Define compositional depth.

Compositional depth refers to the number of sequential operations or moves required to reach a full solution in a given puzzle environment. In the context of the paper, the compositional depth scales with the problem size across different puzzle environments. For example, the Tower of Hanoi puzzle exhibits exponential growth in compositional depth, while the Checker Jumping puzzle displays quadratic scaling. Other puzzles, like River Crossing and Blocks World, show more moderate, near-linear growth with the number of elements involved in the problem. This characterization of compositional depth is crucial for evaluating how language reasoning models handle various sequential reasoning challenges and whether their accuracy correlates with the compositional depth needed to solve the puzzle^[1].