ora-0047

3.3 Universal query theories

Let \(\mathcal Q\) be an effective class of admissible questions. A query may ask for an equation between composites, a factorization, a limit or colimit, a lifting object, a Kan extension, or representability of a presheaf. Define

\[ \operatorname {Th}_{\mathcal Q}(\mathcal C) =\{ (q,\operatorname {Ans}_{\mathcal C}(q)):q\in \mathcal Q\} . \]

Write \(\mathcal C\simeq _{\mathcal Q}\mathcal D\) when their \(\mathcal Q\)-theories agree. This is the operational target of learning.

Definition 3.4 Categorical identification in the limit

An ORACLE learner identifies \(\mathcal C_\star \) from a presentation class \(\Pi \) relative to \(\mathcal Q\) if, for every presentation \((T_t)\in \Pi (\mathcal C_\star )\), there is an \(N\) such that

\[ t\geq N\quad \Longrightarrow \quad \widehat{\mathcal C}_t\simeq _{\mathcal Q}\mathcal C_\star . \]

If the hypotheses may continue to change while this condition holds, the identification is behaviorally correct. If the hypotheses stabilize coherently to one equivalence class, it is explanatory.

A weaker pointwise criterion permits the stabilization time to depend on the query. The distinction between one global \(N\) and the family \((N_q)\) is the categorical analogue of uniform versus pointwise predictive convergence.

The query language itself is typed by the prior. Let \(\mathcal Q_i\) denote questions internal to one core fragment and let \(\mathcal Q_{ij}\) contain questions about its overlap with another. For example, an object query may ask whether two appearances represent the same persisting object, while an object–agent query asks whether an observed path is compatible with a declared goal. A language query may ask whether a continuation belongs to the behavior of an admissible grammar coalgebra; a language–social query asks whether an utterance coherently updates shared social information.

This gives a hierarchy

\[ \mathcal Q_{\mathrm{frag}} =\bigcup _i\mathcal Q_i \ \subseteq \mathcal Q_{\mathrm{pair}} =\mathcal Q_{\mathrm{frag}}\cup \bigcup _{i,j}\mathcal Q_{ij} \ \subseteq \mathcal Q_{\mathrm{core}}. \]

Agreement on each isolated fragment need not imply agreement on \(\mathcal Q_{\mathrm{core}}\): two worlds can contain equivalent object, space, and language components yet attach words to objects or agents to goals differently. Categorical identification is therefore not just simultaneous identification of six marginal theories. It must identify their gluing.