Complexities of Nominal Schemas

In this extended abstract, we review our recent work “Nominal Schemas in Description Logics: Complexities Clarified” [6], to be presented at KR 2014. The fruitful integration of reasoning on both schema and instance level poses a continued challenge to knowledge representation and reasoning. While description logics (DLs) excel at the former task, rule-based formalisms are often more adequate for the latter. An established and highly productive strand of research therefore continues to investigate ways of reconciling both paradigms. A practical breakthrough in this area was the discovery of DL-safe rules, which ensure decidability of reasoning by restricting the applicability of rules to a finite set of elements that are denoted by an individual name [7]. As of today, DL-safe rules are the most widely used DL-rule extension, supported by several mainstream reasoners [3,8]. More recently, nominal schemas have been proposed as an even tighter integration of “DL-safe” instance reasoning with DL schema reasoning [5]. A nominal is a DL concept expression {a} that represents a singleton set containing only (the individual denoted by) a. Nominal schemas replace a by a variable x that ranges over all individual names, so that it might represent arbitrary nominals {a}, where all occurrences of {x} in one axiom represent the same nominal. For example, ∃hasFather. {x}u∃hasMother. ({y}u∃married.{x}) represents the set of all individuals whose father (x) and mother (y) are married to each other, where the parents must be represented by individual names. No standard DL can express this in such a concise way. The interplay with other DL features also makes nominal schemas more expressive than the combination of DLs and DL-safe rules. Nominal schemas have thus caused significant research interest, and several reasoning algorithms that exploit this succinct representation have been proposed [4,10,9,1]. Most recently, it was demonstrated that such algorithms can even outperform other systems for reasoning with DL-safe rules [9]. Surprisingly and in sharp contrast to these successes, many basic questions about the expressivity and complexity of nominal schemas have remained unanswered until recently. A naive reasoning approach is based on grounding, i.e., replacing nominal schemas by nominals in all possible ways, which leads to complexity upper bounds one exponential above the underlying DL. The only tight complexity result so far is that the N2EXPTIME combined complexity of reasoning in the DL SROIQ is not affected by nominal schemas—a result that reveals almost nothing about the computational or expressive impact of nominal schemas in general [5]. Beyond this singular result, it is only known that nominal schemas can simulate Datalog rules of any arity using ∃, u, and the universal role U [2]. In our KR 2014 paper “Nominal Schemas in Description Logics: Complexities Clarified” [6], we give a comprehensive account of the reasoning complexities of a wide range of DLs, considering both combined complexities (w.r.t. the size of the given knowledge base) and data complexities (w.r.t. the size of the ABox only). Figure 1 summarizes our results for combined complexities for DLs with nominal schemas (right; marked by the letter V) in comparison with known complexities of DLs with nominals (left). It turns out that SROIQ is an exception, while most other DLs experience exponential complexity increases due to nominal schemas. The effects on the data complexity are even more striking. The data complexity of standard DLs is either in P (for EL and Horn-DLs, which restrict the use of t and ¬) or in NP. In contrast, the data complexities for all nominal-schema DLs in Fig. 1 are only one exponential below their combined complexity, i.e., EXPTIME or NEXPTIME for most cases.