Aiming to harmonise finite and infinite model reasoning, we initiate the study of partially finite models, where the reasoning task comes with a formula that specifies a part of the model that must be finite. We focus on the problem of partially finite query entailment in description logics (DLs): given a knowledge base (KB), a query, and a distinguished concept, decide whether the query holds in all models of the KB that interpret the distinguished concept as a finite set. To break the ground, we work with the DL S, an extension of the basic DL ALC with transitive roles, which is one of the simplest cases where finite and infinite query entailment diverge. Generalising previous results on the finite and infinite cases, we show that also partially finite entailment of conjunctive queries is in 2-ExpTime for S. The solution involves sophisticated infinite model surgery and goes far beyond combining the arguments for the two special cases. As a direct application, we show how the problem of query containment in the presence of closed predicates can be solved by reduction to partially finite query entailment.

Recent works have explored the use of counting queries coupled with Description Logic ontologies. The answer to such a query in a model of a knowledge base is either an integer or infinity, and its spectrum is the set of its answers over all models. While it is unclear how to compute and manipulate such a set in general, we identify a class of counting queries whose spectra can be effectively represented. Focusing on atomic counting queries, we pinpoint the possible shapes of a spectrum over ALCIF ontologies: they are essentially the subsets of N and infinity closed under addition. For most sublogics of ALCIF, we show that possible spectra enjoy simpler shapes, being [ m, infinity ] or variations thereof. To obtain our results, we refine constructions used for finite model reasoning and notably rely on a cycle-reversion technique for the Horn fragment of ALCIF. We also study the data complexity of computing the proposed effective representation and establish the FP^NP[log]-completeness of this task under several settings.

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.

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.

We study the enumeration of answers to ontology-mediated queries (OMQs) where the ontology is a set of guarded TGDs or formulated in the description logic ELI and the query is a conjunctive query (CQ). In addition to the traditional notion of an answer, we propose and study two novel notions of partial answers that can take into account nulls generated by existential quantifiers in the ontology. Our main result is that enumeration of the traditional complete answers and of both kinds of partial answers is possible with linear-time preprocessing and constant delay for OMQs that are both acyclic and free-connex acyclic. We also provide partially matching lower bounds. Similar results are obtained for the related problems of testing a single answer in linear time and of testing multiple answers in constant time after linear time preprocessing. In both cases, the border between tractability and intractability is characterized by similar, but slightly different acyclicity 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. [Holger Dell et al., 2019]. 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.

Infinite duration games with regular conditions are one of the crucial tools in the areas of verification and synthesis. In this paper we consider a branching variant of such games - the game contains branching vertices that split the play into two independent sub-games. Thus, a play has the form of~an~infinite tree. The winner of the play is determined by a winning condition specified as a set of infinite trees. Games of this kind were used by Mio to provide a game semantics for the probabilistic mu-calculus. He used winning conditions defined in terms of parity games on trees. In this work we consider a more general class of winning conditions, namely those definable by finite automata on infinite trees. Our games can be seen as a branching-time variant of the stochastic games on graphs. We address the question of determinacy of a branching game and the problem of computing the optimal game value for each of the players. We consider both the stochastic and non-stochastic variants of the games. The questions under consideration are parametrised by the family of strategies we allow: either mixed, behavioural, or pure. We prove that in general, branching games are not determined under mixed strategies. This holds even for topologically simple winning conditions (differences of two open sets) and non-stochastic arenas. Nevertheless, we show that the games become determined under mixed strategies if we restrict the winning conditions to open sets of trees. We prove that the problem of comparing the game value to a rational threshold is undecidable for branching games with regular conditions in all non-trivial stochastic cases. In the non-stochastic cases we provide exact bounds on the complexity of the problem. The only case left open is the 0-player stochastic case, i.e. the problem of computing the measure of a given regular language of infinite trees.

Tree pattern queries are being investigated in database theory for more than a decade. They are a fundamental and flexible query mechanism and have been considered in the context of querying tree structured as well as graph structured data. We revisit their containment, validity, and satisfiability problem, both with and without schema information. We present a comprehensive overview of what is known about the complexity of containment and develop new techniques which allow us to obtain tractability- and hardness results for cases that have been open since the early work on tree pattern containment. For the tree pattern queries we consider in this paper, it is known that the containment problem does not depend on whether patterns are evaluated on trees or on graphs. This means that our results also shed new light on tree pattern queries on graphs.