Human-like automatic deductive reasoning has always been one of the most challenging open problems in the interdisciplinary field of mathematics and artificial intelligence. This paper is the third in a series of our works. We built a neural-symbolic system, named FGeo-DRL, to automatically perform human-like geometric deductive reasoning. The neural part is an AI agent based on deep reinforcement learning, capable of autonomously learning problem-solving methods from the feedback of a formalized environment, without the need for human supervision. It leverages a pre-trained natural language model to establish a policy network for theorem selection and employ Monte Carlo Tree Search for heuristic exploration. The symbolic part is a reinforcement learning environment based on geometry formalization theory and FormalGeo, which models geometric problem solving (GPS) as a Markov Decision Process (MDP). In the formal symbolic system, the symmetry of plane geometric transformations ensures the uniqueness of geometric problems when converted into states. Finally, the known conditions and objectives of the problem form the state space, while the set of theorems forms the action space. Leveraging FGeo-DRL, we have achieved readable and verifiable automated solutions to geometric problems. Experiments conducted on the formalgeo7k dataset have achieved a problem-solving success rate of 86.40%.

The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our previous work, we established a geometric formalized system known as FormalGeo. Moreover, we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the fact that FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (theorem predictor), which utilizes the language model to predict the theorem sequences for solving geometry problems. The encoder and decoder components in the transformer architecture naturally establish a mapping between the sequences and embedding vectors, exhibiting inherent symmetry. We compare the effectiveness of various transformer architectures, such as BART or T5, in theorem prediction, and implement pruning in the search process of FGPS, thereby improving its performance when solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solution times and search steps across problems of varying difficulty levels.

Automatic geometric problem-solving is an active and challenging subfield at the intersection of AI and mathematics, where geometric problem parsing plays a critical role. It involves converting geometric diagram and text into certain formal language. Due to the complexity of geometric shapes and the diversity of geometric relationships, geometric problem parsing demands that the parser exhibit cross-modal comprehension and reasoning capabilities. In this paper, we propose an enhanced geometric problem parsing method called FGeo-Parser, which converts problem diagrams and text into the formal language of the FormalGeo. It also supports reverse formalization to generate human-like solutions, reflecting the symmetry between parsing and generating. Specifically, diagram parser leverages the BLIP to generate the construction CDL and image CDL, while text parser employs the T5 to produce the text CDL and goal CDL where these neural networks are both based on a symmetric encoder–decoder architecture. With the assistance of a theorem predictor, these CDLs were automatically parsed and step-by-step reasoning was executed within FGPS. Finally, the reasoning process was input into a solution generator, which subsequently produced a human-like solution process. Additionally, we re-annotated problem diagrams and text based on the FormalGeo7K dataset. The formalization experiments on the new dataset achieved a match accuracy of 91.51% and a perfect accuracy of 56.47%, while the combination with the theorem predictor achieved a problem-solving accuracy of 63.45%.

Geometric problem solving (GPS) has always been a long-standing challenge in the fields of automated reasoning. Its problem representation and solution process embody rich symmetry. This paper is the second in a series of our works. Based on the Geometry Formalization Theory and the FormalGeo geometric formal system, we have developed the Formal Geometric Problem Solver (FGPS) in Python 3.10, which can serve as an interactive assistant or as an automated problem solver. FGPS is capable of executing geometric predicate logic and performing relational reasoning and algebraic computation, ultimately achieving readable, traceable, and verifiable automated solutions for geometric problems. We observed that symmetry phenomena exist at various levels within FGPS and utilized these symmetries to further refine the system’s design. FGPS employs symbols to represent geometric shapes and transforms various geometric patterns into a set of symbolic operation rules. This maintains symmetry in basic transformations, shape constructions, and the application of theorems. Moreover, we also have annotated the formalgeo7k dataset, which contains 6981 geometry problems with detailed formal language descriptions and solutions. Experiments on formalgeo7k validate the correctness and utility of the FGPS. The forward search method with random strategy achieved a 39.71% problem-solving success rate.

Plane geometry problem solving has been a long-term challenge in mathematical reasoning and symbolic artificial intelligence. With the continued advancement of automated methods, the need for large-scale datasets and rigorous evaluation frameworks has become increasingly critical for benchmarking and guiding system development. However, existing resources often lack sufficient scale, systematic difficulty modeling, and quantifiable, process-based evaluation metrics. To address these limitations, we propose FGeo-Eval, a comprehensive evaluation system for plane geometry problem solving, and introduce the FormalGeo30K dataset, an extended dataset derived from FormalGeo7K. The evaluation system includes a problem completion rate metric PCR to assess partial progress, theorem weight computation to quantify knowledge importance, and a difficulty coefficient based on reasoning complexity. By analyzing problem structures and solution dependencies, this system enables fine-grained difficulty stratification and objective performance measurement. Concurrently, FormalGeo30K expands the dataset to 30,540 formally annotated problems, supporting more robust model training and evaluation. Experimental results demonstrate that the proposed metrics effectively evaluate problem difficulty and assess solver capabilities. With the augmented dataset, the average success rate across all difficulty levels for the FGeo-HyperGNet model increases from 77.43% to 85.01%, while the average PCR increases from 88.57% to 91.79%. These contributions provide essential infrastructure for advancing plane geometry reasoning systems, offering standardized benchmarks for model development and guiding optimization of geometry-solving models.

Deciphering the structural hierarchy of musical compositions is indispensable for a range of music analysis applications, encompassing feature extraction, data compression, interpretation, and visualization. In this paper, we introduce a quantitative model grounded in fractal theory to evaluate the significance of individual notes within a musical piece. To analyze the quantized note importance, we adopt a rhythm-based approach and propose a series of detection operators informed by fundamental rhythmic combinations. Employing the Mamba model, we carry out recursive detection operations that offer a hierarchic understanding of musical structures. By organizing the composition into a tree data structure, we achieve an ordered layer traversal that highlights the music piece’s multi-dimensional features. Musical compositions often exhibit intrinsic symmetry in their temporal organization, manifested through repetition, variation, and self-similar patterns across scales. Among these symmetry properties, fractality stands out as a prominent characteristic, reflecting recursive structures both rhythmically and melodically. Our model effectively captures this property, providing insights into the fractal-like regularities within music. It also proves effective in musical phrase boundary detection tasks, enhancing the clarity and visualization of musical information. The findings illustrate the model’s potential to advance the quantitative analysis of music hierarchy, promoting novel methodologies in musicological research.