Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR 2020) Main Track Multi-head Guarded Existential Rules Over Fixed Signatures Georg Gottlob1 , Marco Manna2 , Andreas Pieris3 1Department of Computer Science, University of Oxford 2Department of Mathematics and Computer Science, University of Calabria 3School of Informatics, University of Edinburgh [email protected], [email protected], [email protected] Abstract 2. to understand whether it can be solved by relying on stan- dard database technology via query rewriting. Guarded existential rules form a robust rule-based language for modelling ontologies. The central problem of ontology- It is fair to claim that the above aspects of the problem based query answering, as well as the notion of polynomial are rather well-understood. Concerning the first one, OBQA combined rewritability, have been extensively studied during for guarded existential rules is complete for PTIME in data the last years for this formalism. However, the relevant set- complexity (i.e., when the ontology and the query are fixed), ting where the underlying signature is fixed is far from being for NP when the underlying signature (or schema) is fixed well-understood. All the existing results on ontology-based query answering and polynomial combined rewritability im- with the additional implicit assumption that rule-heads have plicitly assume rule-heads with one atom, whereas existential only one atom (further details about the latter assumption are rules in real ontologies are typically coming with multi-heads given below), for EXPTIME over signatures of fixed arity, consisting of several atoms. We aim to fill this gap. and for 2EXPTIME in combined complexity (Cal`ı, Gottlob, and Lukasiewicz 2012; Cal`ı, Gottlob, and Kifer 2013). Concerning query rewriting, the PTIME-hardness in data 1 Introduction complexity immediately implies that the problem in ques- In ontology-based query answering (OBQA), ontologies are tion is not first-order rewritable. On the other hand, we know used to enrich incomplete data with domain knowledge, en- that it is Datalog rewritable (Bar´ any,´ Benedikt, and ten Cate abling more complete answers to queries, typically conjunc- 2013; Gottlob, Rudolph, and Simkus 2014). At this point, let tive queries (CQs). A notable range of ontology formalisms us stress that the above results refer to the pure approach to for OBQA, which vary in syntax, expressivity and complex- query rewriting, where the rewriting phase does not depend ity, has been developed during the last decade. Two promi- on any database. With the aim of obtaining first-order rewrit- nent families of languages, obtained from this extensive ef- ings, and thus being able to use a standard relational engine fort, are description logics (DL) (Baader et al. 2017), and for OBQA purposes, the work (Gottlob, Manna, and Pieris existential rules (a.k.a. tuple-generating dependencies and 2014) adopted the more refined approach to query rewriting Datalog rules) (Baget et al. 2011; Cal`ı et al. 2010). Many known as the combined approach, originally introduced in of the existing DL-based and rule-based ontology languages the context of DLs (Lutz, Toman, and Wolter 2009), which guarantee good computational and model-theoretic proper- allows us to rewrite also the database in a query-independent ties by posing restrictions on the use of quantifiers. They are way. It was shown that indeed OBQA for guarded existential essentially defined through the relativisation of quantifiers rules is combined first-order rewritable. The really interest- by atomic formulas, similar to the guarded fragment of first- ing result of that work, though, is that in the fixed signature order logic (Andreka,´ van Benthem, and Nemeti´ 1998). It is case the rewriting process takes only polynomial time with generally agreed that guardedness is a paradigm that leads the additional implicit assumption that rule-heads mention to reasonably expressive and robust ontology languages. only one atom. Note that beyond fixed signatures, the com- The main rule-based ontology language that originated bined rewriting process is unlikely to be polynomial. from this paradigm, which is the main concern of this work, is the class of guarded existential rules (Cal`ı, Gottlob, and Multi-head Rules and Fixed Signatures. Despite the thor- Kifer 2013). It consists of sets of first-order sentences of the ough analysis of the OBQA problem for guarded existential form 8x¯8y¯φ(¯x; y¯) ! 9z¯ (¯x; z¯), where φ (the body) and rules, there is a striking gap, which we reveal below, that (the head) are conjunctions of relational atoms, and φ has passed unnoticed until recently when it was brought to our an atom that contains all the universally quantified variables. attention by a colleague of ours (Benedikt 2018). Recall that an existential rule is typically defined as a first- Guarded Existential Rules. The problem of OBQA has order sentence 8x¯8y¯φ(¯x; y¯) ! 9z¯ (¯x; z¯), where φ and been rigorously studied for guarded existential rules during are conjunctions of relational atoms. For OBQA purposes, the last years. The two main research directions were: the fact that in the head we can have a conjunction of atoms 1. to pinpoint its computational complexity, and is usually seen as syntactic sugar. The reason is because we 445 Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR 2020) Main Track can always convert a set Σ of existential rules into a set Σ1 over a fixed signature is constant, which in turn implies that of existential rules with only atom in the head such that Σ we can only have a constant number of different single-head and Σ1, although not logically equivalent, are equivalent for guarded rules (up to variable renaming). This is far from OBQA. This, in fact, relies on a very simple transformation being true for multi-head guarded existential rules since we that replaces each existential rule σ 2 Σ of the form can have an unbounded number of different rules with the same guarded body due to the unguarded multi-heads. 8x¯8y¯ φ(¯x; y¯) ! 9z¯ R1(¯x1; z¯1) ^ · · · ^ Rn(¯xn; z¯n)); Our Results. The main results of this work are as follows: where x¯i ⊆ x¯ and z¯i ⊆ z¯, with the set of existential rules 1. Before delving into the analysis that will provide answers 8x¯8y¯ φ(¯x; y¯) ! 9z¯Auxσ(¯x; z¯)) to our main research questions, we ask ourselves whether 8x¯8z¯(Auxσ(¯x; z¯) ! Ri(¯xi; z¯i)); for i 2 f1; : : : ; ng; multi-head guarded rules are strictly more expressive than single-head ones. We show that, for each schema S (with with Auxσ being a fresh relation not occurring in Σ. Notice at least one binary relation), there exists a set of multi- further that if Σ is guarded, then also Σ1 is guarded. head guarded existential rules over S that cannot be equiv- Due to the above transformation, all the known complex- alently rewritten as a set of single-head guarded existen- ity results on OBQA for guarded existential rules, as well as tial rules over S for OBQA purposes. In simple words, the result on polynomial combined first-order rewritability, if we are not allowed to introduce new relation symbols, have been shown for existential rules with only one atom in multi-heads are not merely syntactic sugar but come with the head. The reason was purely technical, that is, to sim- additional expressive power. plify the technical definitions and proofs. Although this sim- plifying assumption can be made, in general, without affect- 2. We then show that OBQA for multi-head guarded exis- ing the generality of the results, this is not true in the relevant tential rules over fixed signatures is NP-complete. The setting of fixed signatures. This is because, even if we start non-trivial result is the upper bound; the lower bound is from a set of guarded existential rules over a fixed signature, inherited from CQ evaluation. Towards showing the upper the obtained set of single-head guarded rules after the trans- bound, we obtain several results of independent interest: formation mentions an unbounded number of new relations (i) instance checking for multi-head guarded existential each of unbounded arity (see the auxiliary predicates). rules over fixed signatures is in PTIME, (ii) OBQA for As mentioned above, fixing the signature is a relevant set- multi-head guarded existential rules can be polynomially ting, which is close in spirit to data complexity. Typically, reduced to multi-head linear existential rules (i.e., rules the size of the signature is much smaller than the size of the with only one atom in the body) without increasing the database, and therefore, it can be productively assumed to arity of the signature, and (iii) OBQA for multi-head lin- be fixed. Furthermore, since the typical purpose of an ontol- ear existential rules is NP-complete for signatures of fixed ogy is to model a certain domain of interest, the schema is arity. The above are known for single-head rules, but can- actually predetermined by the application domain in ques- not be directly transferred to multi-head ones. tion, which is another justification for considering the signa- 3. Finally, by exploiting the machinery introduced for es- ture fixed. The above, together with the fact that it is more tablishing the NP upper bound, we can show that OBQA convenient to design ontologies using multi-head existential for multi-head guarded existential rules over fixed signa- rules, it suggests that the gap in the analysis of OBQA for tures is polynomially combined first-order rewritable; in guarded existential rules described above is a fundamental fact, the target query language is existential positive first- one, which undoubtedly deserves our attention.