Finding Small Proofs for Description Logic Entailments: Theory and Practice

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.

Finding Good Proofs for Description Logic Entailments using Recursive Quality Measures

“…We analyze these measures not only for polynomial derivers, but this time also consider exponential derivers, thus giving insights on how our complexity results transfer to more expressive logics. In addition to upper bounds for the general class of monotone recursive Φ-measures, we show improved bounds for the specific measures considering depth and tree size, in the latter case improving results from [2]. Overall, we thus obtain a comprehensive picture of the complexity landscape for the problem of finding good proofs for DL and other entailments (see Table 1).…”

“…In practice, proofs and derivation structures are constructed by a reasoning system, and in theoretical investigations, it is common to define proofs by means of a calculus. To abstract from these details, we use the concept of a deriver as in [2], which is a function that, given a theory T and a conclusion η, produces the corresponding derivation structure in which we can look for an optimal proof. However, in practice, it would be inefficient and unnecessary to compute the entire derivation structure beforehand when looking for an optimal proof.…”

“…Intuitively, a Φ-measure m does not increase when the proof gets smaller, either when parts of the proof are removed (to obtain a subproof) or when parts are merged (in a homomorphic image). For example, m size ((V, E, )) := |V | is a Φ-measure, called the size of a proof, and we have already investigated the complexity of the following deicision problem for m size in [2].…”

“…For this reason, there has been considerable work in the Automated Deduction and Logic in AI communities on how to produce "good" proofs for certain purposes, both for full first-order logic, but also for decidable logics such a Description Logics (DLs) [9]. We mention here only a few approaches, and refer the reader to the introduction of our previous work [2] for a more detailed review.…”

“…In our first work in this direction [2], we focused our attention on size as the measure of proof quality. We could show that the above decision problem is NP-complete even for polynomial derivers and unary coding of numbers.…”