Reason with procedures and search.

This is the same reference used in Technical mode. Expand it whenever you need the exact numbers, assumptions or further explanation.

A remembered answer and a reusable procedure

A lookup table can return a multiplication result it has seen before. An algorithm applies a sequence of operations to new inputs. In the source example, 18 boxes × 24 apples − 175 sold becomes 432 − 175 = 257 apples. Multiplication and subtraction supply reusable rules; storing the answer 257 does not by itself supply those rules.

A neural network can learn computations that behave like procedures, but gradient training does not guarantee a robust algorithm. Test changes in input size, names, composition, and required depth. Success on familiar patterns alone does not establish systematic generalization.

Written intermediate steps can let an autoregressive model reuse prior calculations. They can also contain errors. More words are not an independent check of arithmetic. In an exactly defined task, an exact calculator or simulator can provide a stronger check.

A local clue is not a complete plan

A heuristic estimates what looks promising. Our direct rule chooses a legal neighboring square with the smallest Manhattan distance: horizontal distance plus vertical distance to the goal. It considers four local moves, but does not explore their later consequences. Obstacles can make a temporary move away from the goal necessary.

The policy detects a repeated square and stops so that the demonstration cannot loop forever. That memory detects the failure; it does not give the policy a strategy for escaping it. The rule is hand-written, not a trained policy network.

Search keeps alternatives

Breadth-first search keeps a queue of discovered states. It expands a state by checking its four possible actions, records how newly reached states were found, and continues with the earliest queued state. The parent records reconstruct a path when the goal is reached.

With equal move costs and correct deterministic rules, breadth-first search finds a shortest path when given enough work. It may explore many states that are not on the final path. That extra work is the cost of keeping alternatives instead of committing to a local guess.

A queue emptied without reaching the goal rules out a route in this finite graph. A query-budget cutoff does not: unexplored alternatives may remain. In our lab, the budget counts transition queries, not all bookkeeping or wall-clock time. An independent replay then checks every proposed move and the final goal.

Where neural estimates can fit

A policy network can prioritize promising actions, and a value network can estimate future success. Search can use those estimates while an exact simulator handles legal transitions. A verifier checks whether the final proposal satisfies the task. These roles complement one another; an estimate is not an exact rule.

A fixed-depth network performs a fixed sequence of transformations per pass. Additional generated tokens allow additional passes. A recurrent model instead repeatedly updates a hidden state. Both can add sequential computation, but the updates must be useful. We inspect repeated state updates in the refinement lesson.

Sort the jobs in chess

The source exercise asks you to separate pattern recognition, search, exact execution, and verification in move selection. Try assigning each of these jobs before opening the example.