A theoretical review on solving algebra problems
Xinguo Yu, Weina Cheng, Chuanzhi Yang, Ting Zhang · Expert Systems with Applications · 2025
Solving algebra problems (APs) continues to attract significant research interest as evidenced by the large number of algorithms and theories proposed over the past decade. Despite these important research contributions, however, the body of work remains incomplete in terms of theoretical justification and scope. The current contribution intends to fill the gap by developing a review framework that aims to lay a theoretical base, create an evaluation scheme, and extend the scope of the investigation. This paper first develops the State Transform Theory (STT), which emphasizes that the problem-solving algorithms are structured according to states and transforms unlike the understanding that underlies traditional surveys which merely emphasize the progress of transforms. The STT, thus, lays the theoretical basis for a new framework for reviewing algorithms. This new construct accommodates the relation-centric algorithms for solving both word and diagrammatic algebra problems. The latter not only highlights the necessity of introducing new states but also allows revelation of contributions of individual algorithms – obscured in prior reviews without this approach. A review of AP solving algorithms (2014 to date) is subsequently enhanced by applying the STT specifically designed to analyze individual and collective algorithms for states and transforms. Furthermore, the State Transform Analysis (STA) is a core function in the identification of progress in terms of states and transforms. Thirdly, the Perspective Confusion Comparison (PCC) is developed to extend the application of STA to add capabilities for systematic and individual evaluation of transforms, algorithms, and approaches. This is a new evaluation method of being different from other methods that can do more in-depth evaluation for decomposed algorithms for solving problems with multiple types of inputs. Finally, this work identifies several research directions by extracting benefit from theoretical reviews. This work significantly contributes to the advancement of AP-solving by providing the mechanism for identifying and understanding the key contributors to building high-performance problem-solving algorithms at the three levels of transform, algorithm, and approach.