FactorLibrary: From Polynomials to Circuits via Recursive Subgoals

Finding minimal arithmetic circuits for polynomials over finite fields is a combinatorially hard problem central to algebraic complexity theory. We formulate it as a reinforcement learning problem in two directions, bottom-up and top-down. To address the challenge of a fast-growing combinatorial search space, we introduce FactorLibrary, which stores factorizable subexpressions that serve as reusable subgoals across training episodes. We trained a bottom-up agent with Gumbel-PPO-MCTS and two top-down agents with PPO+MCTS and SAC. The PPO+MCTS top-down agent exhibited the most stable performance, finding certified optimal circuits up to complexity $8$ with a success rate of $91.8\%$.

Average Attention Transformers and Arithmetic Circuits

We analyse the computational power of transformer encoders as sequence-to-sequence functions on vectors. We show that average hard attention can be used to simulate arithmetic circuits if they are given as an input to an encoder. The circuit families that can be simulated this way have constant depth while using unbounded addition, binary multiplication and sign gates. The transformers we use have arithmetic circuits instead of feed-forward networks. With typical average attention the functions they compute are also computed by the same class of circuit families. Our results hold for transformers over the reals, rationals and any ring in between the two.