Causal-R: A Causal-Reasoning Geometry Problem Solver for Optimized Solution Exploration

The task of geometry problem solving has been a long-standing focus in the automated mathematics community and draws growing attention due to its complexity for both symbolic and neural models. Although prior studies have explored various effective approaches for enhancing problem solving performances, two fundamental challenges remain unaddressed, which are essential to the application in practical scenarios. First, the multi-step reasoning gap between the initial geometric conditions and ultimate problem goal leads to a great search space for solution exploration. Second, obtaining multiple interpretable and shorter solutions remains an open problem. In this work, we introduce the Causal-Reasoning Geometry Problem Solver to overcome these challenges. Specifically, the Causal Graph Reasoning theory is proposed to perform symbolic reasoning before problem solving. Several causal graphs are constructed according to predefined rule base, where each graph is composed of primitive nodes, causal edges and prerequisite edges. By applying causal graph deduction from initial conditions, the reachability status of nodes are iteratively conveyed by causal edges until reaching the target nodes, representing feasible causal deduction paths. In this way, the search space of solutions is compressed from the beginning, the end and intermediate reasoning paths, while ensuring the interpretability and variety of solutions. To achieve this, we further propose Forward Matrix Deduction which transforms the causal graphs into matrices and vectors, and applies matrix operations to update the status value of reachable nodes in iterations. Finally, multiple solutions can be generated by tracing back from the target nodes after validation. Experiments demonstrate the effectiveness of our method to obtain multiple shorter and interpretable solutions. Code is available after acceptance.