Probabilistic Equivalence Verification Approach for Automatic Mathematical Solution Assessment

Automatic mathematical solution assessment checks the equivalence of mathematical expressions in the user answer and standard solution. It is a challenging problem as the semantics of mathematical expressions are highly symbolic and equivalent mathematical expressions can be expressed in different forms. In this paper, we propose an effective Probabilistic Equivalence Verification (PEV) approach for automatic mathematical solution assessment. The proposed PEV approach is a randomized method based on the probabilistic numerical equivalence testing of two mathematical expressions. It can avoid false negative errors completely while guaranteeing a small probability of false positive errors to occur. The performance results have shown that the proposed PEV approach has outperformed other popular techniques in Computer Algebra Systems such as Maple and Mathematica.