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Logic-based approaches to AI have the advantage that their behavior can in principle be explained to a user. If, for instance, a Description Logic reasoner derives a consequence that triggers some action of the overall system, then one can explain such an entailment by presenting a proof of the consequence in an appropriate calculus. How comprehensible such a proof is depends not only on the employed calculus, but also on the properties of the particular proof, such as its overall size, its depth, the complexity of the employed sentences and proof steps, etc. For this reason, we want to determine the complexity of generating proofs that are below a certain threshold w.r.t. a given measure of proof quality. Rather than investigating this problem for a fixed proof calculus and a fixed measure, we aim for general results that hold for wide classes of calculi and measures. In previous work, we first restricted the attention to a setting where proof size is used to measure the quality of a proof. We then extended the approach to a more general setting, but important measures such as proof depth were not covered. In the present paper, we provide results for a class of measures called recursive, which yields lower complexities and also encompasses proof depth. In addition, we close some gaps left open in our previous work, thus providing a comprehensive picture of the complexity landscape.

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.

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Planning in the presence of background ontologies is a topic of long-standing interest in AI. It combines the problems of (1) belief update complexity and (2) state-space combinatorics. DL-Lite offers an attractive solution to (1), with belief updates possible at the ABox level. Indeed, it has been shown that DL-Lite planning can be compiled into the commonly used planning language PDDL. Yet that compilation was previously found to be infeasible for off-the-shelf planning systems. Here we analyze the reasons for this problem and find that the bottleneck lies in the planner pre-processes, in particular in the naïve DNF transformations used to compile the PDDL input into the planners' internal representations. Consequently, we design a PDDL pre-compiler realizing a polynomial DNF transformation. We leverage a particular PDDL language feature ("derived predicates") to avoid the need for excessive control structure. Our pre-compiler turns out to be quite effective: the previous bottleneck disappears, and experiments on a broad range of benchmarks demonstrate the first practical technology for DL-Lite planning.