Research on Automatic Problem-Solving Technology of Olympic Mathematics in Primary Schools Based on AORBCO Model

Considering the importance of semantic understanding in the process of problem analysis, the Ego individual needs to truly clarify the meaning of the natural language received, which refers to the knowledge described by humans using natural language, thereby updating its own cognition based on this understanding. The degree of the Ego's understanding of natural language relies on the prior knowledge possessed by the current Ego, similar to the process of human enlightenment learning. However, regardless of whether the Ego's understanding is correct, the form of the understanding's representation is expressed in the form of knowledge described using descriptive language. The Ego first processes the received natural language text sentence by sentence, based on the prior knowledge it possesses, splits the sentences, and understands the sentences word by word using the concepts of nouns. The Ego then comprehends the semantics of the entire sentence and finally uses the AORBCO model to represent the semantic information understood by the Ego [1].

The methods of reasoning are the current automatic problem-solving software for elementary geometry. Due to the limitation of data structure and reasoning methods, most of them are based on one-way application, and most of them are forward deduction. The advantage of forward deduction method is that a lot of useful information can be inferred from the known information whether the conclusion can be deduced or not, which is of great significance to inspire students to think; The disadvantage is that the efficiency is not ideal for the reasoning of topics with more known information [2]. The backward inference method is suitable for the situation that there are a lot of known information and there are few goals to prove. Its main advantages are that it is not necessary to use information that has nothing to do with the goal, and it is beneficial to provide explanations to users. The disadvantage is that the selection of sub-goals is blind and affects efficiency. The advantages and disadvantages of reasoning with only one method are obvious, so consider realizing a combined system to make it have the advantages of both forward and backward reasoning systems, which is a two-way reasoning system. Two-way reasoning overcomes the shortcomings of weak purpose of forward push and blind choice of target of backward push, and at the same time combines the advantages of both. The realization technology is relatively more complicated than the single system, and some difficult problems mainly lie in the judgment of the joint point of forward and backward push, the proportion distribution of forward and backward push and so on [3].

The basic idea of realizing two-way reasoning system is: forward reasoning according to known facts but not all the way to the target (otherwise it is a forward reasoning system); At the same time, backward reasoning from the goal is not always until the known facts are reached (otherwise it is a backward reasoning system) [4]. Combining these two kinds of reasoning in some intermediate link between known facts and goals is the condition for the successful termination of two-way reasoning.

At present, there are 77 high-level strategies, which involve the following issues: remainder, sum multiple, sum difference, difference multiple, division, tree planting, averaging, meeting, twoway, catching up, running water, concentration, profit and loss, lifting, separation movement, chicken and rabbit in the same cage, train crossing the bridge, circular meeting, tax payment and interest, discount and profit, etc [10].

Firstly, a large number of topics are collected, and the high-quality topics are selected. Based on this, the topic data set is expanded, and then they are preprocessed and structured. The knowledge map of mathematical basic rules is established to form the domain knowledge base in AORBCO model, including descriptive knowledge and process knowledge, which paves the way for the subsequent generation of strategic knowledge; The AORBCO model (which consists of belief, ability, desire, planning and execution) plans the matching operation and reasoning calculation process of solving problems, and then discusses the existing cloud computing technology from the perspective of artificial intelligence and epistemology, forming the ability of topic classification and rule selection intelligence in AORBCO model; Finally, relying on the classification of topics in AORBCO model and the ability to select intelligent rules, the limitation of solving existing problems can be changed through this model, and Ego individuals can master the planning and implementation according to the existing belief knowledge and ability in this process, and obtain the results of new problems that have changed. The system design approach is shown in the following Figure 2.

Figure 2.

Overall design of the system

Self-learning is a branch of machine learning, especially a kind of unsupervised learning problem [11]. The optimization model of self-learning mechanism refers to the design and optimization of self-learning algorithm in multi-agent system by using the principles and methods of game theory, so that each agent can adjust its strategy according to its own goals and changes in the environment, thus achieving a balanced or coordinated state. In the AORBCO model, it refers to its natural learning mechanism, and uses a shallow neural network to calculate the semantic distance between the recognized topic texts.

As the object of matching algorithm of reasoning engine, rule base needs high accuracy. In order to reduce the time, cost of constructing rule base artificially, a self-learning module is added, and relevant rules are extracted from standard answers in a data-driven way, and the processes of reasoning, construction and optimization are automatically carried out, and finally the automatic construction of rule base is completed. Different from the traditional top-down knowledge base construction, the self-learning mode is used to form the rule base from the data from the bottom up, so that the automatic construction ensures the unification of engine rules and reduces the potential problems that may occur in the matching algorithm. In order to ensure the simplicity and unity of the reasoning system, it is necessary to transform the information of the rule subgraph into a data structure and provide it to the matching reasoning module. The process of rule standardization includes initialization rule classification, rule conclusion triplet, rule description, rule knowledge points and other information. Table 1 below is a data structure with structured rules.

The data of the self-learning system consists of questions and their standard answers, that is, Q=(q,a). Q is the complete question input, A is the complete answer input, and the question and its standard answer are passed into the self-learning module as a set of data. The system will pass the preprocessed result into the inference module, and the input at this time is qt=(qt,at,t), where qt represents the conditional triplet set of the t inference and at represents the result triplet set of the t inference. As a rule, to be evaluated, the results and conditions of reasoning are generated in the generator [12]. Finally, the generated rules are evaluated by the evaluation module, and the rules with high confidence are put into the rule base. The matching process is shown in Figure 3.

Figure 3.

Matching Process

In order to verify the effectiveness and practicality of the automatic problem-solving technology for elementary school mathematics competitions based on the AORBCO model, this study designed a series of experiments. The experimental data is sourced from the NuminaMath-CoT dataset, which contains 860,000 mathematical problems, covering topics from Chinese elementary school mathematics exercises to international mathematical Olympiad questions. To ensure the quality of the problems, this study selected 39,880 questions as the data source and chose 20% of them as the test set. The testing content mainly includes single-instance testing and batch testing.

In the batch testing, the system automatically solved 7976 questions, achieving a success rate of 78.5%. The following Table 4 provides a detailed analysis of the success rates and average solving times for different types of questions:

It can be seen from the above table that the system has the highest success rate in solving basic operations and relational problems, with the shortest average solving time. In contrast, the success rate for special types of questions and skillbased problems is the lowest, with the longest average solving time. This indicates that there is still room for improvement in the system's handling of complex problem types. Success Rate:

This system significantly outperforms Xiaoyuan Search and Homework Help in terms of success rate, improving by 13.5% and 18.5% respectively. This indicates that this system has higher accuracy and reliability when dealing with elementary school mathematics Olympiad problems.

To demonstrate the dynamic characteristics of knowledge weights during the reasoning process, we conducted feature tracking experiments on 500 problem-solving cases. As shown in Figure 7, the knowledge weight (calculated by Formula 1) shows an exponential decay trend during the initial reasoning phase (0-30s), but exhibits periodic reinforcement patterns after rule matching and cognitive optimization modules are activated. Notably, when the reasoning path encounters dead ends (marked by red arrows), the system triggers backtracking mechanisms that significantly enhance the weights of alternative knowledge nodes (average +23.6%).

Figure 7.

Knowledge weight evolution during problem-solving process

D. Summary of Test Results

The "Chicken-Rabbit Cage Problem" was selected for its multi-path solution characteristics (algebraic, enumerative, and substitution methods), moderate reasoning depth (average 6.8 steps), and explicit intermediate variable requirements.

Figure 8 illustrates the phased analysis using a dual-axis timeline (10ms sampling resolution). The primary axis tracks active knowledge nodes (weight threshold θ =40), while the secondary axis monitors weight concentration dynamics. Four distinct phases emerge: Knowledge Activation (0-5s):

Initial filtering reduced active nodes from 12→9.

Rule Matching (5-25s):

Constraint identification increased weight concentration from 54%→61%.

Cognitive Optimization (25-35s):

Path pruning (58→16 paths) boosted concentration to 89%.

Convergence Verification (>35s):

Final validation through algebraic proof.

Figure 8.

Temporal evolution of active nodes (bars) and weight concentration (line) during problem solving

The system demonstrated 72.4% search space reduction through three optimization waves (Table 6). Error recovery analysis revealed 2.4s mean detection latency for pseudo-solutions, with backtracking depth of 2.3 steps to valid checkpoints.

TABLE V.

PHASE TRANSITION PARAMETERS

Phase	Active Nodes	Weight Concentration	Trigger Condition
Initial Activation	12→9	N/A	Knowledge filtering
Rule Matching	9→14	54%→61%	Constraint identification
Cognitive Optimization	14→5	61%→89%	Path pruning activation

Through single-instance testing and batch testing, this system has demonstrated excellent performance in both solving accuracy and efficiency. In single-instance testing, the system successfully solved the problem, with the solving process consistent with the standard answer, taking 30 seconds. In batch testing, the system successfully solved 6260 out of 7976 problems, achieving a success rate of 78.5% and an average solving time of 1 minute and 30 seconds. In comparative testing, this system outperformed existing elementary school mathematics Olympiad solving software in both success rate and average solving time.

Although the system performed excellently in testing, there are still areas that require improvement. For instance, the system takes longer to solve some complex problems, necessitating further optimization of the matching algorithm and reasoning engine. Additionally, the system still makes errors when handling certain special problem types, requiring further expansion of the rule library and optimization of the self-learning module.

From the statistical chart of error situations in various modules of batch testing, it can be seen that the four modules of the system: natural language understanding, knowledge representation, reasoning system, and self-learning exhibit varying pass rates across different problem types. Due to the nature of the problem type, all modules performed poorly on sequences, while planar geometry faced significant issues in natural language understanding due to its complex expressions and multiple references.

The number of rules generated by the selflearning module is positively correlated with the pass rate of the solving system tests. For different modules, self-learning is also related to the performance of the natural language understanding module. The understanding of standard answers affects data quality, and the performance of the reasoning system impacts the rule merging part of the automatically generated rules, resulting in a high number of rules that cannot be merged, leading to insufficient data volume for the system's reasoning results.

Single-instance testing has proven the completeness of the functions of each module of the solving system, and the statistical results of batch testing also reflect a high degree of connectivity among the system's modules. The system has achieved the basic functions specified in the initial phase, with an average solving rate of 73.4%.