Symbol and Rule Discovery in Continuous Control Through Progressive Inductive Abductive Learning

We study the discovery of symbols and rules from continuous-control trajectories, preserving characteristics of motion dynamics while supporting discrete reasoning. Jointly learning continuous factors, value boundaries, and temporal rules creates a large and coupled optimization problem. We address this problem by DynaAbd, a multi-stage progressive inductive abductive learning (inductive ABL) approach. In the first two stages, contrastive spectral initialization and one-step dynamics modeling organize continuous motion into cyclic and dependent factors, capturing initial features in motion dynamics. In the final stage, we introduce an inductive ABL strategy for fine-grained dynamics modeling. Logical induction and abduction refine the symbol discovery and build the corresponding rules based on multi-step logical consistency. Alternating gradient optimization fine-tunes the symbol value boundaries. Our method is experimentally evaluated in polar motion systems and MuJoCo locomotion tasks. The results show that DynaAbd is able to discover latent driving and dependent symbols hidden inside behavior dynamics, meanwhile, discovering the symbolic rules generalized across environments to accelerate policy learning. Our code and data will be open-sourced upon formal release.