It's somewhat off to the side of current AGI concerns, but following on YKY's last message, the question of how to frame the optimal-control interpretation of the Schrodinger equation is interesting...
The result that minimizing expected action in QM is isomorphic to minimizing expected reward in dynamic programming is clear... At the classical-stochastic-dynamics level, one views the multiverse as being operated by some entity that is choosing histories (multiverse-evolution-paths) with probability proportional to their expected action (where expected action serves the role of expected reward). Then at the quantum level, one views the multiverse as being operated by some entity that is assigning histories amplitudes proportional to their expected action (where expected action serves the role of expected reward). The amplitudes are summed and the stationary-action histories include those that in the "classical limit" give you the maximal-expected-action paths. >From a quantum computing view, if one has a classical algorithm that operates according to dynamic programming with a particular utility function U, one would map it into a quantum algorithm designed so that expected action maximization corresponds to expected U maximization. Put together with the paper I'm currently writing on dynamic programming formulation of OpenCog algos, this would seem to give a direct-ish route to QuantumOpenCog, on a math/algo level I mean... The conceptual interpretation of the underlying stochasticity is interesting from a foundations-of-QM perspective. The paper https://www.nature.com/articles/s41598-019-56357-3 argues " The model presented in this paper suggests that the test particle is moving under the influence of an external random spacetime force. This random movement of the particle induces the transition probability distribution. This means that quantum mechanics can be understood as a statistical theory. In literature there are some conjectures what could be the reason for this random force. ... Therefore, one could make the conjecture that quantum mechanics or quantum field theory is only a phenomenological theory and the reason for the statistical nature lies within the stochastic nature of the spacetime itself " Of course the "stochastic nature of the spacetime itself" that they invoke here may not be the bottom of the rabbit hole though. E.g. could the stochastic nature of spacetime be related to the way spacetime is constructed via collective perception/action of the various consciousnesses polyphonically participating in it? On Mon, Feb 15, 2021 at 1:09 PM Ben Goertzel <[email protected]> wrote: >> >> >> >> 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 > -- Ben Goertzel, PhD http://goertzel.org “He not busy being born is busy dying" -- Bob Dylan ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T54594b98b5b98f83-Mfb7503d8bbca92f610fc37ca Delivery options: https://agi.topicbox.com/groups/agi/subscription
