tab-dial-url-environments

15.13.4 The synthetic laboratory behind the ladder

The word environment hides an important change across these five rungs. DIAL-URL–0 and DIAL-URL–1 use small sequential simulators with states, actions, rewards, observations, transitions, and holding times. DIAL-URL–2 through DIAL-URL–4 instead use controlled counter-witness oracles. They do not render a physical scene or expose raw state trajectories. Given a typed experimental request, an oracle returns a noisy binary indication of whether a structural phenomenon was observed. DIAL-URL–2 through –4 therefore isolate theory revision from the difficult upstream problem of estimating a reliable witness from raw data.

It is useful to write a synthetic laboratory as

\[ \mathcal L=(\mathcal T_\star ,\mathcal Q, \{ P(Y_q\mid \mathcal T_\star ):q\in \mathcal Q\} ), \]

where \(\mathcal T_\star \) is the evaluator’s hidden theory, \(\mathcal Q\) is the registered intervention or probe language, and \(Y_q\) is the observable outcome of applying probe \(q\). The learner knows the candidate language and likelihood model but not \(\mathcal T_\star \). It chooses probes, updates a posterior over theories, and either admits a typed extension or abstains. In the first two rungs, the distribution of \(Y_q\) is estimated from exact operators or raw episodes. In DIAL-URL–2 through –4 it is specified directly, like a calibrated instrument in a deliberately simplified physics experiment.

Rung	Hidden experimental world	Evidence visible to the learner	Withheld structure
URL–0	Exact two-state controlled coalgebra	Rewards, observation map, transition distributions, duration support	Minimal behavior type
URL–1	Stochastic two-state episodic simulator	Finite records \((o,a,r,\tau ,o')\)	Whether observation history or holding time is needed
URL–2	Four-hypothesis intervention oracle	Noisy history and duration counter-witness bits	Best experiment and supported registered type
URL–3	Recursive behavior-grammar oracle	Noisy lag, time, and mixture witness bits	Smallest generated functor expression
URL–4	Expression-plus-law oracle	Marginal witnesses and ordered compound witnesses	Constructor pair and orientation of a comparison cell

Table 15.5. The evidential boundary of the DIAL-URL ladder. Only URL–1 begins from sampled sequential records; URL–2 through URL–4 deliberately factor out witness estimation.