Agent Systems: Exploration and Discovery

Problem

A single greedy line of reasoning can miss useful cases, counterexamples, or alternative proof strategies. Open-ended exploration can also spend resources without producing new information.

Intent and structure

seed → branch or mutate → test → retain evidence → select or stop

The candidate representation, test procedure, scoring rule, and termination condition must be explicit. Exploration is a search procedure; it is not itself proof.

Mathematical uses

Use the pattern to generate conjectures, search examples, compare proof strategies, or explore neighboring definitions. Candidate records should retain the assumptions tested, the computation or argument used, and the reason for retaining or discarding the candidate.

Forces and failure modes

Breadth improves discovery but creates combinatorial growth, duplicate candidates, and selection bias. Beam limits, diversity criteria, counterexample tests, budgets, and human review keep the process meaningful.

Design question

What new evidence justifies expanding the search, and what evidence justifies stopping it?

Figure

Figure: candidate branches are tested, recorded, and selected under a budget and stopping rule.