Sometimes one reading is not enough, and each further test uses time or resources. We need a plan that works across possible answers, not just a promising next question. Our question plan changes as answers arrive, while the card's state stays fixed. We are learning about its Being; Becoming would change that state.

Use the four cards again. The goal for the next worked example is to identify the complete card, using exact tests while it stays fixed. A reads colour, B shape, and the library also supplies the question “is this exactly that card?” for each card. We label this four-card example C4. Our earlier lamp machine is labeled M0 and has a different set of tests. Its test set has not changed.

A decision tree is a plan for asking questions. One state is unknown but fixed. Each answer retains the states consistent with it; an informative answer makes that set smaller. Our four states are zero-zero, zero-one, one-zero and one-one. A reads the first digit; B reads the second. The library also contains a test for each individual card: is the state exactly this one? The current support is the set of states consistent with the answers so far. Ask A at the root. Answer zero leaves the first pair, and answer one leaves the second. Both branches belong in the plan, even though an actual run follows only one. Ask B on each remaining pair. On the A-zero branch, B zero identifies zero-zero, and B one identifies zero-one. On the A-one branch, B zero identifies one-zero, and B one identifies one-one. Every branch now ends in a singleton, a support containing exactly one state. That leaf identifies the state.

The depth of a leaf is the number of questions on its path from the root. Here every leaf has depth two, so the worst-case depth is two. The tree has four leaves and three internal question nodes. We do not ask all three questions on one run. We follow one path, asking two. Distinguish the size of the entire plan from the number of questions used on a path.

A plan can choose the next question from the answer already received. After A equals zero, ask whether the state is zero-zero. After A equals one, ask whether it is one-zero. Both branches still identify the state in two questions. That is adaptive choice. We specify a rule for each possible answer history, rather than promising to ask the same full sequence everywhere. A test may become uninformative on a smaller support. On the pair zero-zero and zero-one, A is constant. Asking it again produces no split.

Available means a test belongs to the supplied library. Admissible means it also satisfies the rules at the current support and stage. For a simple separate rule, allow at most one yes-or-no question in total. A is available and legal as the first question. But no one-question plan can distinguish four states, because it has at most two answer leaves. Feasible means there exists a complete admissible plan meeting the goal on every possible branch. A legal next move alone does not prove feasibility.

A specialist research branch studies restrictions that depend on the remaining possibilities. That is optional here. Our main route keeps the distinction between a permitted next question and a complete plan meeting its goal.

Restore the unrestricted library. The two-question plan is feasible. A plan that checks individual states one at a time is also feasible, but its worst branch uses three questions. To say optimal, specify a cost. For worst-case question count, depth two is optimal here. One binary question has at most two leaves, so it cannot identify four states. We have exhibited a two-question solution, meeting that lower bound. Expected depth would additionally require a probability distribution. Structural constraints require their own admissibility check. Keep the library, rules, goal and cost visible whenever comparing strategies.

The card tree worked because those queries were supplied. Return to M0 with only exact READ while the state stays fixed. Starting from u or v, can repeating READ reveal which predictive group applies?

[SILENT HOLD:5SECONDS;NOT SPOKEN]

No. Both give zero on every such reading. If the goal is only to read the current lamp, zero already answers it. If the goal concerns the future after PULSE, more is needed. The next lesson allows actions and checks what they change, including the state being identified.