Large-Scale Reasoning on Expressive Horn Ontologies

Abstract

E cient large-scale reasoning is a fundamental prerequisite for the development of the Semantic Web. In this scenario, it is convenient to reduce standard reasoning tasks to query evaluation over (deductive) databases. From a theoretical viewpoint much has been done. Conversely, from a practical point of view, only a few reasoning services have been developed, which however typically can only deal with lightweight ontologies. To ll the gap, the paper presents owl2dlv, a novel and modern Datalog system for evaluating SPARQL queries over very large OWL 2 knowledge bases. owl2dlv builds on the well-known ASP system dlv by incorporating novel optimizations sensibly reducing memory consumption and a server-like behavior to support multiplequery scenarios. The high potential of owl2dlv for large-scale reasoning is outlined by the results of an experiment on data-intensive benchmarks, and con rmed by the direct interest of a major international industrial player, which has stimulated and partially supported this work.

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Datalog is a powerful, yet simple and elegant rule-based language originally designed in the context of deductive databases for querying relational data. After almost 40 years, however, its scope of applicability and its extensions go de nitely beyond the initial target, so much that now they range from optimization and constraint satisfaction problems [ 19 ] to even ontological design and reasoning in the Semantic Web [ 20,30 ]. Indeed, in the development of the Semantic Web, e cient large-scale reasoning is a fundamental prerequisite. In this scenario, it is convenient to reduce standard reasoning tasks to query evaluation over (deductive) databases. From a theoretical viewpoint much has been done: in many ontological settings, the problem of evaluating a conjunctive query over a knowledge base (KB) consisting of an extensional dataset (ABox) paired with an ontology (TBox) can be reduced to the evaluation of a Datalog query (i.e., a Datalog program, possibly nonrecursive and including strong constrains, paired with a union of conjunctive queries, both constructed only from the original query and the TBox) over the same ABox [ 25,28,35,42,45 ]. Conversely, from a practical viewpoint the situation is not so rosy. Many classical Datalog reasoners, such as clingo [ 29 ] and dlv [ 39 ], are based on one-shot executions performing heavy operations (e.g., loading and indexing) multiple times and hence are rather unsuited. Also, only a few reasoning services with a server-like behavior, such as mastro [ 24 ], ontop [ 23 ], and rdfox [ 41 ], have been developed, which however can only deal with lightweight TBoxes. To ll the gap, the paper presents owl2dlv, a modern Datalog system, based on the aforementioned rewriting approach, for evaluating SPARQL conjunctive queries [ 44 ] over very large OWL 2 knowledge bases [ 26 ].

Reasoning over OWL 2 is generally a very expensive task: fact entailment (i.e., checking whether an individual is an instance of a concept) is already 2NExpTime-hard, while decidability of conjunctive query answering is even an open problem. To balance expressiveness and scalability, the W3C identi ed three tractable pro les |OWL 2 EL, OWL 2 QL, and OWL 2 RL| exhibiting good computational properties: the evaluation of conjunctive queries over KBs falling in these fragments is in PTime in data complexity (query and TBox are considered xed) and in PSpace in combined complexity (nothing is xed) [ 43 ]. To deal with a wide variety of ontologies, owl2dlv implements the Horn-SHIQ fragment of OWL 2, which enjoys good computational properties: conjunctive queries are evaluated in PTime (resp., ExpTime) in data (resp., combined) complexity. Moreover, it is also quite expressive: it generalizes both OWL 2 QL and OWL 2 RL, while capturing all OWL 2 EL constructs except role chain [ 38 ].

From the technical side, owl2dlv builds on the well-known ASP system dlv [ 39 ], and in particular its most recent incarnation dlv2 [ 5 ], by incorporating a server-like modality, which is able to keep the main process alive, receive and process multiple user's requests on demand, and restore its status thanks to an embedded persistency layer. Following the approach proposed by Eiter et al. [ 28 ], a Horn-SHIQ TBox paired with a SPARQL query are rewritten, independently from the ABox, into an equivalent Datalog query.

The high potential of owl2dlv for large-scale ontological reasoning is outlined by the results of an experiment on data-intensive benchmarks, and conrmed by the direct interest of a big international industrial player, which has partially supported this work and also stimulated the evolution of the system with a major challenge: \deal with LUBM-8000 |the well-known LUBM [ 31 ] standard benchmark for ontological reasoning collecting about 1 billion factual assertions upon 8,000 universities| over machines equipped with 256GB RAM and with an average query evaluation time of at most 10 minutes". Eventually, not only the system was able to widely win the general challenge as reported in Figure 1; but, amazingly, the average time taken by owl2dlv on the ten (out of fourteen) bound queries |i.e., queries containing at least one constant| of LUBM-8000 was eventually less than one second (see Section 7 for details).

Number of handled triples using at most 256 GB 1.03 billion triples 1 billion triples 675 millions triples

13 minutes 10 minutes

OWL2DLV (Feb 2019) Challenge target

DLV2 (Aug 2018) 375 gigabytes 256 gigabytes 250.0 gigabytes

Peak of memory over LUBM-8000 (1 billion triples)

Average time over LUBM-8000 2.4 minutes (1 billion triples) OWL 2. As said, we focus on the Horn-SHIQ [ 33 ] fragment of OWL 2. In Description Logic (DL) terminology and notation, a knowledge base (KB) K is a pair (A; T ), where A is a set of (factual) assertions representing extensional knowledge about individuals and T is a set of axioms representing intensional knowledge about the domain of interest. Let NI (individuals), NC f>; ?g (atomic concepts) and NR (role names) be pairwise disjoint discrete sets. A role r is either a role name s or its inverse s . A concept is either an atomic concept or an expression of the form C u D, C t D, :C, 8r:C, 9r:C, > nr:C or 6 nr:C, where C and D are concepts, r is a role, and n 1. General concept inclusions (GCIs), role inclusions (RIs), and transitive axioms (TAs) are respectively of the form C1 v C2, r1 v r2, and Tr (r), where: t is disallowed in C2, > nr and 6 nr are disallowed in C1, and they are disallowed also in C2 in case r is transitive. A Horn-SHIQ TBox is a nite set of GCIs, RIs and TAs satisfying some non-restrictive global conditions [ 34,36 ]. An instance I is a set of assertions of the form C(a) and r(a; b), where C 2 NC, r 2 NR, and a; b 2 NI. An ABox is any nite instance. Common formats for OWL 2 KBs are both Turtle [ 17 ] and RDF/XML [ 26 ]. For example, assertions Woman(Ann), Person(Tom) and likes(Tom; Ann) are encoded in Turtle as: :Ann rdf:type :Woman .

:Tom rdf:type :Person ; :likes :Ann . while the GCI Woman v Person is encoded in RDF/XML as: <owl:Class rdf:about="#Person"/> <owl:Class rdf:about="#Woman">

<rdfs:subClassOf rdf:resource="#Person"/> </owl:Class> SPARQL. It is the standard language in the Semantic Web for querying OWL 2 knowledge bases [ 44 ]. As in databases, the most important class of SPARQL queries are the conjunctive ones, which syntactically are quite similar to SQL queries. For example, to select the individuals that are instances of Person we write SELECT ?X WHERE f ?X rdf:type :Person g. According to the example given in the previous section, however, the answer to this query is di erent when executed over the ABox only (giving just Tom as answer) or by also taking into account the TBox (giving as answer also Ann). More generally, when querying OWL 2 knowledge bases the TBox plays the role of a st-order theory and it has to be taken into account properly, as described next.

OBQA. A model of a KB K = (A; T ) is typically any instance I A satisfying all the axioms of T , written I j= T , where GCIs, RIs and TAs can be regarded as rst-order expressions [ 15 ]. For example, inclusion C u D v E over atomic concepts is equivalent to 8x (E(x) C(x) ^ D(x)). (For a comprehensive picture, we refer the reader to [ 38 ].) The set of all models of K is denoted by mods(K). To comply with the so-called open world assumption (OWA), note that I might contain individuals that do not occur in K. The answers to a query q(x) over an instance I is the set q(I) = fa 2 NIjxj j I j= q(a)g of jxj-tuples of individuals obtained by evaluating q over I. Accordingly, the certain answers to q under OWA is the set cert(K; q) = TI2mods(D; ) q(I): Finally, ontology-based query answering (OBQA) is the problem of computing cert(K; q).

DLV. It is one of the most used logic programming systems based on answer set semantics[ 39 ], a well-known formalism extended by many expressive constructs [ 3 ] useful in several application domains [ 4,8,9,10,11 ]. Recently, it has been redesigned and reengineered to version 2.0, called dlv2 [ 5 ], to enjoy modern evaluation techniques together with development platforms fully complying with the ASP-Core-2 language. Nowadays, it integrates two sub-systems: i-dlv [ 21 ], handling the deductive databases and program grounding, and wasp [ 6 ] for the model search phase. dlv2 has been the basis for the development of owl2dlv. 3

Ontological Reasoning via Datalog

As said, to perform OBQA, owl2dlv follows the approach of Eiter et al. [ 28 ]. From an OWL 2 Horn-SHIQ TBox T and a SPARQL conjunctive query q(x), owl2dlv runs Algorithm 1 to build a Datalog program PT and a union of conjunctive queries Qq;T (x) such that, for each ABox A, the evaluation of Qq;T (x) over A [ PT produces the same answers as the evaluation of q(x) over A [ T .

Algorithm 1: TBox and Query Rewriting

Input: An OWL 2 Horn-SHIQ TBox T together with a query q(x) Output: The Datalog program PT together with the query Qq;T (x) 1. T 0 Normalize(T ); 2. T EmbedTransitivity(T 0); 3. (T ) Saturate(T ); 4. PT RewriteTBox( (T )); 5. Qq;T (x) RewriteQuery(q(x); (T ));

As an example, consider a TBox T consisting of the GCIs CommutingArea v 9linked :Capital , 9linked :Capital v DesirableArea, and Capital v DesirableArea, together with the TA Tr (linkedViaTrain) and the RI linkedViaTrain v linked . According to Algorithm 1 PT is as follows: linked(X,Y) :- linkedViaTrain(X,Y). desirableArea(Y) :- capital(X), linked(Y,X). desirableArea(X) :- capital(X). capital*(Y) :- commutingArea(X), linkedViaTrain(X,Y). capital*(Y) :- capital*(X), linkedViaTrain(X,Y). capital(X) :- capital*(X).

desirableArea(X) :- commutingArea(X).

Moreover, starting from the SPARQL query q(x) reported below

SELECT ?X WHERE { ?X :linkedViaTrain ?Y. ?Y rdf:type :DesirableArea } we obtain the following UCQ Qq;T (x), also encoded as a set of Datalog rules: q(X) :- linkedViaTrain(X,Y), desirableArea(Y). q(X) :- linkedViaTrain(X,Y), capital(Y). q(X) :- linkedViaTrain(X,Y), commutingArea(Y). q(X) :- linkedViaTrain(X,Y), capital*(Y). q(X) :- commutingArea(X).

q(X) :- capital*(X). 4

Query Optimization

The pair (Qq;T (x); PT ) returned by Algorithm 1 is further optimized by a pruning strategy followed by the so-called Magic Sets rewriting |the latter is already in use in dlv2 but it has been further improved due to the speci c nature of PT . The result of this phase consists of the pair (opt(Qq;T (x)); opt(PT )). Pruning Strategy. Pairs of GCIs of the form C1 v C2 and C2 v C1 give rise to Datalog queries containing rules with multiple predicates having the same extensions: C1(X) : C2(X) and C2(X) : C1(X). The same happens with RIs. During the evaluation of the query, however, this can be considerably expensive. Hence, we adopt the following pruning strategy. Let E = fE1; :::; Ekg be a set of equivalent concepts or roles. First, we remove from PT all the rules of the form Ei(X) : Ej(X) with 1 < i k and 1 j k. Second, let PT1 be the subset of PT containing only rules of the form E1(X) : Ej(X) with 2 j k, for each i 2 f2; :::; kg, we replace each occurrence of Ei by E1 both in Qq;T (x) and in each rule of PT that does not belong to PT1 . An analogous technique is applied over RIs of the form r v s and s v r by taking into account, in this case, that the rst argument of r (resp., s) maps the second one of s (resp., r). Magic Sets Rewriting. Datalog systems usually implement a bottom-up algorithm that iteratively derives new facts by matching bodies with already known facts. Queries are answered on the xpoint of the algorithm producing the canonical model of the program. In contrast, a typical top-down algorithm for query answering looks for a rule from which some answers to the input query might be derived; if this kind of rule is found, its body atoms are considered as subqueries and the procedure is iterated. This way, only parts of the program that are relevant for answering the query are evaluated. Magic Sets [ 7 ] aim at combining the bene ts of the two algorithms; in fact the program is rewritten to simulate a top-down query evaluation via a bottom-up algorithm. Basically, Magic Sets introduce rules de ning additional atoms, named magic atoms, to identify relevant atoms for answering the query, namely atoms reachable by a top-down query evaluation. The bottom-up evaluation is then limited by adding magic atoms in the bodies of the original rules. Consider the Datalog query ancestor(X,Y) :- parent(X,Y). ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y).

ancestor(mario,X)? The Magic Sets rewriting starts with the query seed m#ancestor#bf(mario), modi es the rules de ning the intentional predicate ancestor, and introduces magic rules for every occurrence of intentional predicates in the modi ed rules: m#ancestor#bf(mario). ancestor(X,Y) :- m#ancestor#bf(X), parent(X,Y). ancestor(X,Y) :- m#ancestor#bf(X),parent(X,Z),ancestor(Z,Y). m#ancestor#bf(Z) :- m#ancestor#bf(X), parent(X,Z).

ancestor(mario,X)? The new program is specialized for answering the original query as its bottomup evaluation only materializes descendants of mario, rather than the full ancestor relation. For large programs, however, many \irrelevant" rules may be introduced, which in turn unavoidably overload the reasoning. To optimize the rewriting, owl2dlv performs two novel steps: (1) Eliminate rules that have a magic atom with predicate m#p# if 6= f f and m#p#f f also occurs in the rewritten program; and (2) Remove every rule r1 whenever it is subsumed by some other rule r2 (r1 v r2), namely there is a variable substitution that maps the head (resp., body) of r2 to the head (resp., body) of r1. To avoid the quadratic number of checks, owl2dlv associates each rule with a suitable hash value of size 64 bits. Then, r1 v r2 is checked only if the bit-a-bit equation hash(r1) & hash(r2) == hash(r2) is satis ed. 5

OWL2DLV: Design and Implementation

The owl2dlv architecture is depicted in Figure 2. The system features four main modules: Loading, Rewriting, Query Answering, and Command Interpreter. Clients interact with the system through the latter one, which takes user commands and requests to the internal modules to execute the corresponding behavior, and provide the output to the client. This module allows to \keep alive" the system, and execute multiple commands (e.g., loading, warmup, data updates, and query evaluations) without having to instantiate a new process for each client request. The Command Interpreter can be controlled either via command line (e.g., bash shell in Linux) or from external applications through a Java API. The remaining three modules are discussed below.

Loading. It handles the input of the system. In particular, it processes an OWL 2 ABox encoded in Turtle format via the ABox Loader. To guarantee a high performance both in the scanner and in the parser phase, owl2dlv employs Flex & Bison [ 1,2 ] in the implementation of the o cial Turtle grammar and the parsing procedure. The RDFS datatypes xsd : double, xsd : integer, and xsd : string are e ciently and e ectively handled; nevertheless, the datatypes check is optional. The result of the parsing phase are Datalog-like facts; with all the pre xed names expanded, predicate names encoded using IRIs and terms encoded using strings (and eventually integers). They are then stored in the owl2dlv data structures handled by the Data Manager. Finally, it worth noting that a scalability test over the LUBM dataset [ 31 ] shows that this module evolves linearly both in time and memory. Concerning the TBox, owl2dlv supports OWL 2 Horn-SHIQ ontologies encoded in RDF/XML. The input is parsed by the TBox Loader using the well-known OWL API [ 32 ] and loaded in DL-like data structures that are suitable for the purposes of Algorithm 1. During the parsing, every range restriction on one of the aforementioned datatype properties is also kept in a suitable data structure, later exploited by other modules. Finally, the system supports a set of SPARQL conjunctive queries via the Query Loader. Also in this case, the input is parsed via the OWL API.

Rewriting. It is responsible to implement Algorithm 1 via the Datalog Rewriter submodule and optimize |as described in Section 4 by applying the pruning strategy and the Magic Sets rewriting| its output (Qq;T (x); PT ) via the Optimizer. The Datalog Rewriter is also in charge of producing the datatypes directives: for each data property enforcing a datatype d in the range of a role r, the directive #datatypes( r/2 ,f2:dg) is added to PT . This syntax is inherited from dlv2, where r/2 says that r is of arity 2, and 2:d that the second argument of r must be of type d. Moreover, for each range assertion for a role r2 over a datatype d and for each role inclusion r1 v r2, a range assertion for r1 over d is added to PT , until a xpoint is reached.

Query Answering. This module, which is in charge of answering to Datalog queries over the input ABox, consists of three submodules. The Data Manager handles the extensional part of each input predicate and organizes it e ciently by means of indexed data structures; as data may undergo through updates, the module is also responsible for managing data additions and deletions. The Query Manager handles query rewritings, along with meta-information about the scheduling, guiding the evaluation of each input query. These two submodules may need to exchange info: the Data Manager may ask for information about the structure of the Datalog queries to single out an optimal indexing policy, while the Query Manager may need statistics about data distribution to de ne an optimal scheduling. Query answering is carried out by the Datalog Engine, which represents an extension of i-dlv [ 21 ] (the grounder of dlv2). The overall evaluation procedure is based on a bottom-up process based on a semi-nave approach empowered with optimizations working in synergy [ 21,22 ] and extended via techniques speci cally devised to manage e ciently large sizes of data as we brie y summarize below. The evaluation process has been endowed with a fruitful memory-releasing policy that, on the basis of structural information over the program at hand, anticipates the release of memory occupied by internal data structures as soon as these are no longer needed. Speci cally, data structures intended to represent the intensional part of a predicate p, are removed once all rules depending on p have been fully grounded. Moreover, we devised an optimized data retrieval strategy to reduce both the memory needed to store data and the time required to retrieve them. In particular, performance improvements have been achieved by re-implementing internal data structures and by optimizing crucial points of the retrieval task that, when frequently executed on large sizes, may negatively a ect the performance. The system features also a persistence mechanism allowing to serialize input data handled by the Data Manager on the disk in order to enable a faster reloading of input data. 6

OWL2DLV: Functionalities and Application Scenarios

When dealing with query answering in large-scale contexts, it is fairly common that the system at hand is required to repeatedly query on demand a certain KB where the ABox slowly changes over time by preserving, however, its structural properties. This features a number of scenarios that might signi cantly di er, depending on what is known before the reasoning starts. To maximize the performance, each scenario requires an appropriate query answering strategy, or setting, acting as a tuning mechanism that determines an e ective indexing schema, along with a compatible body-ordering (when possible) for all rules. This is performed by the Query Answering module during the so-called warmup phase, executed just after the loading phase has been completed. The resulting overhead, although not negligible, is paid only once. We now summarize some of the most common scenarios while assuming that an initial ABox A is known. Informed. Both the TBox T and some template queries (i.e., prototypical conjunctive queries where some arguments bounded by constants are marked in order to indicate that such constants might change at query time) are known. In this case, for each template query q(x), the system performs a preliminary run of opt(Qq;T (x)) over A [ opt(PT ). This step pre-computes indices and body orderings that will be of use when the system will be actually queried; roughly, it saves the choices that dlv2 would make in case of a \one-shot" execution. Responsive. The TBox T is known, while no information is available about the incoming queries. Here, a more general strategy is adopted: the system performs a single preliminary run over A [ PT (without any query) and, similarly to the previous case, it stores information about body ordering and indexing for all rules of PT that dlv2 would choose in a one-shot execution. Then, by predicting how PT is generally modi ed by the pruning and the Magic Sets rewriting, owl2dlv enriches the set of created indices accordingly.

Dynamic. Nothing is available. Clearly, no preliminary run for pre-computing body orderings can be performed. Depending on the memory availability, this drives the system to opt either for an aggressive indexing policy where all possible indices are computed over input data or a parsimonious (yet \blind") indexing policy where only the rst attribute of each predicate is indexed, while delegating to the query answering phase the creation of possibly needed further indices. This setting is also used as the default whenever the user tries to specify a setting not compatible with the actual scenario. 7

OWL2DLV: Performance

We report the results of an experimental evaluation of owl2dlv over LUBM [ 31 ] and DBpedia [ 14 ]. Note that, by focusing on the few ready-to-use OWL 2 reasoning services with a server-like behavior, neither mastro [ 24 ] nor ontop [ 23 ] nor rdfox [ 41 ] fully support query answering in both domains; in particular all of them do not process some of the axioms in the LUBM TBox. Further experiments with computationally intensive benchmarks, such as LUBM9 [ 40 ] and UOBM [ 37 ], will be part of an extended version of this paper. Moreover, a comparison against mastro, ontop and rdfox on lightweight ontologies as well as a comparison against modern Datalog-based systems like vadalog [ 18 ] and graal [ 16 ] for query existential rules [ 12,13 ] is also in our agenda. Benchmarks. LUBM is the prime choice of our industrial partner for the challenge. It describes a very-large real-world application domain encoded in OWL 2 Horn-SHIQ with customizable and repeatable synthetic data. The benchmark incorporates 14 SPARQL queries, 10 of which are bound (i.e., containing at least a constant). When rewritten together with the TBox, each LUBM query gives rise to a Datalog query consisting of about 130 rules. Data generation is carried out by the LUBM data generator tool (UBA) whose main parameter is the number of universities to consider: 8,000 in our case, for a total number of about 1 billion triples. This dataset is next referred to as LUBM-8000. Concerning DBpedia, it is a well-known KB created with the aim of sharing on the Web the multilingual knowledge collected by Wikimedia projects in a machine-readable format. For this benchmarks, we inherited a set of queries from an application conceived to query DBpedia in natural language applying the approach [ 27 ]. When rewritten together with the TBox, each DBpedia query gives rise to a Datalog query of almost 5400 rules. The latest release of the o cial DBpedia dataset consists of 13 billion pieces of multilingual information (RDF triples). Here, we focus on the Max Memory Loading Warmup Query Answering (all) Query Answering (bound) information extracted from the English edition of Wikipedia that is composed by about half a billion triples (https://wiki.dbpedia.org/public-sparql-endpoint). Results and Discussion. The machine used for testing is a Dell Linux server with an Intel Xeon Gold 6140 CPU composed of 8 physical CPUs clocked at 2.30 GHz, with 297GB of RAM. According to the challenge, a memory limit of 256GB has been set during all tests. Table 1 shows the results of our analysis; bound queries are reported in bold. The upper part of the table reports times needed by owl2dlv to answer queries under the three scenarios discussed in Section 6; the second part shows extra statistics about peaks of memory, loading and warmup times, and average answering times computed over all queries and over bound queries only. In the informed scenario {where we assume that the TBox and the template queries are known in advance{ we obtain the best performance. Despite the large ABox, on LUBM almost all bound queries are answered in less than 0:1 seconds with an average time of about 0:7 seconds, while on DBpedia the average evaluation time over all queries is about 0:2 seconds. These results con rm the e ectiveness of all enhancements herein discussed and of the Magic Sets technique. In the responsive scenario {where only the TBox is known in advance{ the system performance is comparable with the one obtained in the informed scenario, although in general the evaluation time is a bit higher. These results con rm that the warmup policy of this setting has a positive impact on the system performance although queries are not known. Finally, in the dynamic scenario {where nothing is known{ the parsimonious indexing policy is adopted since 256GB are not enough to use the aggressive one. Uniformly, the same policy is also used for DBpedia although not expressly needed. This produces a general gain in the warmup phase later unavoidably paid during the query evaluation due to some missing index that has to be computed on-the- y.

Acknowledgments. This work has been partially supported by Samsung under project \Enhancing the DLV system for large-scale ontology reasoning", by MISE under project \S2BDW" (F/050389/01-03/X32) { \Horizon2020" PON I&C2014-20 and by Regione Calabria under project \DLV LargeScale" (CUP J28C17000220006) { POR Calabria 2014-20.

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