INDUCTION: Finite-Structure Concept Synthesis in First-Order Logic

Induction is the search for a general rule that explains observations. We study logical induction in finite relational worlds: each problem gives small structures over a fixed vocabulary, labels objects belonging to an unknown unary concept, and asks for one first-order formula $\varphi$(x) that accounts for those labels across worlds. Finite domains make formulas mechanically checkable by exact evaluation and SMT. We introduce INDUCTION, a benchmark for finite-structure concept synthesis with three regimes: FULLOBS (full observation), where all facts are observed; CI (contrastive induction), where YES/NO worlds require discriminative hypotheses; and EC (existential completion), where validity is defined by world-local completion of unknown facts. We evaluate frontier language models, include symbolic synthesis baselines, and score both validity and formula size. Prompted models show real but incomplete capability, with sharp difficulty gradients and hard structural families. Held-out evaluation shows that compact formulas generalize far better than bloated ones; parsimony separates concept recovery from finite-world fit.