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4Vanessa Kosoy4dIn the past I considered the learning-theoretic approach to AI theory
[https://www.alignmentforum.org/posts/5bd75cc58225bf0670375575/the-learning-theoretic-ai-alignment-research-agenda]
as somewhat opposed to the formal logic approach popular in MIRI
[https://www.alignmentforum.org/posts/3qXE6fK47JhSfkpnB/do-sufficiently-advanced-agents-use-logic]
(see also discussion
[https://www.alignmentforum.org/posts/3qXE6fK47JhSfkpnB/do-sufficiently-advanced-agents-use-logic#fEKc88NbDWZavkW9o]
):
* Learning theory starts from formulating natural desiderata for agents,
whereas "logic-AI" usually starts from postulating a logic-based model of the
agent ad hoc.
* Learning theory naturally allows analyzing computational complexity whereas
logic-AI often uses models that are either clearly intractable or even
clearly incomputable from the onset.
* Learning theory focuses on objects that are observable or
finite/constructive, whereas logic-AI often considers objects that
unobservable, infinite and unconstructive (which I consider to be a
philosophical error).
* Learning theory emphasizes induction whereas logic-AI emphasizes deduction.
However, recently I noticed that quasi-Bayesian reinforcement learning
[https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v]
and Turing reinforcement learning
[https://www.alignmentforum.org/posts/3qXE6fK47JhSfkpnB/do-sufficiently-advanced-agents-use-logic#fEKc88NbDWZavkW9o]
have very suggestive parallels to logic-AI. TRL agents have beliefs about
computations they can run on the envelope: these are essentially beliefs about
mathematical facts (but, we only consider computable facts and computational
complexity plays some role there). QBRL agents reason in terms of hypotheses
that have logical relationships between them: the order on functions corresponds
to implication, taking the minimum of two functions corresponds to logical
"and", taking the concave hull of two func
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37dTheorem: Fuzzy beliefs (as in
https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v
[https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v]
) form a continuous DCPO. (At least I'm pretty sure this is true. I've only
given proof sketches so far)
The relevant definitions:
A fuzzy belief over a set X is a concave function ϕ:ΔX→[0,1] such that sup(ϕ)=1
(where ΔX is the space of probability distributions on X). Fuzzy beliefs are
partially ordered by ϕ≤ψ⟺∀μ∈ΔX:ϕ(μ)≥ψ(μ) . The inequalities reverse because we
want to think of "more specific"/"less fuzzy" beliefs as "greater", and these
are the functions with lower values; the most specific/least fuzzy beliefs are
ordinary probability distributions, which are represented as the concave hull of
the function assigning 1 to that probability distribution and 0 to all others;
these should be the maximal fuzzy beliefs. Note that, because of the
order-reversal, the supremum of a set of functions refers to their pointwise
infimum.
A DCPO (directed-complete partial order) is a partial order in which every
directed subset has a supremum.
In a DCPO, define x<<y to mean that for every directed set D with supD≥y, ∃d∈D
such that d≥x. A DCPO is continuous if for every y , y=sup{x∣x<<y}.
Lemma: Fuzzy beliefs are a DCPO.
Proof sketch: Given a directed set D , (supD)(μ)=min{d(μ)∣d∈D} is convex, and
{μ∣(supD)(μ)=1}=⋂d∈D{μ∣d(μ)=1}. Each of the sets in that intersection are
non-empty, hence so are finite intersections of them since D is directed, and
hence so is the whole intersection since ΔX is compact.
Lemma: ϕ<<ψ iff {μ∣ψ(μ)=1} is contained in the interior of {μ∣ϕ(μ)=1} and for
every μ such that ψ(μ)≠1, ϕ(μ)>ψ(μ).
Proof sketch: If supD≥ψ, then ⋂d∈D{μ∣d(μ)=1}⊆{μ∣ψ(μ)=1} , so by compactness of
ΔX and directedness of D, there should be d∈D such that
{μ∣d(μ)=1}⊆int({μ∣ϕ(μ)=1}). Similarly, for each μ such that ψ(μ)≠1, there should
be dμ∈D s