ may have a probabilistic > interpretation.... > Oh wow! Yes that is a very pertinent observation. I recall reading the relation btw the dynamic programming functional equation and the Schrodinger equation eons ago but you have stimulated those dormant neural assemblies extremely pertinently, thanks YKY !! This actually connects my in-progress paper on dynamic programming formulation of OpenCog algos with my recent paper on mapping paraconsistent into quantum logic It suggests that if we pass from paraconsistent/probabilistic to quantum probabilities, this likely has the effect of mapping the standard OC algos into corresponding quantum algos, wherein maximal utility classical solutions map into stationary-action solutions of the equations corresponding to the quantum algos... Which does not help much w/ my current Hyperon work but is pretty cool... ben ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T54594b98b5b98f83-Mabfa4e8df0dbdf59508129ce Delivery options: https://agi.topicbox.com/groups/agi/subscription
