Satisfiability Checking and Conjunctive Query Answering in Description Logics with Global and Local Cardinality Constraints

Franz Baader, Bartosz Bednarczyk, Sebastian Rudolph
2019 International Workshop on Description Logics  
We consider an expressive description logic (DL) in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we show that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints
more » ... appropriately restricted. locally, i.e., they refer to the role successors of an individual under consideration. It was shown in [1] that pure concept satisfiability in ALCSCC is a PSpace-complete problem, and concept satisfiability w.r.t. a general TBox is ExpTime-complete. This shows that the more expressive number restrictions do not increase the complexity of reasoning since reasoning in ALCQ has the same complexity [17, 19] . On the other hand, we have extended the terminological formalism of the well-known description logic ALC from general TBoxes to more general cardinality constraints expressed in QFBAPA [4], which we called extended cardinality constraints (ECBoxes). These constraints are global since they refer to all individuals in the interpretation domain. It was shown in [4] that reasoning w.r.t. ECBoxes is in NExpTime even if the numbers occurring in the constraints are encoded in binary. A NExpTime lower bound follows from a result of Tobies [18] for a restricted form of cardinality constraints, where the cardinality of a concept can only be compared with a fixed number. This complexity can be lowered to ExpTime if a restricted form of cardinality constraints (RCBoxes) is used. Such RCBoxes are still powerful enough to express statistical knowledge bases [13] . An obvious way to generalize these two approaches is to combine the two extensions, i.e., to consider extended cardinality constraints, but now on ALCSCC concepts rather than just ALC concepts. This combination was investigated in [2] , where a NExpTime upper bound was established for reasoning in ALCSCC w.r.t. ECBoxes. It is also shown in [2] that reasoning w.r.t. RCBoxes stays in ExpTime also for ALCSCC. Here we go one step further by allowing for a tighter integration of global and local constraints. The resulting logic, which we call ALCSCC ++ , allows, for example, to relate the number of role successors of a given individual with the overall number of elements of a certain concept. We show that, from a worst-case complexity point of view, this extended expressivity comes for free if we consider classical reasoning problems. Concept satisfiability in ALCSCC ++ has the same complexity as in ALC and ALCSCC with global cardinality constraints: it is NExpTime-complete. Yet, for effective conjunctive query answering this logic turns out to be too expressive. We show that conjunctive query entailment w.r.t. ALCSCC ++ knowledge bases is, in fact, undecidable. In contrast, we can show that conjunctive query entailment w.r.t. ALCSCC RCBoxes is decidable. We assume the reader to be sufficiently familiar with all the standard notions of description logics [3, 5, 15] . The logic ALCSCC ++ As in [1, 4] , we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) to express our constraints. In this logic, one can build set terms by applying Boolean operations (intersection ∩, union ∪, and complement • c ) to set variables as well as the constants ∅ and U. Set terms s, t can then be used to state set constraints, which are equality and inclusion constraints of the form s = t, s ⊆ t, where s, t are set terms. Presburger Arithmetic
dblp:conf/dlog/BaaderBR19 fatcat:tc2h7f4imjhtrnvel7zhbd6dcy