[Paper Review] Granularity-Adaptive Proof Presentation

Marvin Schiller, Christoph Benzmueller|ArXiv.org|Mar 2, 2009

Logic, programming, and type systems6 references4 citations

TL;DR

This paper proposes a machine learning-based framework for adaptive proof presentation in automated theorem proving, modeling proof granularity from example proofs and generating human-readable proofs at user-specific levels of detail. By classifying proof steps as appropriate, too big, or too small using cognitive and logical features, the system dynamically adjusts explanation granularity to match user knowledge and didactic goals.

ABSTRACT

When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitable for human use. A challenge therefore is to develop user- and goal-adaptive proof presentation techniques that obey common mathematical practice. We present a flexible and adaptive approach to proof presentation that exploits machine learning techniques to extract a model of the specific granularity of proof examples and employs this model for the automated generation of further proofs at an adapted level of granularity.

Motivation & Objective

• Address the gap in automated theorem provers that typically present proofs at a fixed, low-level granularity unsuitable for human learners.
• Model proof granularity adaptively based on user-specific didactic goals and assumed knowledge.
• Enable automated generation of proofs at varying levels of detail that align with common mathematical practice.
• Support intelligent tutoring systems by dynamically adjusting proof step size to match student understanding.
• Overcome limitations of static proof layers in systems like Omega-mega by learning optimal granularity from proof examples.

Proposed method

• Treat proof granularity as a binary classification task: classify each proof step as 'appropriate', 'too big', or 'too small'.
• Define a feature space combining mathematical/logical aspects (e.g., number of assertion applications, concept types) and cognitive factors (e.g., user background knowledge).
• Use standard machine learning techniques to train a model on proof examples, learning how to predict optimal step size.
• Integrate the learned model into the Omega-mega proof assistant to generate proof presentations at user-adapted granularity levels.
• Leverage assertion-level inference (based on CoRe calculus) to ensure each step is justified by a mathematical fact, avoiding low-level technical steps.
• Apply the model to generate proof presentations that match the granularity observed in human-written proofs, such as the set theory example in Figure 1.

Experimental results

Research questions

• RQ1 How can proof granularity be modeled adaptively to match a user’s knowledge and didactic goals?
• RQ2 What features are most predictive of an appropriate proof step size in mathematical proofs?
• RQ3 Can machine learning effectively extract a granularity model from proof examples to guide automated proof presentation?
• RQ4 To what extent does the proposed framework improve the usability of automated proofs for human learners compared to fixed-granularity presentations?
• RQ5 How can proof presentation be made flexible and adaptive while preserving logical correctness and mathematical rigor?

Key findings

• The framework successfully models proof granularity by learning from example proofs, enabling dynamic adjustment of step size based on user context.
• Proof steps involving multiple inference applications—such as the transition from line 9 to 10 in Figure 1—can be classified as 'too big' when they combine multiple concepts without explicit mention.
• The system identifies that steps combining concepts from different domains (e.g., naive set theory and propositional logic) are more likely to be perceived as too large if not properly segmented.
• The model captures that steps relying on previously applied concepts and operating on a common subexpression are more likely to be judged as appropriate.
• The framework demonstrates that machine learning can effectively learn and apply human-like granularity preferences from proof examples.
• The approach enables automated proof generation at varying levels of detail, improving accessibility and usability for educational applications.