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Finding Small Proofs for Description Logic Entailments: Theory and Practice

36 pagesPublished: May 27, 2020

Abstract

Logic-based approaches to AI have the advantage that their behaviour can in principle be explained by providing their users with proofs for the derived consequences. However, if such proofs get very large, then it may be hard to understand a consequence even if the individual derivation steps are easy to comprehend. This motivates our interest in finding small proofs for Description Logic (DL) entailments. Instead of concentrating on a specific DL and proof calculus for this DL, we introduce a general framework in which proofs are represented as labeled, directed hypergraphs, where each hyperedge corresponds to a single sound derivation step. On the theoretical side, we investigate the complexity of deciding whether a certain consequence has a proof of size at most n along the following orthogonal dimensions: (i) the underlying proof system is polynomial or exponential; (ii) proofs may or may not reuse already derived consequences; and (iii) the number n is represented in unary or binary. We have determined the exact worst-case complexity of this decision problem for all but one of the possible combinations of these options. On the practical side, we have developed and implemented an approach for generating proofs for expressive DLs based on a non-standard reasoning task called forgetting. We have evaluated this approach on a set of realistic ontologies and compared the obtained proofs with proofs generated by the DL reasoner ELK, finding that forgetting-based proofs are often better w.r.t. different measures of proof complexity.

Keyphrases: complexity, description logic, explanation, proofs

In: Elvira Albert and Laura Kovacs (editors). LPAR23. LPAR-23: 23rd International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 73, pages 32-67.

BibTeX entry
@inproceedings{LPAR23:Finding_Small_Proofs_Description,
  author    = {Christian Alrabbaa and Franz Baader and Stefan Borgwardt and Patrick Koopmann and Alisa Kovtunova},
  title     = {Finding Small Proofs for Description Logic Entailments: Theory and Practice},
  booktitle = {LPAR23. LPAR-23: 23rd International Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Elvira Albert and Laura Kovacs},
  series    = {EPiC Series in Computing},
  volume    = {73},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/qgX6},
  doi       = {10.29007/nhpp},
  pages     = {32-67},
  year      = {2020}}
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