A method for the automated generation of proof exercises with comparable levels of proving complexity

2026-03-07

AI Summary

This study addresses the critical challenge of controlling proof difficulty consistency in the automatic generation of mathematical proof exercises. It proposes a method based on a cut-free semantic tableau proof system within first-order logic, which is free of logical symbols and endowed with analyticity and structural properties. This framework enables the mechanized extraction of rules that capture the cognitive effort required for informal proofs and supports a formal model of proof complexity. Leveraging this approach, the system can generate novel exercises whose difficulty closely matches that of a given problem, making it suitable for discrete mathematics courses. This work represents the first integration of tableau-based structural analysis with formal complexity modeling to achieve controllable, automated generation of proof exercises tailored to educational contexts.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Web data generation and simulation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The automated generation of exercises may substantially reduce the time educators devote to manual exercise design. A major obstacle to the integration of such automation into teaching practice, however, lies in the ability to control the difficulty of mechanically generated exercises. This paper presents a method for the automated generation of proof exercises with comparable levels of proving complexity. The method takes as input a proof exercise together with a set of rules that yield a proof of the exercise, and produces as output a set of proof exercises whose proving complexity is comparable to that of the input exercise. The approach focuses on mathematical proof exercises formulated in first-order languages, covering topics typically addressed in undergraduate Discrete Mathematics courses. We assess the proving complexity of these exercises by considering the effort required to solve them through informal proofs, and argue that this effort can be formally captured through cut-based tableau proofs that are free of logical symbols. The rules governing such proofs are obtained through a mechanizable extraction procedure introduced in this paper. By exploiting the analytic nature of these rules and the structure inherent in proofs constructed via tableau rules, we derive a computational procedure implementing the proposed method.

Problem

• automated exercise generation
• proof complexity
• mathematical proofs
• difficulty control
• first-order logic

Innovation

• automated exercise generation
• proof complexity
• tableau proofs
• first-order logic
• mechanizable rule extraction