Simplifying complex symbolic expressions is hard because the search space of possible transformations is enormous and the reward is sparse — most rewriting steps leave the expression equally complex. Reinforcement learning struggles because the signal that a simplification succeeded arrives only at the end of a long trajectory of manipulations. Supervised learning struggles because there is no labeled dataset of (complex expression, simple form) pairs with the intermediate steps shown.
Shih (arXiv:2603.11164, 2026) solves the data problem by reversing the direction. Start with simple expressions. Apply random complicating transformations — expand, substitute, rearrange. Record each step. The result is a dataset of scrambling trajectories: start simple, end complex, every intermediate state labeled. Now reverse these trajectories. Each one becomes an oracle path from complex to simple, with every step visible.
The training data is free because complication is easy. Expanding a product, substituting a variable, applying an identity in the wrong direction — these are all cheap operations that produce arbitrarily complex expressions from simple ones. The hard direction (simplification) is the reverse of the easy direction (complication), and the easy direction generates its own supervision.
Applied to dilogarithm reduction and spinor-helicity amplitude simplification in particle physics — expressions with over 200 terms — the trained policy achieves 100% full simplification on 5-point gluon tree-level amplitudes, outperforming both reinforcement learning and regression-based methods.
The structural point: the asymmetry between complication and simplification is what makes the problem solvable. If simplification were as easy as complication, there would be no problem. If complication were as hard as simplification, there would be no training data. The difficulty gradient between the two directions IS the resource. The method works precisely because the problem is hard in one direction and easy in the other — and the easy direction leaves a trail that teaches the hard direction.