Journal of Automated Reasoning

, Volume 55, Issue 3, pp 295–306 | Cite as

Improving Legibility of Formal Proofs Based on the Close Reference Principle is NP-Hard

  • Karol Pąk
Open Access


Proof development in proof assistants such as HOL, Coq, Mizar, etc. is an activity where authors usually produce proofs by typing out proof scripts or system tactics. Quite frequently, however, authors also have to read existing proof scripts, either to imitate smart proof pieces, or to refactor fragments of reasoning to make some theorem stronger, more easily applicable and so on. Therefore, it is important to develop techniques to improve legibility of proofs, since it directly affects productivity of script writers. To analyze the legibility of natural deduction proofs, we investigate proof graphs that represent the flow of information in given reasoning. Our analysis of the information flow leads to methods of improving proof readability based on Behaghel’s First Law, which states that in legible text relevant pieces of information must occur close to each other. The presented method maximizes the number of close connections between premises and steps that use these steps as justification. In this paper we show that our optimization method is NP-hard.


Natural deduction Legibility NP-completeness 


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Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BiałystokBiałystokPoland

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