sec-uocl-theory-learning
6.5 Learning generators, relations, and models
The basic UOCL target can be enriched once more. A learner may identify not only a category of encountered objects and arrows, but a compact algebraic theory explaining how that category is generated. This is stronger than memorizing a growing composition table and different from learning tangent structure. It asks for reusable syntax, equations, and functorial semantics.
A theory-bearing UOCL hypothesis is a tuple
where \(\mathbb S\) is a sketch, \(\mathfrak D\) is its preservation doctrine, \(\mathbb A\) is the presented algebraic theory, \(\mathcal C\) is a candidate world category, and \(M\) is a structure-preserving interpretation. Its queries may address the world, the presentation, the completed theory, the interpretation, or comparison maps among them.
The added levels separate four questions:
Which configurations and transformations have been encountered?
Which generators suffice to express them?
Which relations identify apparently different construction paths?
In which semantic worlds do those formal operations have valid models?
For the LEGO learner of Section 1.5, a newly observed tower can be assimilated by factoring its construction through known joining and tensor operations. Discovering that a separation reverses an attachment adds an equation, or perhaps a partial-inverse domain, to the sketch. That is theory-level accommodation.
Let \(\mathcal O_t\subseteq \mathcal C\) be the observation subcategory generated by the transcript at time \(t\). A theory-bearing hypothesis is generatively adequate at \(t\) when every declared object and arrow of \(\mathcal O_t\) is in the essential image of the interpretation of a well-typed expression in \(\mathbb A\), and every equality registered by the transcript is respected by \(M\). It is factorization-complete for a query doctrine when every queried future composite in the doctrine admits such an expression.
Generative adequacy is presentation-relative and deliberately one-sided. It says the learned operations can express what has been observed; it does not say the inferred relations are complete, the generators are minimal, or the physical interpretation is faithful. Those are separate probes. A compact but false theory may generate every training example by equating too many paths.
Rivest–Schapire diversity representations provide a finite-state instance of this principle (Section 5.4.1). Their compact object is generated by action-transformed tests modulo observational equivalence, even when the associated global state space is enormous. This suggests a useful UOCL design bias: learn a presentation of the probes and their compositional update laws before attempting to enumerate the objects they distinguish. Algebraic theories become operationally valuable here because they generate future experiments, not merely because they compress past observations.
Particle physics supplies a richer instance (Section 0.8). There the learned presentation is not an enumeration of physical states: particle types, interaction vertices, symmetries, and relations generate response predictions for entire families of collider probes. The experimental response functor in the declaration \(\mathfrak C_{\mathrm{coll}}\) is the semantic model against which the theory is assimilated or repaired. Proposition 6.5 then says how exact evidence narrows the observational quotient without pretending that a finite probe family determines a unique presentation.
Let \(F:\mathbb S\to \mathbb S'\) be a map of sketches that induces, naturally over every registered semantic category \(\mathcal C\), an equivalence
Then no UOCL query that factors through these model categories can distinguish \(\mathbb S\) from \(\mathbb S'\).
Each admitted query sends equivalent model objects and their morphisms to equivalent answers. Naturality makes the comparison stable under change of registered semantic category. Hence both presentations lie in the same observational class for every query factoring through model semantics.
This proposition supplies the right success criterion: learn a theory up to the equivalence visible in its model semantics, not a privileged string of generators. It also exposes a new nonidentifiability result. If two nonequivalent presentations have indistinguishable models under all available semantic probes, interaction cannot choose between them without enlarging the doctrine.
Prefer the smallest warranted generating theory to an inventory of observed composites, but treat minimality, completeness of relations, and semantic faithfulness as separate claims. A compact presentation is an explanation only relative to its model and query doctrine.