Carsten Lutz’s research while affiliated with University of Leipzig and other places

We study extensions of description logics from the widely used EL family with operators that make it possible to speak about different levels of abstraction. We analyze the computational complexity of reasoning and show that often, this complexity is significantly lower than in the corresponding extension of the more expressive description logic ALC. By slightly varying the semantics, we also obtain a case that admits reasoning in polynomial time.

We study extensions of expressive decidable fragments of first-order logic with circumscription, considering in particular the two-variable fragment FO^2, its extension C^2 with counting quantifiers, and the guarded fragment GF. We prove that if only unary predicates are minimized (or fixed) during circumscription, then decidability of logical consequence is preserved. For FO^2 the complexity increases from NExp to NExp^NP-complete, for GF it (remarkably!) increases from 2Exp to Tower-complete, and for C^2 it remains open. We also consider querying circumscribed knowledge bases whose ontology is a GF sentence, showing that the problem is decidable for unions of conjunctive queries, Tower-complete in combined complexity, and elementary in data complexity. Already for atomic queries and ontologies that are sets of guarded existential rules, however, for every k > 0 there is an ontology and query that are k-Exp-hard in data complexity.

We study extensions of expressive decidable fragments of first-order logic with circumscription, in particular the two-variable fragment FO$^2$, its extension C$^2$ with counting quantifiers, and the guarded fragment GF. We prove that if only unary predicates are minimized (or fixed) during circumscription, then decidability of logical consequence is preserved. For FO$^2$ the complexity increases from $\textrm{coNexp}$ to $\textrm{coNExp}^\textrm{NP}$-complete, for GF it (remarkably!) increases from $\textrm{2Exp}$ to $\textrm{Tower}$-complete, and for C$^2$ the complexity remains open. We also consider querying circumscribed knowledge bases whose ontology is a GF sentence, showing that the problem is decidable for unions of conjunctive queries, $\textrm{Tower}$-complete in combined complexity, and elementary in data complexity. Already for atomic queries and ontologies that are sets of guarded existential rules, however, for every $k \geq 0$ there is an ontology and query that are k-$\textrm{Exp}$-hard in data complexity.

A fitting algorithm for conjunctive queries (CQs) is an algorithm that takes as input a collection of data examples and outputs a CQ that fits the examples. In this column, we propose a set of desirable properties of such algorithms and use this as a guide for surveying results from the authors' recent papers published in PODS 2023, IJCAI 2023, and Inf. Proc. Letters 2024. In particular, we explain and compare several concrete fitting algorithms, and we discuss complexity and size bounds for constructing fitting CQs with desirable properties.

We study the complexity of answer counting for ontology-mediated queries and for querying under constraints, considering conjunctive queries and unions thereof (UCQs) as the query language and guarded TGDs as the ontology and constraint language, respectively. Our main result is a classification according to whether answer counting is fixed-parameter tractable (FPT), W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, lifting a recent classification for UCQs without ontologies and constraints due to Dell et al. The classification pertains to various structural measures, namely treewidth, contract treewidth, starsize, and linked matching number. Our results rest on the assumption that the arity of relation symbols is bounded by a constant and, in the case of ontology-mediated querying, that all symbols from the ontology and query can occur in the data (so-called full data schema). We also study the meta-problems for the mentioned structural measures, that is, to decide whether a given ontology-mediated query or constraint-query specification is equivalent to one for which the structural measure is bounded.

Circumscription is one of the main approaches for defining non-monotonic description logics (DLs) and the decidability and complexity of traditional reasoning tasks, such as satisfiability of circumscribed DL knowledge bases (KBs) are well understood. For evaluating conjunctive queries (CQs) and unions thereof (UCQs), in contrast, not even decidability had been established. In this paper, we prove decidability of (U)CQ evaluation on circumscribed DL KBs and obtain a rather complete picture of both the combined complexity and the data complexity, for DLs ranging from ALCHIO via EL to various versions of DL-Lite. We also study the much simpler atomic queries (AQs).

Ontologies often require knowledge representation on multiple levels of abstraction, but description logics (DLs) are not well-equipped for supporting this. We propose an extension of DLs in which abstraction levels are first-class citizens and which provides explicit operators for the abstraction and refinement of concepts and roles across multiple abstraction levels, based on conjunctive queries. We prove that reasoning in the resulting family of DLs is decidable while several seemingly harmless variations turn out to be undecidable. We also pinpoint the precise complexity of our logics and several relevant fragments.

We propose bounded fitting as a scheme for learning description logic concepts in the presence of ontologies. A main advantage is that the resulting learning algorithms come with theoretical guarantees regarding their generalization to unseen examples in the sense of PAC learning. We prove that, in contrast, several other natural learning algorithms fail to provide such guarantees. As a further contribution, we present the system SPELL which efficiently implements bounded fitting for the description logic ELHr based on a SAT solver, and compare its performance to a state-of-the-art learner.

We study the enumeration of answers to ontology-mediated queries when the ontology is formulated in a description logic that supports functional roles and the query is a CQ. In particular, we show that enumeration is possible with linear preprocessing and constant delay when a certain extension of the CQ (pertaining to functional roles) is acyclic and free-connex acyclic. This holds both for complete answers and for partial answers. We provide matching lower bounds for the case where the query is self-join free.