15.13.1 Assimilation and accommodation over behavior functors

15.13.1 Assimilation and accommodation over behavior functors

Coalgebraic assimilation. Fix \(F\). The learner estimates \(\xi \), observations, rewards, policies, value semantics, or a behaviorally minimal quotient. Bellman evaluation can then be studied through its algebraic fixed point, while metric coinduction provides a quantitative proof principle for contractive realizations [ Kozen and Ruozzi , 2009 ] . This direction may encompass difficult statistical learning, but it does not change the declared type of system.

Coalgebraic proto-accommodation. Persistent failure may instead justify

The repair could add an observation channel, holding time, nondeterministic branching, termination effect, continuous measure structure, or a law governing how two effects distribute over one another. Sokolova’s grammar of probabilistic behavior functors supplies a principled initial language for such changes: identity and constant functors, action-indexed exponentials, powerset and distribution functors, and products, coproducts, and compositions constructed from them [ Sokolova , 2011 ] . Natural transformations between behavior functors provide typed system translations. A distributive law may be required when the repair combines effects rather than merely adjoining a component. The resulting proposal becomes accommodation only after its realization, transport, and consequences pass the registered admission contract.

The DIAL directions are therefore

Their mixed obstruction asks whether learning and theory transport agree. If \(\eta :F\Rightarrow F^{+}\) is a candidate translation, compare learning an \(F\)-coalgebra and then transporting it with transporting the declaration first and learning in the enlarged class. Here transport of \(\xi :X\to FX\) along an actual natural transformation means the \(F^{+}\)-coalgebra \(\eta _X\circ \xi :X\to F^{+}X\). A persistent discrepancy in held-out behavior localizes a mixed defect, but it does not by itself show that \(F\) is inadequate: estimation error, optimization failure, or a bad translation may also break the comparison. Only after those alternatives are controlled does the defect support a candidate change of behavioral type.

This comparison is not automatically a Lie bracket. Parameters of coalgebras inside one smooth chart may admit infinitesimal vector fields and mixed brackets. Changing the functor itself may be a finite declaration change. DIAL-URL therefore uses infinitesimal diagnosis within a chart, but finite sketch extension, evidence transport, and typed admission when it crosses a model-class boundary.

. The DIAL-URL objective is not only to learn a policy or an environment model, but to determine when the behavioral type used to formulate those objects must itself be revised.