Reasoning in the OWL 2 Full Ontology Language using First-Order Automated Theorem Proving

Reasoning in the O WL 2 F ull On tology Language using First-Order Automated Theorem Pro ving Mic hael Sc hneider 1 ? and Geoﬀ Sutcliﬀe 2 1 FZI Researc h Center for Information T ec hnology , Germany 2 Univ ersity of Miami, USA Abstract. OWL 2 has been standardized by the W orld Wide W eb Con- sortium (W3C) as a family of ontology languages for the Seman tic W eb. The most expressive of these languages is OWL 2 F ull, but to date no reasoner has b een implemen ted for this language. Consistency and en- tailmen t c hecking are known to be undecidable for OWL 2 F ull. W e ha ve translated a large fragmen t of the OWL 2 F ull semantics in to ﬁrst- order logic, and used automated theorem pro ving systems to do reasoning based on this theory . The results are promising, and indicate that this approac h can b e applied in practice for eﬀectiv e OWL reasoning, b ey ond the capabilities of current Seman tic W eb reasoners. This is an extende d version of a pap er with the same title that has b een published at CADE 2011, LNAI 6803, pp. 446–460. The extended v ersion provides app endices with additional resources that were used in the rep orted ev aluation. Key w ords: Semantic W eb, OWL, First-order logic, A TP 1 In tro duction The W eb On tology Language OWL 2 [16] has b een standardized by the W orld Wide W eb Consortium (W3C) as a family of on tology languages for the Seman tic W eb. OWL 2 includes OWL 2 DL [10], the OWL 2 RL/RDF rules [9], as well as O WL 2 F ull [12]. The fo cus of this work is on reasoning in O WL 2 F ull, the most expressiv e of these languages. So far, O WL 2 F ull has largely been ignored b y the researc h comm unity , and no reasoner has been implemen ted for this language. O WL 2 F ull does not enforce an y of the n umerous syn tactic restrictions of the description logic-st yle language OWL 2 DL. Rather, OWL 2 F ull treats arbitrary RDF graphs [7] as v alid input on tologies, and can safely be used with w eakly structured RDF data as is typically found on the W eb. F urther, O WL 2 F ull pro vides for reasoning outside the scop e of O WL 2 DL and the OWL 2 RL/RDF rules, including sophisticated reasoning based on meta-mo deling. In addition, O WL 2 F ull is seman tically fully compatible with RDFS [5] and also with the O WL 2 RL/RDF rules, and there is even a strong semantic corresp ondence [12] ? P artially supported b y the pro jects SEALS (Europ ean Commission, EU-IST-2009- 238975) and THESEUS (German F ederal Ministry of Economics and T echnology , FK OIMQ07019). 2 Mic hael Schneider and Geoﬀ Sutcliﬀe with OWL 2 DL, roughly stating that any OWL 2 DL conclusion can be reﬂected in O WL 2 F ull. This makes OWL 2 F ull largely interoperable with the other O WL 2 languages, and allows an OWL 2 F ull reasoner to b e combined with most existing OWL reasoners to provide higher syn tactic ﬂexibilit y and semantic expressivit y in reasoning-enabled applications. Due to its combination of ﬂexibility and expressivity , OWL 2 F ull is compu- tationally undecidable with regard to consistency and en tailment c hecking [8]. While there cannot b e an y complete decision pro cedure for O WL 2 F ull, the question remains to what extent practical OWL 2 F ull reasoning is p ossible. This pap er presents the results of a series of exp erimen ts ab out reasoning in O WL 2 F ull using ﬁrst-order logic (FOL) theorem pro ving. A large fragmen t of the O WL 2 F ull semantics has b een translated in to a FOL theory , and auto- mated theorem proving (A TP) systems hav e b een used to do reasoning based on this theory . The primary fo cus of these exp erimen ts was on the question of what can b e achiev ed at all; a future study may shift the fo cus to eﬃciency asp ects. The basic idea used in this work is not new. An early application of this approac h to a preliminary version of RDF and a precursor of O WL was reported b y Fikes et al. [2]. That w ork fo cused on iden tifying technical problems in the original language sp eciﬁcations, rather than on practical reasoning. Hay es [4] pro vided fairly complete translations of RDF(S) and O WL 1 F ull into Common Logic, but did not rep ort on any reasoning exp erimen ts. This gap was ﬁlled b y Hawk e’s reasoner Surnia [3], whic h applied an A TP system to an FOL ax- iomatisation of O WL 1 F ull. F or unkno wn reasons, how ever, Surnia p erformed rather p oorly on reasoning tests [17]. Comparable studies ha ve b een carried out for A TP-based OWL DL reasoning, as for Ho olet [15], an OWL DL rea- soner implemented on top of a previous v ersion of the V ampire A TP system ( http://www.vprover.org ). The work of Horrocks and V oronko v [6] addresses reasoning ov er large ontologies, which is crucial for practical Semantic W eb rea- soning. Finally , [1] rep orts on some historic kno wledge represen tation systems using A TP for description logic-st yle reasoning, such as Krypton in the 1980s. All these previous eﬀorts are outdated, in that they refer to precursors of O WL 2 F ull, and app ear to ha ve b een discon tinued after publication. The w ork rep orted in this pap er refers to the current speciﬁcation of OWL 2 F ull, and mak es a more extensiv e exp erimen tal ev aluation of the FOL-based approach than any previous work. Several asp ects of OWL 2 F ull reasoning ha ve b een studied: the degree of language co verage of OWL 2 F ull; semantic conclusions that are c haracteristic sp eciﬁcally of OWL 2 F ull; reasoning on large data sets; and the ability of ﬁrst-order systems to detect non-en tailments and consisten t on tologies in OWL 2 F ull. The FOL-based results ha ve b een com p ared with the results of a selection of well-kno wn Semantic W eb reasoners, to determine whether the FOL-based approach is able to add signiﬁcan t v alue to the state-of- the-art in Seman tic W eb reasoning. This pap er is organized as follows: Section 2 provides an in tro duction to the tec hnologies used in this pap er. Section 3 describ es the FOL-based reasoning approac h. Section 4 describ es the ev aluation setting, including the test data, Reasoning in OWL 2 F ull using First-Order A TP 3 the reasoners, and the computers used in the exp erimen ts. The main part of the paper is Section 5, which presents the results of the exp erimen ts. Section 6 concludes, and giv es an outlo ok on p ossible future w ork. The app endices presen t the ra w result data underlying the ev aluation results (A); the complete test suite of “characteristic OWL 2 F ull conclusions” that has b een used in the ev aluation (B); and an example sho wing ho w RDF data and the semantics of O WL 2 F ull ha ve b een translated into the ﬁrst-order logic formalism (C). 2 Preliminaries 2.1 RDF and OWL 2 F ull O WL 2 F ull is speciﬁed as the language that uses the OWL 2 RDF-Based Se- man tics [12] to interpret arbitrary RDF graphs. RDF graphs are deﬁned by the RDF Abstract Syn tax [7]. The OWL 2 RDF-Based Semantics is deﬁned as a seman tic extension of the RDF Seman tics [5]. According to the RDF Abstract Syn tax, an RDF gr aph G is a set of RDF triples: G = { t 1 , . . . , t n } . Each RDF triple t is giv en as an ordered ternary tuple t = s p o of RDF no des . The RDF no des s , p , and o are called the subje ct , pr e dic ate , and obje ct of the triple t , respectively . Each RDF no de is either a URI , a (plain, language-tagged or t yp ed) liter al , or a blank no de . The RDF Semantics is deﬁned on top of the RDF Abstract Syntax as a mo del theory for arbitrary RDF graphs. F or an interpr etation I and a domain U , a URI denotes an individual in the domain, a literal denotes a concrete data v alue (also considered a domain element), and a blank no de is used as an existen tially quan- tiﬁed v ariable indicating the existence of some domain element. The meaning of a triple t = s p o is a truth v alue of the relationship h I ( s ) , I ( o ) i ∈ IEXT( I ( p )), where IEXT is a mapping from domain elements that are pr op erties to associated binary relations. The meaning of a graph G = { t 1 , . . . , t n } is a truth v alue deter- mined b y the conjunction of the meaning of all the triples, taking in to account the existential semanti cs of blank nodes occurring in G . If an RDF graph G is true under an in terpretation I , then I satisﬁes G . An RDF graph G is c onsistent if there is an interpretation I that satisﬁes G . An RDF graph G entails another RDF graph H if every interpretation I that satisﬁes G also satisﬁes H . Whether an in terpretation satisﬁes a given graph is primarily determined by a collection of mo del-theoretic semantic c onditions that constrain the mapping IEXT. There are diﬀerent sets of mo del-theoretic semantic conditions for the diﬀeren t semantics deﬁned b y the RDF Semantics sp eciﬁcation. F or example, the semantics of class subsumption in RDFS is deﬁned mainly by the seman tic condition deﬁned for the RDFS vocabulary term rdfs:subClassOf : h c, d i ∈ IEXT( I ( rdfs:subClassOf )) ⇒ c, d ∈ I C ∧ ICEXT( c ) ⊆ ICEXT( d ) where “ c ” and “ d ” are universally quantiﬁed v ariables. Analogous to the map- ping IEXT, the mapping ICEXT asso ciates classes with subsets of the domain. The t wo mappings are resp onsible for the metamo deling c ap abilities of RDFS 4 Mic hael Schneider and Geoﬀ Sutcliﬀe and its seman tic extensions: Although the quantiﬁers in the RDFS semantic conditions range ov er exclusively domain elemen ts, whic h keeps RDFS in the realm of ﬁrst-order logic, the asso ciations provided by the tw o mappings allow domain elemen ts (prop erties and classes) to indirectly refer to sets and binary relations. This enables a limited but useful form of higher order-style mo deling and reasoning. The OWL 2 RDF-Base d Semantics , i.e. the seman tics of O WL 2 F ull, extends the RDF Seman tics speciﬁcation b y additional seman tic conditions for the O WL- sp eciﬁc vocabulary terms, such as owl:unionOf and owl:disjointWith . 2.2 F OL, the TPTP language, and A TP The translation of the O WL 2 F ull semantics is to classical unt yp ed ﬁrst-order logic. The concrete syn tax is the TPTP language [14], which is the de facto stan- dard for state-of-the-art A TP systems for ﬁrst-order logic. The A TP systems used in the ev aluation w ere taken from their web sites (see Section 4.3) or from the arc hives of the 5th IJCAR A TP System Competition, CASC-J5 ( http://www. tptp.org/CASC/J5/ ). Most of the systems are also av ailable online as part of the SystemOnTPTP service ( http://www.tptp.org/cgi- bin/SystemOnTPTP/ ). 3 Approac h Eac h of the mo del-theoretic seman tic conditions of the O WL 2 F ull semantics is translated in to a corresp onding F OL axiom. The result is an axiomatization of OWL 2 F ull. The RDF graphs to reason ab out are also conv erted into F OL form ulae. In the case of c onsistency che cking there is a single RDF graph that is conv erted in to a FOL axiom, for which satisﬁability needs to b e c heck ed. In the case of entailment che cking , there is a premise graph that is conv erted in to a F OL axiom, and a conclusion graph that is conv erted in to a F OL conjecture. The F OL form ulae (those representing the input RDF graphs and those building the FOL axiomatization of the OWL 2 F ull seman tics) are passed to an A TP system, which tries to prov e the conclusion or establish consistency . W e apply a straight-forw ard tr anslation of the semantic c onditions , making use of the fact that all seman tic conditions ha ve the form of F OL form ulae. A se- man tic relationship of the form “ h s, o i ∈ IEXT( p )” that app ears within a seman- tic condition is conv erted into an atomic F OL formula of the form “iext( p, s, o )”. Lik ewise, a relationship “ x ∈ ICEXT( c )” is conv erted in to “icext( c, x )”. Apart from this, the basic logical structure of the semantic conditions is retained. F or example, the seman tic condition specifying RDFS class subsumption sho wn in Section 2.1 is translated in to ∀ c, d : [ iext( rdfs:subClassOf , c, d ) ⇒ ( ic( c ) ∧ ic( d ) ∧ ∀ x : (icext( c, x ) ⇒ icext( d, x )) ) ] The tr anslation of RDF gr aphs amounts to conv erting the set of triples “ s p o ” in to a conjunction of corresp onding “iext( p, s, o )” atoms. A URI o ccurring in an Reasoning in OWL 2 F ull using First-Order A TP 5 RDF graph is conv erted into a constan t. An RDF liter al is con v erted into a func- tion term, with a constant for the literal’s lexical form as one of its argumen ts. Diﬀeren t functions are used for the diﬀeren t kinds of literals: function terms for plain liter als ha ve arit y 1; function terms for language-tagge d liter als hav e a con- stan t representing the language tag as their second argument; function terms for typ e d liter als ha ve a constant for the datat yp e URI as their second argumen t. F or each blank no de , an existen tially quantiﬁed v ariable is introduced, and the scop e of the corresp onding existential quan tiﬁer is the whole conjunction of the “iext” atoms. F or example, the RDF graph :x rdf:type foaf:Person . :x foaf:name "Alice"^^xsd:string . whic h con tains the blank no de “ :x ”, the typed literal “ "Alice"^^xsd:string ”, and the URIs “ rdf:type ”, “ foaf:Person ”, and “ foaf:name ”, is translated into the FOL formula ∃ x : [ iext( rdf:type , x, foaf:Person ) ∧ iext( foaf:name , x, literal typed ( Alice , xsd:string )) ] 4 Ev aluation Setting This section describ es the ev aluation setting: the OWL 2 F ull axiomatization, the test cases, the reasoners, and the computing resources. Supplementary mate- rial including the axiomatizations, test data, raw results, and the soft ware used for this pap er can b e found online at: http://www.fzi.de/downloads/ipe/schneid/cade2011- schneidsut- owlfullatp.zip . 4.1 The F OL Axiomatization and RDF Graph Conv ersion F ollowing the approach describ ed in Section 3, most of the normativ e semantic conditions of the OWL 2 F ull semantics hav e b een con verted into the correspond- ing FOL axioms, using the TPTP language [14]. The main omission is that most of the seman tics concerning r e asoning on datatyp es has not b een treated, as we w ere only interested in ev aluating the “logical core” of the language. All other language features of OWL 2 F ull were cov ered in their full form, with a restriction that was suﬃcient for our tests: while O WL 2 F ull has many size-parameterized language features, for example the intersection of arbitrarily many classes, our axiomatization generally supp orts these language feature sc hemes only up to a size of 3. The resulting F OL axiomatization consists of 558 formulae. The ax- iom set is fully ﬁrst-order with equality , but equality accounts for less than 10% of the atoms. The ﬁrst-order A TP systems used (see Section 4.3) conv ert the form ulae to clause normal form. The resultan t clause set is non-Horn. Almost all the clauses are range-restricted, which can result in reasoning that pro duces mostly ground clauses. In addition, a con verter from RDF graphs to F OL form ulae was implemented. This allow ed the use of RDF-enco ded OWL test data in the experiments, without time consuming and error prone manual conv ersion. 6 Mic hael Schneider and Geoﬀ Sutcliﬀe 4.2 T est Data Tw o complementary test suites were used for the exp erimen ts: one test suite to ev aluate the degree of language cov erage of O WL 2 F ull, and another suite consisting of characteristic conclusions for OWL 2 F ull reasoning. F or scalability exp erimen ts a large set of RDF data was also used. The Language Cov erage T est Suite. F or the language cov erage exp erimen ts, the test suite describ ed in [13] was used. 3 The test suite was created sp eciﬁcally as a conformance test suite for O WL 2 F ull and its main sub languages, including RDFS and the OWL 2 RL/RDF rules. The test suite consists of one or more test cases for each of the semantic conditions of the OWL 2 RDF-Based Semantics, i.e., the test suite pro vides a systematic co verage of O WL 2 F ull at a speciﬁcation lev el. Most of the test cases are p ositiv e entailmen t and inconsistency tests, but there are also a few negative entailmen t tests and p ositiv e consistency tests. The complete test suite consists of 736 test cases. A large fraction of the test suite deals with datat yp e reasoning. As the FOL axiomatization has almost no supp ort for datatype reasoning, only the test cases that co ver the “logical core” of OWL 2 F ull w ere used. F urther, only the p ositiv e entailmen t and inconsistency tests were used. The resultant test suite has 411 test cases. O WL 2 F ull-characteristic T est Cases. In order to inv estigate the extent of the reasoning p ossible using the F OL axiomatization, a set of test cases that are c haracteristic conclusions of OWL 2 F ull w as created. “Characteristic” means that the test cases represent OWL 2 F ull reasoning that cannot normally b e exp ected from either OWL 2 DL reasoning or from reasoners implementing the OWL 2 RL/RDF rules. The test suite consists of 32 tests, with 28 en- tailmen t tests and 4 inconsistency tests. There are test cases probing semantic consequences from meta-modeling, annotation prop erties, the unrestricted use of complex prop erties, and consequences from the use of OWL v o cabulary terms as regular en tities (sometimes called “syn tax reﬂection”). Bulk RDF Data. F or the scalabilit y exp erimen ts, a program that generates RDF graphs of arbitrary size (“bulk RDF data”) was written. The data consist of RDF triples using URIs that do not conﬂict with the URIs in the test cases. F urther, no O WL vocabulary terms are used in the data sets. This ensures that adding this bulk RDF data to test cases do es not aﬀect the reasoning results. 4.3 Reasoners This section lists the diﬀeren t reasoning systems that w ere used in the exp eri- men ts. The idea behind the selection w as to ha ve a small n umber of represen- 3 There is an oﬃcial W3C test suite for OWL 2 at http://owl.semanticweb.org/ page/OWL_2_Test_Cases (2011-02-09). How ever, it does not cov er OWL 2 F ull suﬃ- cien tly w ell, and w as not designed in a systematic wa y that allo ws easy determination of which parts of the language sp eciﬁcation are not supp orted by a reasoner. Reasoning in OWL 2 F ull using First-Order A TP 7 tativ e systems for (i) ﬁrst-order proving, (ii) ﬁrst-order mo del ﬁnding, and (iii) O WL reasoning. Details of the A TP systems can b e found on their web sites, and (for most) in the system descriptions on the CASC-J5 w eb site. The O WL reasoners were tested to pro vide comparisons with existing state of the art Se- man tic W eb reasoners. Unless explicitly stated otherwise, the systems were used in their default mo des. Systems for ﬁrst-order theorem proving – V ampire 0.6 ( http://www.vprover.org ). A p o werful sup erposition-based A TP system, including strategy scheduling. – V ampire-SInE 0.6 A v ariant of V ampire that alwa ys runs the SInE strat- egy ( http://www.cs.man.ac.uk/ ~ hoderk/sine/desc/ ) to select axioms that are exp ected to be relev ant. – iProv er-SInE 0.8 ( http://www.cs.man.ac.uk/ ~ korovink/iprover ). An instan tiation-based A TP system, using the SInE strategy , and including strategy scheduling. Systems for ﬁrst-order mo del ﬁnding – Parado x 4.0 ( http://www.cse.chalmers.se/ ~ koen/code/ ). A ﬁnite mo del ﬁnder, based on conv ersion to propositional form and the use of a SA T solver. – DarwinFM 1.4.5 ( http://goedel.cs.uiowa.edu/Darwin ). A ﬁnite mo del ﬁnder, based on con version to function-free ﬁrst-order logic and the use of the Darwin A TP system. Systems for OWL reasoning – Pellet 2.2.2 ( http://clarkparsia.com/pellet ). An OWL 2 DL reasoner that implements a tableaux-based decision procedure. – HermiT 1.3.2 ( http://hermit- reasoner.com ). An OWL 2 DL reasoner that implements a tableaux-based decision procedure. – F aCT++ 1.5.0 ( http://owl.man.ac.uk/factplusplus ). An O WL 2 DL reasoner that implemen ts a tableaux-based decision pro cedure. – BigOWLIM 3.4 ( http://www.ontotext.com/owlim ). An RDF en tailment- rule reasoner that comes with predeﬁned rule sets. The O WL 2 RL/RDF rule set ( owl2-rl ) w as used. The commercial “BigO WLIM” v ariant of the reasoning engine w as applied, b ecause it pro vides inconsistency chec king. – Jena 2.6.4 ( http://jena.sourceforge.net ). A Ja v a-based RDF frame- w ork that supp orts RDF en tailment-rule reasoning and comes with prede- ﬁned rule sets. The most expressive rule set, OWL MEM RULE INF , w as used. – Parliamen t 2.6.9 ( http://parliament.semwebcentral.org ). An RDF triple store with some limited OWL reasoning capabilities. P arliament can- not detect inconsistencies in on tologies. 8 Mic hael Schneider and Geoﬀ Sutcliﬀe 4.4 Ev aluation En vironment T esting was done on computers with a 2.8GHz Intel Pen tium 4 CPU, 2GB mem- ory , running Linux FC8. A 300s CPU time limit was imp osed on each run. 5 Ev aluation Results This section presen ts the results of the following reasoning exp erimen ts: a lan- guage c over age analysis , to determine the degree of conformance to the language sp eciﬁcation of OWL 2 F ull; “char acteristic” OWL 2 F ul l r e asoning exp erimen ts to determine the extent to whic h distinguishing OWL 2 F ull reasoning is p ossible; some basic sc alability testing ; and several mo del ﬁnding exp eriments to deter- mine whether ﬁrst-order mo del ﬁnders can b e used in practice for the recognition of non-entailmen ts and consistent ontologies. The follo wing mark ers are used in the result tables to indicate the outcomes of the exp erimen ts: – suc c ess (‘ + ’) : a test run that pro vided the correct result. – wr ong (‘ − ’) : a test run that provided a wrong result, e.g., when a reasoner claims that an entailmen t test case is a non-en tailment. – unknown (‘ ? ’) : a test run that did not provide a result, e.g., due to a pro cessing error or time out. This section also presents comparative ev aluation results for the OWL rea- soners listed in Section 4.3. This illustrates the degree to which OWL 2 F ull reasoning can already b e achiev ed with existing O WL reasoners, and the added v alue of our reasoning approach compared to existing Semantic W eb technology . This means, for example, that an OWL 2 DL reasoner will pro duce a wrong result if it classiﬁes an O WL 2 F ull entailmen t test case as a non-en tailment. Ho wev er, this negative ev aluation result refers to only the lev el of conformance with resp ect to O WL 2 F ull reasoning, i.e., the reasoner may still b e a complian t implemen tation of OWL 2 DL. 5.1 Language Co v erage This exp erimen t used the FOL axiomatization with the 411 test cases in the language co verage suite describ ed in Section 4.2. The results of the exp erimen t are shown in T able 1. iProv er-SInE succeeded on 93% of the test cases, and V ampire succeeded on 85%. It needs to b e men tioned that the results w ere not p erfectly stable. Ov er several runs the num b er of successes v aried for iProv er- SInE b et ween 382 and 386. This is caused b y small v ariations in the timing of strategy changes within iPro ver-SInE’s strategy scheduling. Figure 1 shows the run time b eha vior of the tw o systems, with the times for successes sorted into increasing order. Both systems take less than 1s for the ma jorit y of their successes. Although V ampire succeeded on less cases than iPro ver-SInE, it is typically faster in the case of a success. Reasoning in OWL 2 F ull using First-Order A TP 9 0.01 0.1 1 10 100 50 100 150 200 250 300 350 400 CPU time in seconds (log scale) Solution number Vampire iProver-SInE Fig. 1. Language cov erage: ordered system times of A TPs. An analysis of the 28 test cases for which both V ampire and iPro ver-SInE did not succeed revealed that 14 of them require supp ort for O WL 2 F ull lan- guage features not cov ered b y the F OL axiomatization, including certain forms of datat yp e reasoning and support for the RDF con tainer v o cabulary [5]. A fu- ture v ersion of the axiomatization will enco de these parts of the OWL 2 F ull seman tics, whic h migh t lead to improv ed results. F or eac h of the remaining 14 test cases, subsets of axioms suﬃcient for a solution were hand-selected from the FOL axiomatization. These axiom sets were generally very small, with up to 16 axioms, and in most cases less than 10 axioms. iProv er-SInE succeeded on 13 of these 14 test cases. The remaining test case is a considerably complex one, inv olving the semantics of qualiﬁed cardinalit y restrictions. It w as solv ed b y V ampire. Thus, all test cases were solv ed except for the 14 that are beyond the current axiomatization. F or comparison, the OWL reasoners listed in Section 4.3 were also tested. The results are sho wn in T able 2. The OWL 2 DL reasoners P ellet and HermiT b oth succeeded on about 60% of the test cases. A comparison of the individual results show ed that the tw o reasoners succeeded mostly on the same test cases. In terestingly , although most of the test cases are formally in v alid O WL 2 DL on tologies, reasoning rarely resulted in a pro cessing error. Rather, in ca. 40% of the cases, the reasoners wrongly rep orted a test case to b e a non-en tailment or a consisten t ontology . The third O WL 2 DL reasoner, F aCT++, signaled a pro cessing error more often, and succeeded on less than 50% of the test cases. The OWL 2 RL/RDF rule reasoner BigO WLIM succeeded on roughly 70% of the test cases. Although the num b er of successful tests w as larger than for Reasoner Success W rong Unkno wn V ampire 349 0 62 iPro ver-SInE 383 0 28 T able 1. Language cov erage: A TPs with OWL 2 F ull axiom set. 10 Mic hael Schneider and Geoﬀ Sutcliﬀe Reasoner Success W rong Unkno wn P ellet 237 168 6 HermiT 246 157 8 F aCT++ 190 45 176 BigO WLIM 282 129 0 Jena 129 282 0 P arliament 14 373 24 T able 2. Language cov erage: OWL reasoners. 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 PE + + + − − − − − + + − − − − + − − − − + + − − − − + − − ? − − − HE + ? + − − ? − + + + − − − − + − − − − + + − − + ? + − − ? − − − F A + ? ? ? ? ? ? − ? + − − − ? + ? − − − + + ? ? ? ? + − ? ? − − ? BO + − − + − − + + − − + + − − + − − + + − − − − − − − − − − − − − JE + − − − − + + + − − + − − − − − − + − − − − + − − + − − − − − + P A + − − − − − − + − − ? − − − − − − − ? − − − − − − − − − − ? ? − T able 3. Characteristic conclusions: O WL reasoners. PE=P ellet, HE=HermiT, F A=F aCT++, BO=BigOWLIM, JE=Jena, P A=Parliamen t. all the O WL 2 DL reasoners, there was a considerable num b er of test cases for whic h the OWL 2 DL reasoners w ere successful but not BigOWLIM, and vice v ersa. The Jena OWL reasoner, which is an RDF entailmen t rule reasoner like BigO WLIM, succeeded on ab out only 30% of the test cases, which is largely due to missing supp ort for OWL 2 features. Finally , P arliament succeeded on only 14 of the test cases. In particular, it did not solv e any of the inconsistency test cases. The lo w success rate reﬂects the style of “light-w eight reasoning” used in man y reasoning-enabled RDF triple stores. 5.2 Characteristic O WL 2 F ull Conclusions The test suite of characteristic OWL 2 F ull conclusions fo cuses on seman tic consequences that are t ypically b ey ond the scop e of O WL 2 DL or RDF rule reasoners. This is reﬂected in T able 3, whic h presen ts the results for the OWL reasoning systems. The column num b ers corresp ond to the test case num b ers in the test suite. In general, the OWL reasoners sho w signiﬁcantly w eaker perfor- mance on this test suite than on the language cov erage test suite. Note that the successful test cases for the O WL 2 DL reasoners (P ellet, HermiT and F aCT++) ha ve only little ov erlap with the successful test cases for the RDF rule reasoners (BigO WLIM and Jena). P arliament succeeded on only t wo test cases. The ﬁrst tw o rows of T able 4 show that the A TP systems achiev ed muc h b etter results than the O WL reasoners, using the complete O WL 2 F ull ax- iomatization. iProv er-SInE succeeded on 28 of the 32 test cases, and V ampire succeeded on 23. As was done for the language co verage test cases, small sub- sets of axioms suﬃcient for each of the test cases w ere hand-selected from the Reasoning in OWL 2 F ull using First-Order A TP 11 0.01 0.1 1 10 100 5 10 15 20 25 30 CPU time in seconds (log scale) Solution number Vampire Small iProver-SInE Small Vampire Complete iProver-SInE Complete Fig. 2. Characteristic conclusions: ordered system times of A TPs. F OL axiomatization. As the last tw o rows of T able 4 show, both A TP systems succeeded on all these simpler test cases. 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 V A/c + + + + + + + + + ? + ? ? + + + + + + ? ? ? + + ? + ? ? + + + + IS/c + + + + + + + + + + + ? ? + + + + + + ? ? + + + + + + + + + + + V A/s + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + IS/s + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + T able 4. Characteristic conclusions: A TPs with complete and small axiom sets. V A/c=V ampire/complete, IS/c=iProv er-SInE/complete, V A/s=V ampire/small, IS/s=iPro ver-SInE/small. Figure 2 sho ws the runtime b eha vior of the t wo systems. F or the complete axiomatization, V ampire either succeeds in less than 1s or does not succeed. In con trast, iProv er’s p erformance degrades more gracefully . The reasoning times using the small-suﬃcient axiom sets are generally up to several magnitudes lo w er than for the complete axiomatization. In the ma jority of cases they are below 1s. 5.3 Scalabilit y The Semantic W eb consists of huge data masses, but single reasoning results presumably often dep end on only a small fraction of that data. As a basic test of the A TP systems’ abilities to ignore irrelev ant bac kground axioms, a set of one million “bulk RDF axioms” (as describ ed in Section 4.2) was added to the test cases of characteristic O WL 2 F ull conclusions. This was done using the complete F OL axiomatization, and also the small-suﬃcien t sets of axioms for eac h test case. T able 5 sho ws the results. The default version of V ampire pro duced v ery p oor results, as is sho wn in the ﬁrst and fourth rows of the table. (Strangely , V ampire had tw o more successes with the complete axiomatization than with the small-suﬃcien t axiom sets. That can be attributed to diﬀerences in the strategies selected for the diﬀeren t axiomatizations.) In contrast, as shown in the second, 12 Mic hael Schneider and Geoﬀ Sutcliﬀe 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 V A/c + + + ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? VS/c + + + + + + ? + ? ? + ? ? ? + + ? + + ? ? ? + ? ? + ? ? ? + ? + IS/c + + + + + + + + + + + ? ? + + + + + + ? ? + + + + + + + + + + + V A/s + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? VS/s + + + + + + ? + + + + + ? + + + ? + + ? ? + + + + + + ? + + ? + IS/s + + + + + + + + + + + + ? + + + + + + + + + + + + + + + + + + + T able 5. Scalabilit y: A TPs with complete and small axiom sets, 1M RDF triples. V A/c=V ampire/complete, VS/c=V ampire-SInE/complete, IS/c=iPro ver-SInE/complete, V A/s=V ampire/small, VS/s=V ampire-SInE/small, IS/s=iPro ver-SInE/small. 10 100 5 10 15 20 25 30 CPU time in seconds (log scale) Solution number Vampire Small Vampire-SInE Small iProver-SInE Small Vampire Complete Vampire-SInE Complete iProver-SInE Complete Fig. 3. Scalability: ordered system times of A TPs, 1M RDF triples. third, ﬁfth and sixth ro ws, the v ersion of V ampire-SInE and iPro ver-SInE did m uch better. The use of the SInE strategy for selecting relev ant axioms clearly helps. Figure 3 sho ws the runtime behavior of the systems. The bulk axioms evi- den tly add a constant o verhead of about 20s to all successes, whic h is b eliev ed to b e taken parsing the large ﬁles. In an application setting this migh t b e done only once at the start, so that the time w ould b e amortized o ver m ultiple reasoning tasks. The step in iProv er’s p erformance at the 20th problem is an artifact of strategy scheduling. The bulk axioms were designed to hav e no connection to the F OL axiomati- zation or the RDF graphs. As suc h, simple analysis of inference chains from the conjecture [11] would b e suﬃcient to determine that the bulk axioms could not b e used in a solution. This simplistic approach is metho dologically an appropri- ate w ay to start testing robustness against irrelev ant axioms, and potentially not to o far oﬀ the reality of Seman tic W eb reasoning. How ever, future work using axioms that are not so obviously redundant w ould prop erly exercise the p o wer of the SInE approach to axiom selection. 5.4 Mo del Finding This section presents the results from exp erimen ts concerning the detection of non-en tailments and consisten t ontologies w.r.t. O WL 2 F ull and tw o of its sub languages: ALCO F ull [8] and RDFS [5]. ALCO F ull is interesting b ecause it is Reasoning in OWL 2 F ull using First-Order A TP 13 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 P A/A + + + + + + ? + + + + + ? ? ? ? + ? ? ? + + ? + P A/R + + + + + + + + + + + + + + + + + + + + + + + + + + + + + DF/R + + + + + + + + + + + + + + + + + + + + + + + + + + + + + T able 6. Model Finding: A TPs with ALCO F ull and RDFS axiom sets. The black entries indicate p ositiv e en tailments or inconsisten t on tologies. P A/A=P aradox/ALCO F ull, P A/R=P aradox/RDFS, DF/R=DarwinFM/RDFS. a small fragmen t of OWL 2 F ull that is known to b e undecidable [8]. RDFS is in teresting b ecause it is a minimally meaningful language that shares the main c haracteristics of O WL 2 F ull. The RDFS axioms included the “extensional” seman tic extension, as non-normatively deﬁned in Section 4.2 of [5]. Similarly , the original deﬁnition of ALCO F ull was extended to include extensional RDFS. No report is given for the OWL reasoners, as only the OWL 2 DL reasoners ha ve mo del-ﬁnding capabilities, and not for any of the three languages considered here. Consistency c hecking for an RDF graph G w.r.t. some on tology language L corresp onds to consistency c hecking for the com bination of a complete axioma- tization of L and the FOL translation of G . Hence, a minimum requirement is to conﬁrm that the FOL axiomatization of O WL 2 F ull is consistent. Unfortu- nately , for the O WL 2 F ull axiomatization no mo del ﬁnder was able to conﬁrm consistency . 4 F or the ALCO F ull axioms, P aradox found a ﬁnite mo del of size 5 in ca. 5s CPU time, while DarwinFM timed out. Parado x w as then used on the charac- teristic OWL 2 F ull test cases, with the OWL 2 F ull axiomatization replaced b y the ALCO F ull axioms. As ALCO F ull is a sub language of OWL 2 F ull, 24 of the 32 test cases are either non-entailmen ts or consisten t ontologies, out of whic h 15 were correctly recognized by P aradox. iProv er-SInE w as used to conﬁrm that the remaining 8 test cases are p ositiv e entailmen ts or inconsisten t on tologies. The results are shown in the ﬁrst ro w of T able 6. F or the RDFS axioms, analogous experiments w ere done. Parado x found a ﬁnite mo del of the axioms, of size 1, in ab out 1s. The consistency was conﬁrmed b y DarwinFM in less than 1s. With the O WL 2 F ull axiomatization replaced b y the RDFS axioms, 29 of the 32 characteristic test cases are non-en tailments or consistent ontologies. Parado x and Darwin conﬁrmed all of these, mostly in ca. 1s, with a maximum time of ca. 2s. iProv er-SInE conﬁrmed that the remaining 3 test cases are p ositiv e entailmen ts or inconsistent ontologies. These results are sho wn in the second and third rows of T able 6. 4 This raised the question of whether our p ositiv e entailmen t reasoning results were p erhaps due to an inconsisten t axiomatization. How ever, none of the theorem prov ers w as able to establish inconsistency . In addition, the model ﬁnders conﬁrmed the consistency of all the small-suﬃcient axiom sets mentioned in Section 5.2. Hence, it is at least ensured that those p ositiv e reasoning results are achiev able from consistent subsets of the OWL 2 F ull axiomatization. 14 Mic hael Schneider and Geoﬀ Sutcliﬀe An interesting observ ation made during the model ﬁnding exp erimen ts was that ﬁnite mo del ﬁnders were eﬀective, e.g., the results of Parado x and Dar- winFM ab o ve. In contrast, other model ﬁnders such as iProv er-SA T (a v ariant of iProv er tuned for mo del ﬁnding) and Darwin (the plain Mo del Evolution core of DarwinFM) w ere less eﬀectiv e, e.g., taking 80s and 37s resp ectiv ely to conﬁrm the satisﬁability of the RDFS axiom set. 6 Conclusions and F uture W ork This pap er has describ ed ho w ﬁrst order A TP systems can b e used for reasoning in the O WL 2 F ull ontology language, using a straight-forw ard translation of the underlying mo del theory in to a FOL axiomatization. The results were obtained from tw o complementary test suites, one for language cov erage analysis and one for probing characteristic conclusions of OWL 2 F ull. The results indicate that this approach can b e applied in practice for eﬀective OWL reasoning, and oﬀers a viable alternativ e to current Semantic W eb reasoners. Some scalability testing w as done by adding large sets of seman tically unrelated RDF data to the test case data. While the A TP systems that include the SInE strategy eﬀectively ignored this redundan t data, it was surprising that other A TP systems did not use simple reachabilit y analysis to detect and ignore this bulk data – this suggests an easy w ay for dev elop ers to adapt their systems to such problems. In contrast to the successes of the A TP systems proving theorems, model ﬁnders were less successful in identifying non-entailmen ts and consistent ontolo- gies w.r.t. OWL 2 F ull. Ho wev er, some successes w ere obtained for ALCO F ull. Since ALCO F ull is an undecidable sub-language of OWL 2 F ull, there is hope that the failures were not due to undecidabilit y but rather due to the large num- b er of axioms. This needs to b e inv estigated further. Mo del ﬁnding for RDFS w orked quite eﬃciently , which is interesting b ecause we do not know of any to ol that detects RDFS non-entailmen ts. In the future we plan to extend the approach to datatype reasoning, which is of high practical relev ance in the Semantic W eb. It may b e p ossible to take adv an- tage of the t yp ed ﬁrst-order or t yp ed higher-order form of the TPTP language to eﬀectiv ely encode the datatypes, and reason using A TP systems that take adv an- tage of the type information. Another topic for further research is to dev elop tech- niques for identifying parts of the FOL axiomatization that are relev ant to a giv en reasoning task. It is hop ed that by taking in to account OWL 2 F ull sp eciﬁc kno wl- edge, more precise axiom selection than oﬀered b y the generic SInE approac h will b e possible. An imp ortan t area of developmen t will b e query answering, i.e., the ability to obtain explicit answers to users’ questions. F or future O WL 2 F ull reasoners this will b e a very relev ant reasoning task, particularly with resp ect to the curren t extension of the standard RDF query language SP ARQL to- w ards “entailmen t regimes” ( http://www.w3.org/TR/sparql11- entailment ). This topic is also of gro wing in terest in the A TP comm unity , with a prop osal b eing considered for expressing questions and answers in the TPTP language ( http://www.tptp.org/TPTP/Proposals/AnswerExtraction.html ). Reasoning in OWL 2 F ull using First-Order A TP 15 References 1. Baader, F., McGuinness, D.L., Nardi, D., Patel-Sc hneider, P .F. (eds.): The De- scription Logic Handbo ok: Theory , Implementation, and Applications. Cambridge Univ ersity Press, Cambridge (2003) 2. Fikes, R., McGuinness, D., W aldinger, R.: A First-Order Logic Semantics for Se- man tic W eb Markup Languages. T ech. Rep. KSL-02-01, Knowledge Systems Lab- oratory , Stanford Universit y , Stanford, CA 94305 (January 2002) 3. Hawk e, S.: Surnia. 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Pro ceedings of the 6th In ternational W orkshop on O WL: Ex- p eriences and Directions (OWLED 2009). CEUR W orkshop Pro ceedings, vol. 529 (2009), access: http://ceur- ws.org/Vol- 529/owled2009_submission_19.pdf 14. Sutcliﬀe, G.: The TPTP Problem Library and Asso ciated Infrastructure. The FOF and CNF Parts, v3.5.0. Journal of Automated Reasoning 43(4), 337–362 (2009) 15. Tsarko v, D., Riazanov, A., Bechhofer, S., Horrocks, I.: Using V ampire to Reason with OWL. In: Pro ceedings of the Third In ternational Seman tic W eb Conference (ISW C 2004). LNCS, v ol. 3298, pp. 471–485. Springer (2004) 16. W3C O WL W orking Group (ed.): OWL 2 W eb Ontology Language: Do cumen t Ov erview. W3C Recommendation (27 October 2009), access: http://www.w3.org/ TR/owl2- overview/ 17. W3C W ebOnt OWL W orking Group: OWL 1 T est Results (9 March 2004), http: //www.w3.org/2003/08/owl- systems/test- results- out 16 Mic hael Schneider and Geoﬀ Sutcliﬀe A Detailed Raw Result Data This app endix provides detailed raw result data that underlies the exp erimen tal results rep orted in Section 5. The data is given in tables, which presen t the names of the test cases in the ﬁrst column, and the results of individual reasoning exp erimen ts in the other columns. The result of each exp erimen t consists of one of the p ossible outcomes deﬁned at the b eginning of Section 5 and, optionally , the duration of the reasoning exp erimen t, giv en as the n umber of seconds it to ok. All result data presented here is also av ailable in electronic form as part of the supplementary material for this pap er (see the do wnload link at the beginning of Section 4). A.1 Language Co v erage Results The following tables pro vide the raw result data that underlies the aggregated results for the language co verage exp erimen ts, as rep orted in Section 5.1. T able 7 con tains the result data that w as obtained from ev aluating the Semantic W eb reasoners and from the FOL theorem prov ers when used with the complete O WL 2 F ull axiomatization; the corresp onding aggregated results were reported in T ables 2 and 1, resp ectiv ely . The remaining tables provide the raw results from testing the F OL reasoners on those test cases where they had failed originally , no w using small but suﬃcient subaxiomatizations that were manually crafted for each of the test cases. T able 7: Result data of the language coverage exp erimen ts for Semantic W eb reasoners and for FOL theorem prov ers when used with the complete OWL 2 F ull axiomatization. Time v al- ues ha ve only b e measured for the FOL reasoners. PE=Pellet, HE=HermiT, F A=F aCT++, BO=BigOWLIM, JE=Jena, P A=Parliament, V A=V ampire, IS=iProv er-SInE T est Case PE HE F A BO JE P A V A IS rdfbased-sem-bo ol-complemen t-data - - ? - - - + (0.07) + (0.16) rdfbased-sem-bo ol-complemen t-ext + + + - - - + (0.33) + (0.61) rdfbased-sem-bo ol-complemen t-inst + + + + - ? + (0.28) + (0.29) rdfbased-sem-bo ol-demorgan + + + - - - ? (300.00) ? (300.00) rdfbased-sem-bo ol-in tersection-data-lo calize + + ? - - - + (0.60) + (38.46) rdfbased-sem-bo ol-in tersection-ext + + + - - - ? (300.00) ? (300.00) rdfbased-sem-bo ol-in tersection-inst-comp + + + + + - + (0.83) + (69.73) rdfbased-sem-bo ol-in tersection-inst-expr + + + + + - + (0.60) + (74.89) rdfbased-sem-bo ol-in tersection-lo calize + + + + + - + (0.87) + (73.54) rdfbased-sem-bo ol-in tersection-term + + + + + - + (1.07) + (81.20) rdfbased-sem-bo ol-tollens + + + - - - + (0.41) + (0.57) rdfbased-sem-bo ol-union-data-localize + + ? - - - + (0.54) + (79.34) rdfbased-sem-bo ol-union-ext + + + - - - ? (300.00) ? (300.00) rdfbased-sem-bo ol-union-inst-comp + + + + + - + (0.71) + (7.52) rdfbased-sem-bo ol-union-inst-expr + + + - - - ? (300.00) + (74.41) rdfbased-sem-bo ol-union-localize + + + + + - + (0.85) + (79.08) rdfbased-sem-bo ol-union-term + + + + + - + (1.10) + (82.89) rdfbased-sem-chain-def + + + + - - + (0.50) + (75.35) rdfbased-sem-chain-ext ? + + - - - ? (300.00) ? (300.00) rdfbased-sem-chain-localize + + + - - - + (0.50) + (31.96) rdfbased-sem-chain-subprop ? + + - - - ? (300.00) + (7.58) rdfbased-sem-char-asymmetric-ext + + + - - - + (0.36) + (108.43) rdfbased-sem-char-asymmetric-inst + + + + - ? + (0.19) + (0.09) rdfbased-sem-char-asymmetric-term + + + + - ? + (0.24) + (0.09) rdfbased-sem-char-functional-ext + + + - + - + (0.36) + (108.24) rdfbased-sem-char-functional-inst + + + + + + + (0.37) + (0.40) rdfbased-sem-char-in versefunc-data + + - + + + + (0.38) + (0.40) rdfbased-sem-char-in versefunc-ext + + + - + - + (0.47) + (109.98) rdfbased-sem-char-in versefunc-inst + + + + + + + (0.41) + (0.39) rdfbased-sem-char-in versefunc-term + + + - + - + (1.08) + (6.65) rdfbased-sem-char-irreflexiv e-ext + + + - - - + (0.44) + (78.75) rdfbased-sem-char-irreflexiv e-inst + + + + - ? + (0.03) + (0.08) rdfbased-sem-char-irreflexiv e-term + + + - - ? + (0.03) + (0.17) rdfbased-sem-char-reflexiv e-ext - - ? - - - + (0.40) + (10.01) Reasoning in OWL 2 F ull using First-Order A TP 17 rdfbased-sem-char-reflexiv e-inst + + + - - - + (0.04) + (0.15) rdfbased-sem-char-symmetric-ext + + + - - - + (0.82) + (131.46) rdfbased-sem-char-symmetric-inst + + + + + + + (0.18) + (0.09) rdfbased-sem-char-transitiv e-ext + + + - - - ? (300.00) + (131.03) rdfbased-sem-char-transitiv e-inst + + + + + + + (0.39) + (0.11) rdfbased-sem-char-transitiv e-term - ? ? - - - + (0.41) + (2.30) rdfbased-sem-class-alldifferent-ext - - ? + - - + (0.03) + (0.16) rdfbased-sem-class-alldifferent-t yp e + + + + - - + (0.02) + (0.17) rdfbased-sem-class-alldisjointclasses-ext - - ? + - - + (0.03) + (0.10) rdfbased-sem-class-alldisjointclasses-t yp e + + + + - - + (0.02) + (0.11) rdfbased-sem-class-alldisjointproperties-ext - - ? + - - + (0.03) + (0.10) rdfbased-sem-class-alldisjointproperties-type + + + + - - + (0.02) + (0.10) rdfbased-sem-class-annotation-ext - - ? + - - + (0.03) + (0.08) rdfbased-sem-class-annotation-type + + + + - - + (0.03) + (0.08) rdfbased-sem-class-annotationproperty-type + + + + + - + (0.02) + (0.08) rdfbased-sem-class-asymmetricproperty-ext - - ? + - - + (0.37) + (0.18) rdfbased-sem-class-asymmetricproperty-type + + + + - - + (0.03) + (0.08) rdfbased-sem-class-axiom-ext - - ? + - - + (0.03) + (0.09) rdfbased-sem-class-axiom-type + + + + - - + (0.03) + (0.08) rdfbased-sem-class-datarange-ext - - ? + - - + (0.32) + (0.16) rdfbased-sem-class-datarange-type + + + + - - + (0.02) + (0.08) rdfbased-sem-class-datatype-ext - - ? + - - + (0.28) + (0.16) rdfbased-sem-class-datatype-type + + + + + - + (0.04) + (0.08) rdfbased-sem-class-datatypeproperty-type + + + + + - + (0.03) + (0.12) rdfbased-sem-class-deprecatedclass-ext - - ? + - - + (0.32) + (0.15) rdfbased-sem-class-deprecatedclass-type + + + + - - + (0.02) + (0.08) rdfbased-sem-class-deprecatedproperty-ext - - ? + - - + (0.41) + (0.18) rdfbased-sem-class-deprecatedproperty-type + + + + - - + (0.03) + (0.09) rdfbased-sem-class-functionalproperty-ext - - ? + + - + (0.34) + (0.42) rdfbased-sem-class-functionalproperty-type + + + + + - + (0.02) + (0.14) rdfbased-sem-class-inv ersefunctionalprop ert y-ext - - ? + + - + (0.41) + (0.39) rdfbased-sem-class-inv ersefunctionalprop ert y-type + + + + + - + (0.02) + (0.15) rdfbased-sem-class-irreflexiveproperty-ext - - ? + - - + (0.40) + (0.18) rdfbased-sem-class-irreflexiveproperty-t yp e + + + + - - + (0.03) + (0.09) rdfbased-sem-class-literal-type + + + + - - + (0.03) + (0.08) rdfbased-sem-class-namedindividual-ext - + ? + - - + (0.36) + (0.10) rdfbased-sem-class-namedindividual-type + + + + - - + (0.02) + (0.08) rdfbased-sem-class-negativeproperty assertion-ext - - ? + - - + (0.02) + (0.08) rdfbased-sem-class-negativeproperty assertion-type + + + + - - + (0.03) + (0.09) rdfbased-sem-class-nothing-ext + + + + + ? + (0.03) + (0.12) rdfbased-sem-class-nothing-term + + + + - - + (0.42) + (0.41) rdfbased-sem-class-nothing-type + + - + + - + (0.03) + (0.10) rdfbased-sem-class-ob jectproperty-ext - - ? + + - + (0.33) + (0.18) rdfbased-sem-class-ob jectproperty-t yp e + + + + + - + (0.03) + (0.08) rdfbased-sem-class-ontology-t yp e + + + + + - + (0.02) + (0.09) rdfbased-sem-class-ontologyproperty-t yp e + + + + + - + (0.02) + (0.09) rdfbased-sem-class-owlclass-ext - - ? + + - + (0.36) + (0.15) rdfbased-sem-class-owlclass-t yp e + + + + + - + (0.02) + (0.08) rdfbased-sem-class-property-ext - - ? + - - + (0.38) + (0.18) rdfbased-sem-class-property-type + + + + + - + (0.02) + (0.08) rdfbased-sem-class-rdfsclass-ext - - ? + - - + (0.42) + (0.16) rdfbased-sem-class-rdfsclass-type + + + + + - + (0.03) + (0.08) rdfbased-sem-class-reflexiveproperty-ext - - ? + - - + (0.38) + (0.16) rdfbased-sem-class-reflexiveproperty-t yp e + + + + - - + (0.02) + (0.09) rdfbased-sem-class-resource-ext - + ? + - - + (0.31) + (0.13) rdfbased-sem-class-resource-type + + + + + - + (0.02) + (0.08) rdfbased-sem-class-restriction-ext - - ? + + - + (0.40) + (0.24) rdfbased-sem-class-restriction-type + + + + + - + (0.02) + (0.16) rdfbased-sem-class-symmetricproperty-ext - - ? + + - + (0.44) + (0.19) rdfbased-sem-class-symmetricproperty-type + + + + + - + (0.02) + (0.08) rdfbased-sem-class-thing-ext - - ? - + - + (0.03) + (0.08) rdfbased-sem-class-thing-term + + + + + - + (0.35) + (0.22) rdfbased-sem-class-thing-type + + - + + - + (0.03) + (0.09) rdfbased-sem-class-transitiveproperty-ext - - ? + + - + (0.32) + (0.18) rdfbased-sem-class-transitiveproperty-t yp e + + + + + - + (0.03) + (0.08) rdfbased-sem-enum-data-localize + + - - - - + (0.56) + (108.48) rdfbased-sem-enum-ext + + + - - - ? (300.00) ? (300.00) rdfbased-sem-enum-inst-closed + + - - - - + (0.48) + (107.23) rdfbased-sem-enum-inst-included + + + + + - + (0.48) + (74.77) rdfbased-sem-eqdis-different-ext + + + - - - + (0.36) + (1.99) rdfbased-sem-eqdis-different-irrflxv - - - - - ? + (0.04) + (0.14) rdfbased-sem-eqdis-different-sameas + + + + + ? + (0.30) + (0.15) rdfbased-sem-eqdis-different-sym + + + - + - + (0.04) + (0.20) rdfbased-sem-eqdis-disclass-eqclass + + + + + ? + (0.34) + (0.42) rdfbased-sem-eqdis-disclass-ext + + + - - - + (0.38) + (110.98) rdfbased-sem-eqdis-disclass-inst + + + + + ? + (0.34) + (0.21) rdfbased-sem-eqdis-disclass-irrflxv - ? - + + ? + (0.06) + (0.20) rdfbased-sem-eqdis-disclass-sym + + + - + - + (0.41) + (0.09) rdfbased-sem-eqdis-disjointunion-composite ? + + - - - ? (300.00) + (136.00) rdfbased-sem-eqdis-disjointunion-disjoin t + + + - - - + (0.85) + (79.84) rdfbased-sem-eqdis-disjointunion-localize + + + - - - + (1.13) + (39.35) rdfbased-sem-eqdis-disjointunion-union + + + - - - ? (300.00) + (116.83) rdfbased-sem-eqdis-disprop-eqprop + + + + - ? + (0.34) + (0.09) rdfbased-sem-eqdis-disprop-ext + + + - - - + (0.49) + (108.55) 18 Mic hael Schneider and Geoﬀ Sutcliﬀe rdfbased-sem-eqdis-disprop-inst + + + + - ? + (0.31) + (0.07) rdfbased-sem-eqdis-disprop-irrflxv - - - + - ? + (0.27) + (0.08) rdfbased-sem-eqdis-disprop-sym + + + - - - + (0.41) + (0.09) rdfbased-sem-eqdis-eqclass-ext + + + - + - + (1.80) + (30.22) rdfbased-sem-eqdis-eqclass-inst + + + + + - + (0.41) + (0.28) rdfbased-sem-eqdis-eqclass-rflxv + + + + + - + (0.42) + (0.57) rdfbased-sem-eqdis-eqclass-subclass-1 + + + + + - + (0.41) + (0.68) rdfbased-sem-eqdis-eqclass-subclass-2 + + + + + - + (0.45) + (0.60) rdfbased-sem-eqdis-eqclass-subst + + + + - - + (0.44) + (3.19) rdfbased-sem-eqdis-eqclass-sym + + + + + - + (0.41) + (0.33) rdfbased-sem-eqdis-eqclass-trans + + + + + - + (0.41) + (1.89) rdfbased-sem-eqdis-eqprop-ext + + + - - - ? (300.00) + (116.80) rdfbased-sem-eqdis-eqprop-inst + + + + + - + (0.33) + (0.07) rdfbased-sem-eqdis-eqprop-rflxv + + + + - - + (0.43) + (0.72) rdfbased-sem-eqdis-eqprop-subprop-1 + + + + + - + (0.45) + (0.37) rdfbased-sem-eqdis-eqprop-subprop-2 + + + + + - ? (300.00) + (0.58) rdfbased-sem-eqdis-eqprop-subst + + + + + - + (0.46) + (0.75) rdfbased-sem-eqdis-eqprop-sym + + + + + - + (0.43) + (0.10) rdfbased-sem-eqdis-eqprop-trans + + + + + - ? (300.00) + (0.13) rdfbased-sem-eqdis-sameas-ext + + ? - - - + (0.43) + (10.88) rdfbased-sem-eqdis-sameas-rflxv + + + + - - + (0.04) + (0.51) rdfbased-sem-eqdis-sameas-subst - - ? + + - + (0.34) + (0.53) rdfbased-sem-eqdis-sameas-sym + + ? + + - + (0.05) + (0.29) rdfbased-sem-eqdis-sameas-trans + + ? + + - + (0.09) + (0.62) rdfbased-sem-facet-def ? ? ? - - - ? (300.00) ? (285.37) rdfbased-sem-facet-empty ? - ? - - - ? (300.00) ? (300.00) rdfbased-sem-facet-localize - - - - - - ? (285.09) ? (300.00) rdfbased-sem-facet-sub - - ? - - - ? (300.00) ? (182.76) rdfbased-sem-facet-unknown - - - - - ? ? (300.00) ? (235.41) rdfbased-sem-inv-ext + + + - - - ? (300.00) + (278.71) rdfbased-sem-inv-inst + + + + + + + (0.35) + (0.07) rdfbased-sem-inv-sym + + + - + - + (0.34) + (0.10) rdfbased-sem-inv-trans + + + - - - ? (300.00) + (1.05) rdfbased-sem-key-def + + ? + - - ? (300.00) + (74.85) rdfbased-sem-key-ext ? + ? - - - ? (300.00) ? (284.78) rdfbased-sem-key-localize + + ? - - - + (0.56) + (74.17) rdfbased-sem-ndis-alldifferent-b w + + + - - - + (0.44) + (82.32) rdfbased-sem-ndis-alldifferent-b w-distinctmembers + + + - - - + (0.92) + (80.11) rdfbased-sem-ndis-alldifferent-fw - - - + - ? + (0.40) + (72.50) rdfbased-sem-ndis-alldifferent-fw-distinctmem b ers + + + - + ? + (0.56) + (67.88) rdfbased-sem-ndis-alldisjointclasses-b w + + + - - - ? (300.00) + (74.11) rdfbased-sem-ndis-alldisjointclasses-fw + + + + - ? + (0.41) + (73.69) rdfbased-sem-ndis-alldisjointclasses-localize + + + + - - + (0.39) + (36.59) rdfbased-sem-ndis-alldisjointproperties-bw - ? ? - - - ? (300.00) + (73.86) rdfbased-sem-ndis-alldisjointproperties-fw + + + + - ? + (0.57) + (72.29) rdfbased-sem-ndis-alldisjointproperties-localize + + + + - - + (0.46) + (13.37) rdfbased-sem-npa-dat-bw + + + - - - ? (300.00) ? (300.00) rdfbased-sem-npa-dat-dnpa + + + - - - ? (300.00) ? (214.25) rdfbased-sem-npa-dat-fw - - - + - ? + (0.41) + (0.17) rdfbased-sem-npa-dat-localize + + - - - - + (0.31) + (0.56) rdfbased-sem-npa-dat-npa + + + - - - + (0.36) + (0.88) rdfbased-sem-npa-ind-bw + + + - - - + (0.11) + (2.75) rdfbased-sem-npa-ind-fw - - - + - ? + (0.35) + (0.16) rdfbased-sem-parts-annotationproperties-instance - - ? - - - + (0.02) + (0.09) rdfbased-sem-parts-annotationproperties-sup er + + + + - - + (0.03) + (0.10) rdfbased-sem-parts-classes-instance - - ? + + - + (0.02) + (0.08) rdfbased-sem-parts-classes-super - - ? + + - + (0.03) + (0.08) rdfbased-sem-parts-datatypeproperties-instance - - ? - - - + (0.05) + (0.66) rdfbased-sem-parts-datatypeproperties-sup er + + + + + - + (0.03) + (0.14) rdfbased-sem-parts-datatypes-instance - - ? - - - + (0.03) + (0.35) rdfbased-sem-parts-datatypes-super + + ? + + - + (0.02) + (0.15) rdfbased-sem-parts-individuals-nonempty - - ? - - - + (0.03) + (0.07) rdfbased-sem-parts-literals-super - - ? + + - + (0.02) + (0.08) rdfbased-sem-parts-ontologies-super - - ? + + - + (0.03) + (0.07) rdfbased-sem-parts-ontologyproperties-instance + + + - - - + (0.16) + (0.66) rdfbased-sem-parts-ontologyproperties-super + + + + + - + (0.03) + (0.10) rdfbased-sem-parts-properties-instance - - ? - - - + (0.02) + (0.09) rdfbased-sem-parts-properties-sup er - - ? + + - + (0.03) + (0.08) rdfbased-sem-prop-allv aluesfrom-ext - - ? + - - + (0.43) + (0.87) rdfbased-sem-prop-allv aluesfrom-type + + + + - - + (0.03) + (0.14) rdfbased-sem-prop-annotatedproperty-ext - - ? + - - + (0.35) + (0.25) rdfbased-sem-prop-annotatedproperty-type + + + + - - + (0.03) + (0.08) rdfbased-sem-prop-annotatedsource-ext - - ? + - - + (0.36) + (0.24) rdfbased-sem-prop-annotatedsource-type + + + + - - + (0.02) + (0.08) rdfbased-sem-prop-annotatedtarget-ext - - ? + - - + (0.36) + (0.26) rdfbased-sem-prop-annotatedtarget-type + + + + - - + (0.03) + (0.08) rdfbased-sem-prop-assertionproperty-ext - - ? + - - + (0.37) + (1.10) rdfbased-sem-prop-assertionproperty-type + + + + - - + (0.03) + (0.12) rdfbased-sem-prop-backw ardcompatiblewith-ext + + - + + - + (0.42) + (0.44) rdfbased-sem-prop-backw ardcompatiblewith-type-annot + + - + - - + (0.03) + (0.08) rdfbased-sem-prop-backw ardcompatiblewith-type-onto - - ? + - - + (0.03) + (0.08) rdfbased-sem-prop-bottomdataprop ert y-ext-hi + + ? + - - + (0.40) + (0.75) rdfbased-sem-prop-bottomdataprop ert y-ext-lo + + ? - - - + (0.36) + (0.52) rdfbased-sem-prop-bottomdataprop ert y-term - - ? - - - + (0.35) + (1.59) Reasoning in OWL 2 F ull using First-Order A TP 19 rdfbased-sem-prop-bottomdataprop ert y-type + + - + - - + (0.02) + (0.12) rdfbased-sem-prop-bottomob jectproperty-ext-hi + + + + - - + (0.32) + (0.24) rdfbased-sem-prop-bottomob jectproperty-ext-lo + + ? - - - + (0.39) + (0.40) rdfbased-sem-prop-bottomob jectproperty-term + + ? - - - + (0.35) + (0.18) rdfbased-sem-prop-bottomob jectproperty-type + + - + - - + (0.03) + (0.12) rdfbased-sem-prop-cardinality-ext - - ? + - - + (0.58) + (1.61) rdfbased-sem-prop-cardinality-t yp e + + + + - - + (0.03) + (0.17) rdfbased-sem-prop-comment-ext + + - + - - + (0.02) + (0.08) rdfbased-sem-prop-comment-t yp e + + - + - - + (0.02) + (0.08) rdfbased-sem-prop-complementof-ext - - ? + - - + (0.38) + (0.65) rdfbased-sem-prop-complementof-t yp e + + + + - - + (0.03) + (0.08) rdfbased-sem-prop-datatypecomplementof-ext - - ? + - - + (0.43) + (0.62) rdfbased-sem-prop-datatypecomplementof-t yp e + + + + - - + (0.02) + (0.08) rdfbased-sem-prop-deprecated-ext + + - + - - + (0.35) + (0.40) rdfbased-sem-prop-deprecated-type + + - + - - + (0.02) + (0.08) rdfbased-sem-prop-differentfrom-ext - + ? + - - + (0.36) + (0.41) rdfbased-sem-prop-differentfrom-t yp e + + + + + - + (0.03) + (0.15) rdfbased-sem-prop-disjointunionof-ext - - ? + - - + (0.35) + (1.59) rdfbased-sem-prop-disjointunionof-t yp e + + + + - - + (0.02) + (0.09) rdfbased-sem-prop-disjoint with-ext - - ? + + - + (0.42) + (0.45) rdfbased-sem-prop-disjoint with-type + + + + + - + (0.02) + (0.08) rdfbased-sem-prop-distinctmembers-ext - - ? + - - + (0.42) + (1.91) rdfbased-sem-prop-distinctmembers-type + + + + - - + (0.03) + (0.15) rdfbased-sem-prop-equiv alentclass-ext - - ? + + - + (0.17) + (0.58) rdfbased-sem-prop-equiv alentclass-type + + + + + - + (0.02) + (0.08) rdfbased-sem-prop-equiv alentproperty-ext - - ? + - - + (0.37) + (0.89) rdfbased-sem-prop-equiv alentproperty-type + + + + - - + (0.03) + (0.08) rdfbased-sem-prop-haskey-ext - - ? + - - + (0.42) + (2.19) rdfbased-sem-prop-haskey-t yp e + + + + - - + (0.03) + (0.15) rdfbased-sem-prop-hasself-ext - - ? + - - + (0.41) + (0.65) rdfbased-sem-prop-hasself-type + + + + - - + (0.02) + (0.15) rdfbased-sem-prop-hasv alue-ext - - ? + - - + (0.36) + (0.64) rdfbased-sem-prop-hasv alue-type + + + + - - + (0.02) + (0.14) rdfbased-sem-prop-imports-ext - - ? + + - + (0.41) + (0.48) rdfbased-sem-prop-imports-type - - ? + + - + (0.02) + (0.08) rdfbased-sem-prop-incompatiblewith-ext + + - + + - + (0.38) + (0.45) rdfbased-sem-prop-incompatiblewith-type-annot + + - + - - + (0.03) + (0.08) rdfbased-sem-prop-incompatiblewith-type-onto - - ? + - - + (0.03) + (0.10) rdfbased-sem-prop-intersectionof-ext - - ? + - - + (0.43) + (1.08) rdfbased-sem-prop-intersectionof-t yp e + + + + + - + (0.03) + (0.10) rdfbased-sem-prop-inv erseof-ext - - ? + - - + (0.37) + (0.89) rdfbased-sem-prop-inv erseof-type + + + + - - + (0.03) + (0.08) rdfbased-sem-prop-isdefinedby-ext + + - + - - + (0.02) + (0.08) rdfbased-sem-prop-isdefinedby-t yp e + + - + - - + (0.03) + (0.08) rdfbased-sem-prop-label-ext + + - + - - + (0.02) + (0.08) rdfbased-sem-prop-label-type + + - + - - + (0.02) + (0.08) rdfbased-sem-prop-maxcardinality-ext - - ? + - - + (0.49) + (1.14) rdfbased-sem-prop-maxcardinality-t yp e + + + + - - + (0.02) + (0.16) rdfbased-sem-prop-maxqualifiedcardinality-ext - - ? + - - + (0.58) + (1.33) rdfbased-sem-prop-maxqualifiedcardinality-t yp e + + + + - - + (0.03) + (0.17) rdfbased-sem-prop-members-ext - - ? + - - + (0.31) + (1.11) rdfbased-sem-prop-members-type + + + + - - + (0.03) + (0.17) rdfbased-sem-prop-mincardinality-ext - - ? + - - + (0.58) + (5.87) rdfbased-sem-prop-mincardinality-t yp e + + + + - - + (0.03) + (0.17) rdfbased-sem-prop-minqualifiedcardinality-ext - - ? + - - + (0.62) + (6.54) rdfbased-sem-prop-minqualifiedcardinality-t yp e + + + + - - + (0.02) + (0.17) rdfbased-sem-prop-onclass-ext - - ? + - - + (0.30) + (1.04) rdfbased-sem-prop-onclass-type + + + + - - + (0.03) + (0.16) rdfbased-sem-prop-ondatarange-ext - - ? + - - + (0.43) + (1.01) rdfbased-sem-prop-ondatarange-type + + + + - - + (0.02) + (0.17) rdfbased-sem-prop-ondatatype-ext - - ? + - - + (0.35) + (0.51) rdfbased-sem-prop-ondatatype-type + + + + - - + (0.02) + (0.09) rdfbased-sem-prop-oneof-ext - - ? + - - + (0.33) + (1.01) rdfbased-sem-prop-oneof-type + + + + + - + (0.02) + (0.14) rdfbased-sem-prop-onproperty-ext - - ? + - - + (0.35) + (1.23) rdfbased-sem-prop-onproperty-type + + + + + - + (0.02) + (0.17) rdfbased-sem-prop-priorversion-ext + + - + + - + (0.37) + (0.45) rdfbased-sem-prop-priorversion-t yp e-annot + + - + - - + (0.02) + (0.09) rdfbased-sem-prop-priorversion-t yp e-on to - - ? + - - + (0.02) + (0.09) rdfbased-sem-prop-propertychainaxiom-ext - - ? + - - + (0.33) + (1.54) rdfbased-sem-prop-propertychainaxiom-t yp e + + + + - - + (0.02) + (0.10) rdfbased-sem-prop-propertydisjoint with-ext - - ? + - - + (0.41) + (1.14) rdfbased-sem-prop-propertydisjoint with-type + + + + - - + (0.02) + (0.09) rdfbased-sem-prop-qualifiedcardinality-ext - - ? + - - + (0.65) + (1.30) rdfbased-sem-prop-qualifiedcardinality-t yp e + + + + - - + (0.02) + (0.18) rdfbased-sem-prop-sameas-ext - + ? + - - + (0.36) + (0.40) rdfbased-sem-prop-sameas-type + + + + + - + (0.03) + (0.15) rdfbased-sem-prop-seealso-ext + + - + - - + (0.02) + (0.07) rdfbased-sem-prop-seealso-type + + - + - - + (0.02) + (0.09) rdfbased-sem-prop-somev aluesfrom-ext - - ? + - - + (0.37) + (0.83) rdfbased-sem-prop-somev aluesfrom-type + + + + - - + (0.02) + (0.15) rdfbased-sem-prop-sourceindividual-ext - - ? + - - + (0.37) + (0.64) rdfbased-sem-prop-sourceindividual-type + + + + - - + (0.02) + (0.12) rdfbased-sem-prop-targetindividual-ext - - ? + - - + (0.42) + (0.34) 20 Mic hael Schneider and Geoﬀ Sutcliﬀe rdfbased-sem-prop-targetindividual-type + + + + - - + (0.02) + (0.09) rdfbased-sem-prop-targetv alue-ext - - ? + - - + (0.44) + (1.08) rdfbased-sem-prop-targetv alue-type + + + + - - + (0.03) + (0.12) rdfbased-sem-prop-topdataprop ert y-ext-hi + + ? + - - + (0.44) + (0.99) rdfbased-sem-prop-topdataprop ert y-ext-lo + + + - - - + (0.04) + (0.44) rdfbased-sem-prop-topdataprop ert y-term - - ? - - - + (0.40) + (2.26) rdfbased-sem-prop-topdataprop ert y-type + + - + - - + (0.02) + (0.11) rdfbased-sem-prop-topob jectproperty-ext-hi + + + + - - + (0.32) + (0.21) rdfbased-sem-prop-topob jectproperty-ext-lo + + + - - - + (0.04) + (0.11) rdfbased-sem-prop-topob jectproperty-term + + ? - - - + (0.38) + (0.20) rdfbased-sem-prop-topob jectproperty-type + + - + - - + (0.03) + (0.14) rdfbased-sem-prop-unionof-ext - - ? + - - + (0.39) + (4.83) rdfbased-sem-prop-unionof-type + + + + - - + (0.03) + (0.09) rdfbased-sem-prop-versioninfo-ext + + - + - - + (0.34) + (0.42) rdfbased-sem-prop-versioninfo-t yp e + + - + + - + (0.02) + (0.08) rdfbased-sem-prop-versioniri-ext - - ? + - - + (0.43) + (0.47) rdfbased-sem-prop-versioniri-t yp e - - ? + - - + (0.03) + (0.09) rdfbased-sem-prop-withrestrictions-ext - - ? + - - + (0.44) + (1.85) rdfbased-sem-prop-withrestrictions-type + + + + - - + (0.03) + (0.08) rdfbased-sem-rdf-container-high val-axiom + + + + - - ? (300.00) ? (181.83) rdfbased-sem-rdf-container-initv al-axiom + + + + - - + (0.02) + (0.07) rdfbased-sem-rdf-list-axiom + + + + + - + (0.02) + (0.17) rdfbased-sem-rdf-reify-axiom + + + + + - + (0.02) + (0.21) rdfbased-sem-rdf-type-axiom + + + + + - + (0.03) + (0.07) rdfbased-sem-rdf-type-cond + + + + + - + (0.03) + (0.19) rdfbased-sem-rdf-v alue-axiom + + + + - - + (0.02) + (0.07) rdfbased-sem-rdf-xmlliteral-type - - ? - - - ? (300.00) ? (181.71) rdfbased-sem-rdfs-annotate-axiom + + - + - - + (0.02) + (0.08) rdfbased-sem-rdfs-class + + + + + - + (0.19) + (0.12) rdfbased-sem-rdfs-container-cond - - ? - + - + (0.03) + (0.11) rdfbased-sem-rdfs-container-high val-axiom - - ? + - - ? (300.00) ? (220.78) rdfbased-sem-rdfs-container-initv al-axiom - - ? + - - + (0.02) + (0.07) rdfbased-sem-rdfs-container-static-axiom - - ? + - - + (0.02) + (0.08) rdfbased-sem-rdfs-data-cond - - ? - - - + (0.03) + (0.10) rdfbased-sem-rdfs-datatype-axiom - - ? + + - + (0.02) + (0.07) rdfbased-sem-rdfs-domain-axiom - - ? + + - + (0.02) + (0.07) rdfbased-sem-rdfs-domain-cond + + + + + - + (0.19) + (0.23) rdfbased-sem-rdfs-list-axiom - - ? + - - + (0.04) + (0.15) rdfbased-sem-rdfs-plain-notag-type - - ? - - - ? (300.00) ? (196.34) rdfbased-sem-rdfs-plain-tagged-type - - ? - - - ? (300.00) ? (202.44) rdfbased-sem-rdfs-range-axiom - - ? + + - + (0.02) + (0.07) rdfbased-sem-rdfs-range-cond + + + + + - + (0.20) + (0.21) rdfbased-sem-rdfs-reify-axiom - - ? + - - + (0.03) + (0.06) rdfbased-sem-rdfs-resource - - ? - - - + (0.02) + (0.14) rdfbased-sem-rdfs-subclass-axiom - - ? + + - + (0.02) + (0.07) rdfbased-sem-rdfs-subclass-cond + + + + + + + (0.34) + (0.35) rdfbased-sem-rdfs-subclass-resource - - ? - + - + (0.03) + (0.10) rdfbased-sem-rdfs-subclass-rflxv - + ? + + - + (0.03) + (0.11) rdfbased-sem-rdfs-subclass-trans + + + + - + + (0.33) + (0.10) rdfbased-sem-rdfs-subprop-axiom - - ? + + - + (0.02) + (0.08) rdfbased-sem-rdfs-subprop-cond + + + + + + + (0.33) + (0.10) rdfbased-sem-rdfs-subprop-rflxv - + ? + + - + (0.02) + (0.10) rdfbased-sem-rdfs-subprop-trans + + + + + + + (0.33) + (0.09) rdfbased-sem-rdfs-type-axiom - - ? + - - + (0.02) + (0.07) rdfbased-sem-rdfs-v alue-axiom - - ? + - - + (0.02) + (0.07) rdfbased-sem-rdfs-xmlliteral-axiom-type + + + + + - + (0.02) + (0.07) rdfbased-sem-rdfs-xmlliteral-axiom-v alue - - ? + - - + (0.02) + (0.06) rdfbased-sem-rdfs-xmlliteral-illtyped - ? ? - - ? ? (300.00) ? (300.00) rdfbased-sem-rdfsext-domain-ext + + + - - - + (1.11) + (65.94) rdfbased-sem-rdfsext-domain-subprop + + + + + - + (0.25) + (2.40) rdfbased-sem-rdfsext-domain-superclass + + + + + - + (0.39) + (39.92) rdfbased-sem-rdfsext-range-ext + + + - + - ? (300.00) + (43.53) rdfbased-sem-rdfsext-range-subprop + + + + + - + (0.43) + (6.21) rdfbased-sem-rdfsext-range-superclass + + + + + - + (0.32) + (11.90) rdfbased-sem-rdfsext-subclass-ext + + + - - - + (0.87) + (107.13) rdfbased-sem-rdfsext-subprop-ext + + + - - - + (0.49) + (111.26) rdfbased-sem-restrict-allv alues-cmp-class - - ? + - - ? (300.00) + (56.84) rdfbased-sem-restrict-allv alues-cmp-prop - - ? + - - ? (300.00) + (73.76) rdfbased-sem-restrict-allv alues-inst-ob j - - ? + + - + (0.77) + (5.21) rdfbased-sem-restrict-allv alues-inst-sub j - - ? - + - + (1.05) + (14.89) rdfbased-sem-restrict-exactcard-inst-ob j-t wo - ? ? - - - + (0.79) + (1.08) rdfbased-sem-restrict-exactcard-inst-sub j-t wo - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-exactqcr-data-localize + + - - - - + (0.66) + (0.78) rdfbased-sem-restrict-exactqcr-inst-ob j-t wo - ? ? - - - ? (300.00) + (15.62) rdfbased-sem-restrict-exactqcr-inst-sub j-t wo - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-hasself-cmp-prop - - ? - - - ? (300.00) + (12.41) rdfbased-sem-restrict-hasself-inst-ob j - - ? - - - + (0.49) + (0.76) rdfbased-sem-restrict-hasself-inst-sub j - - ? - - - + (0.82) + (0.46) rdfbased-sem-restrict-hasv alue-cmp-prop - - ? + - - + (27.08) + (2.00) rdfbased-sem-restrict-hasv alue-inst-ob j - - ? + + - + (0.42) + (0.86) rdfbased-sem-restrict-hasv alue-inst-sub j - - ? + + - + (0.67) + (0.40) rdfbased-sem-restrict-maxcard-cmp-card - - ? - - - ? (300.00) + (112.57) rdfbased-sem-restrict-maxcard-cmp-prop - - ? - - - ? (300.00) ? (285.29) rdfbased-sem-restrict-maxcard-inst-ob j-one - - - + + - + (0.92) + (0.85) Reasoning in OWL 2 F ull using First-Order A TP 21 rdfbased-sem-restrict-maxcard-inst-ob j-zero - - - + + ? + (0.65) + (0.84) rdfbased-sem-restrict-maxcard-inst-sub j-one - - ? - + - ? (300.00) + (276.12) rdfbased-sem-restrict-maxcard-inst-sub j-zero - - ? - - - + (0.67) + (1.61) rdfbased-sem-restrict-maxqcr-cmp-card - - ? - - - ? (300.00) + (111.11) rdfbased-sem-restrict-maxqcr-cmp-class - - ? - - - ? (300.00) + (76.42) rdfbased-sem-restrict-maxqcr-cmp-prop - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-maxqcr-data-localize + + - - - - + (0.65) + (0.77) rdfbased-sem-restrict-maxqcr-inst-ob j-one - - - + - - ? (300.00) + (74.83) rdfbased-sem-restrict-maxqcr-inst-ob j-zero - - - + - ? + (0.81) + (63.29) rdfbased-sem-restrict-maxqcr-inst-sub j-one - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-maxqcr-inst-sub j-zero - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-mincard-cmp-card - - ? - - - ? (300.00) + (0.67) rdfbased-sem-restrict-mincard-cmp-prop - - ? - - - ? (300.00) + (8.93) rdfbased-sem-restrict-mincard-inst-ob j-one - - ? - - - + (0.43) + (0.70) rdfbased-sem-restrict-mincard-inst-sub j-one - - ? - + - + (0.67) + (0.42) rdfbased-sem-restrict-minqcr-cmp-card - - ? - - - ? (300.00) + (74.69) rdfbased-sem-restrict-minqcr-cmp-class - - ? - - - ? (300.00) + (72.80) rdfbased-sem-restrict-minqcr-cmp-prop - - ? - - - ? (300.00) + (2.05) rdfbased-sem-restrict-minqcr-data-localize + + - - - - + (0.60) + (0.70) rdfbased-sem-restrict-minqcr-inst-ob j-one - - ? - - - + (0.76) + (10.16) rdfbased-sem-restrict-minqcr-inst-sub j-one - - ? - - - + (0.79) + (3.96) rdfbased-sem-restrict-somev alues-cmp-class - - ? + - - ? (300.00) + (2.86) rdfbased-sem-restrict-somev alues-cmp-prop - - ? + - - ? (300.00) + (74.43) rdfbased-sem-restrict-somev alues-inst-ob j - - ? - - - + (0.37) + (3.47) rdfbased-sem-restrict-somev alues-inst-sub j - - ? + + - + (0.78) + (1.19) rdfbased-sem-restrict-term-cardqcr - - ? - - - ? (300.00) + (145.98) rdfbased-sem-restrict-term-dataqcr - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-term-minmaxexact - - ? - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-term-minmaxthing - - - - - - ? (300.00) ? (300.00) rdfbased-sem-restrict-term-sameall - - ? - - - ? (300.00) + (74.84) rdfbased-sem-restrict-term-selfsome - - ? - - - + (27.57) + (1.20) rdfbased-sem-restrict-term-somehas - - ? - - - ? (300.00) + (118.06) rdfbased-sem-restrict-term-someqcr - - ? - - - ? (300.23) + (11.39) rdfbased-sem-simple-bnode-iri + + ? - - - + (0.02) + (0.07) rdfbased-sem-simple-bnode-literal + ? ? - - - + (0.03) + (0.24) rdfbased-sem-simple-bnode-rename + + ? - - - + (0.02) + (0.07) rdfbased-sem-simple-bnode-same + + ? - - - + (0.03) + (0.07) rdfbased-sem-simple-emptygraph-an y + + + + + + + (0.03) + (0.05) rdfbased-sem-simple-emptygraph-self + + + + + + + (0.01) + (0.07) rdfbased-sem-simple-subgraph-any + + + + + + + (0.02) + (0.06) rdfbased-sem-simple-subgraph-self + + + + + + + (0.02) + (0.06) T able 8: Result data of the language coverage exp erimen ts for the FOL theorem prover iProver- SInE , used with the small-sufficient OWL 2 F ull subaxiomatizations on those test cases where it and V ampire had failed when using the complete axiomatization. T est Case iProv er-SInE rdfbased-sem-bo ol-demorgan + (36.39) rdfbased-sem-bo ol-in tersection-ext + (0.64) rdfbased-sem-bo ol-union-ext + (0.54) rdfbased-sem-chain-ext + (35.00) rdfbased-sem-enum-ext + (0.17) rdfbased-sem-restrict-exactcard-inst-sub j-t wo + (55.83) rdfbased-sem-restrict-exactqcr-inst-sub j-t wo ? (300.00) rdfbased-sem-restrict-maxcard-cmp-prop + (0.18) rdfbased-sem-restrict-maxqcr-cmp-prop + (0.30) rdfbased-sem-restrict-maxqcr-inst-sub j-one + (1.30) rdfbased-sem-restrict-maxqcr-inst-sub j-zero + (2.34) rdfbased-sem-restrict-term-dataqcr + (0.59) rdfbased-sem-restrict-term-minmaxexact + (39.47) rdfbased-sem-restrict-term-minmaxthing + (2.03) T able 9: Result data of the language coverage experiments for the FOL theorem prover V ampir e , used with the small-sufficient OWL 2 F ull subaxiomatizations on those test cases where iProv er- SInE failed when using the small-sufficient axiomatizations. T est Case V ampire rdfbased-sem-restrict-exactqcr-inst-sub j-t wo + (2.90) 22 Mic hael Schneider and Geoﬀ Sutcliﬀe A.2 O WL 2 F ull-Characteristic Conclusions and Scalabilit y Results The following tables pro vide the ra w result data that underlies the results for the scalability exp erimen ts, as rep orted in Section 5.3. All exp erimen ts were conducted using the test suite of characteristic O WL 2 F ull conclusions, as in- tro duced in Section 4.2 (see Appendix B for more detailed information ab out the test suite). There is one table per combination of a F OL reasoner ( iPr over-SInE , V ampir e in auto mo de, and V ampir e using the SInE strategy) and either the complete OWL 2 F ull axiomatization or the small-suﬃcient subaxiomatizations for the diﬀerent test cases. While Section 5.3 lists only the results for bulk RDF data of size 1 million triples, the tables here also sho w results for several in ter- mediate sizes: 1200, 10,000, and 100,000 triples. In addition, the results for no bulk data (0 triples) are presen ted, which were the base for the results rep orted in Section 5.2 for the test suite of characteristic O WL 2 F ull conclusions. No result data for the characteristic conclusion tests is given here for the Seman tic W eb reasoners, since the data provided in Section 5.2 is already complete for them. The ﬁrst column of each table giv es the name of the test case, and the remaining columns giv es the results for the diﬀeren t bulk data sizes. T able 10: Scalability results for the theorem prover iPr over-SInE using the complete OWL 2 F ull axiomatization. T est Case 0 1200 10k 100k 1M 001 Subgraph Entailmen t + (0.15) + (0.17) + (0.34) + (2.16) + (21.98) 002 Existential Blank Nodes + (0.08) + (0.10) + (0.27) + (2.15) + (21.81) 003 Blank Nodes for Literals + (0.08) + (0.10) + (0.27) + (2.06) + (22.51) 004 Axiomatic T riples + (1.23) + (1.27) + (1.44) + (3.43) + (23.31) 005 Everything is a Resource + (3.03) + (3.03) + (2.56) + (5.19) + (25.07) 006 Literal V alues represented by URIs and Blank No des + (11.66) + (11.65) + (11.77) + (13.61) + (34.00) 007 Equal Classes + (74.40) + (61.38) + (50.34) + (76.84) + (96.51) 008 Inverse F unctional Data Prop erties + (0.41) + (0.42) + (0.59) + (2.41) + (22.66) 009 Existential Restriction Entailmen ts + (2.35) + (2.35) + (2.48) + (4.36) + (24.15) 010 Negative Prop ert y Assertions + (89.45) + (91.03) + (90.05) + (92.60) + (111.97) 011 Entity Types as Classes + (0.30) + (0.32) + (0.49) + (2.30) + (22.20) 012 T emplate Class ? (300.00) + (144.50) ? (300.00) ? (300.00) ? (300.00) 013 Cliques ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 014 Harry belongs to some Species + (33.65) + (41.90) + (32.35) + (54.93) + (98.11) 015 Reflective T autologies I + (0.16) + (0.17) + (0.34) + (2.21) + (22.06) 016 Reflective T autologies I I + (0.95) + (0.96) + (1.12) + (2.95) + (22.83) 017 Builtin Based Definitions + (5.31) + (5.19) + (5.36) + (20.17) + (34.27) 018 Mo dified Logical V o cabulary Semantics + (0.72) + (0.74) + (0.92) + (2.73) + (22.66) 019 Disjoint Annotation Properties + (0.19) + (0.22) + (0.38) + (2.25) + (22.06) 020 Logical Complications ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 021 Comp osite Enumerations ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 022 List Member Access + (123.01) + (122.19) + (122.41) + (123.19) + (143.69) 023 Unique List Comp onen ts + (4.12) + (4.07) + (4.26) + (6.02) + (25.88) 024 Cardinality Restrictions on Complex Properties + (14.89) + (14.06) + (13.84) + (16.78) + (37.30) 025 Cyclic Dependencies between Complex Prop erties + (117.92) + (118.07) + (120.24) + (120.02) + (136.68) 026 Inferred Property Characteristics I + (111.18) + (63.40) + (109.99) + (113.43) + (130.79) 027 Inferred Property Characteristics II + (122.01) + (120.51) + (121.49) + (120.86) + (143.95) 028 Inferred Property Characteristics II I + (3.56) + (3.58) + (3.66) + (5.59) + (25.48) 029 Ex F also Quo dlibet + (74.35) + (74.62) + (74.86) + (76.59) + (96.43) 030 Bad Class + (18.07) + (18.42) + (18.79) + (25.70) + (45.60) 031 Large Universe + (42.44) + (39.85) + (51.68) + (53.45) + (99.83) 032 Datatype Relationships + (2.05) + (2.04) + (2.23) + (4.08) + (23.86) Reasoning in OWL 2 F ull using First-Order A TP 23 T able 11: Scalabilit y results for the theorem prov er V ampire (auto mo de) using the complete OWL 2 F ull axiomatization. T est Case 0 1200 10k 100k 1M 001 Subgraph Entailmen t + (0.02) + (0.04) + (0.21) + (2.31) + (71.14) 002 Existential Blank Nodes + (0.02) + (0.03) + (0.21) + (2.35) + (216.63) 003 Blank Nodes for Literals + (0.02) + (0.04) + (0.21) + (1.90) + (73.91) 004 Axiomatic T riples + (0.43) + (0.46) + (0.67) + (2.49) ? (300.00) 005 Everything is a Resource + (0.03) + (0.05) + (0.22) + (2.38) ? (300.00) 006 Literal V alues represented by URIs and Blank No des + (0.18) + (0.21) + (0.38) + (2.34) ? (300.00) 007 Equal Classes + (0.35) + (0.40) + (0.54) + (2.41) ? (300.00) 008 Inverse F unctional Data Prop erties + (0.40) + (0.44) + (0.56) + (2.45) ? (300.00) 009 Existential Restriction Entailmen ts + (0.39) + (0.40) + (0.57) + (2.34) ? (284.91) 010 Negative Prop ert y Assertions ? (285.53) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 011 Entity Types as Classes + (0.18) + (0.20) + (0.38) + (2.30) ? (300.00) 012 T emplate Class ? (285.57) ? (300.00) ? (284.78) ? (300.00) ? (213.28) 013 Cliques ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 014 Harry belongs to some Species + (1.15) + (1.18) + (1.33) + (2.97) ? (300.57) 015 Reflective T autologies I + (0.03) + (0.05) + (0.21) + (1.99) + (223.44) 016 Reflective T autologies I I + (0.56) + (0.57) + (0.73) + (2.35) ? (300.00) 017 Builtin Based Definitions + (0.38) + (0.43) + (0.59) + (2.20) ? (300.22) 018 Mo dified Logical V o cabulary Semantics + (0.16) + (0.17) + (0.34) + (2.25) ? (301.90) 019 Disjoint Annotation Properties + (0.43) + (0.46) + (0.61) + (2.17) ? (300.00) 020 Logical Complications ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 021 Comp osite Enumerations ? (300.00) ? (300.00) ? (300.00) ? (300.69) ? (300.00) 022 List Member Access ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 023 Unique List Comp onen ts + (0.46) + (0.51) + (0.63) + (2.24) ? (300.00) 024 Cardinality Restrictions on Complex Properties + (0.71) + (0.73) + (0.85) + (2.47) ? (300.00) 025 Cyclic Dependencies between Complex Prop erties ? (300.69) ? (300.31) ? (300.00) ? (300.00) ? (300.00) 026 Inferred Property Characteristics I + (0.47) + (0.48) + (0.64) + (2.28) ? (300.00) 027 Inferred Property Characteristics II ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 028 Inferred Property Characteristics II I ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.36) 029 Ex F also Quo dlibet + (0.92) + (0.93) + (1.13) + (2.69) ? (300.00) 030 Bad Class + (0.39) + (0.45) + (0.57) + (2.18) ? (300.38) 031 Large Universe + (0.44) + (0.46) + (0.62) + (2.22) ? (300.00) 032 Datatype Relationships + (0.65) + (0.64) + (0.81) + (2.35) ? (300.46) T able 12: Scalability results for the theorem prover V ampire-SInE using the complete OWL 2 F ull axiomatization. T est Case 0 1200 10k 100k 1M 001 Subgraph Entailmen t + (0.04) + (0.06) + (0.20) + (1.84) + (20.19) 002 Existential Blank Nodes + (0.04) + (0.06) + (0.21) + (1.81) + (19.28) 003 Blank Nodes for Literals + (0.02) + (0.04) + (0.19) + (1.82) + (19.34) 004 Axiomatic T riples + (4.99) + (5.04) + (5.21) + (6.77) + (24.38) 005 Everything is a Resource + (0.06) + (0.07) + (0.23) + (1.86) + (20.32) 006 Literal V alues represented by URIs and Blank No des + (9.63) + (9.74) + (9.66) + (11.43) + (29.27) 007 Equal Classes ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 008 Inverse F unctional Data Prop erties + (3.13) + (3.14) + (3.37) + (5.06) + (23.54) 009 Existential Restriction Entailmen ts ? (300.00) ? (301.49) ? (300.00) ? (300.00) ? (300.00) 010 Negative Prop ert y Assertions ? (300.26) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 011 Entity Types as Classes + (0.08) + (0.10) + (0.25) + (1.87) + (19.47) 012 T emplate Class ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 013 Cliques ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 014 Harry belongs to some Species ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 015 Reflective T autologies I + (0.06) + (0.08) + (0.22) + (1.86) + (19.37) 016 Reflective T autologies I I + (10.82) + (10.61) + (10.93) + (12.94) + (30.77) 017 Builtin Based Definitions ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 018 Mo dified Logical V o cabulary Semantics + (0.27) + (0.29) + (0.44) + (2.09) + (19.45) 019 Disjoint Annotation Properties + (3.12) + (3.22) + (3.31) + (4.96) + (22.95) 020 Logical Complications ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 021 Comp osite Enumerations ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 022 List Member Access ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 023 Unique List Comp onen ts + (6.97) + (6.93) + (6.92) + (8.87) + (26.57) 024 Cardinality Restrictions on Complex Properties ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 025 Cyclic Dependencies between Complex Prop erties ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 026 Inferred Property Characteristics I + (11.97) + (12.16) + (11.97) + (14.30) + (31.64) 027 Inferred Property Characteristics II ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 028 Inferred Property Characteristics II I ? (300.00) ? (301.10) ? (300.00) ? (300.00) ? (300.00) 029 Ex F also Quo dlibet ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 030 Bad Class + (6.33) + (6.24) + (6.41) + (8.29) + (25.92) 031 Large Universe ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 032 Datatype Relationships + (0.09) + (0.11) + (0.25) + (1.94) + (19.53) 24 Mic hael Schneider and Geoﬀ Sutcliﬀe T able 13: Scalability results for the theorem prov er iPr over-SInE using the small-sufficient OWL 2 F ull subaxiomatizations. T est Case 0 1200 10k 100k 1M 001 Subgraph Entailmen t + (0.04) + (0.07) + (0.24) + (2.03) + (22.87) 002 Existential Blank Nodes + (0.05) + (0.07) + (0.24) + (2.09) + (21.85) 003 Blank Nodes for Literals + (0.05) + (0.07) + (0.26) + (2.14) + (22.07) 004 Axiomatic T riples + (0.10) + (0.12) + (0.30) + (2.12) + (21.97) 005 Everything is a Resource + (0.05) + (0.08) + (0.26) + (2.07) + (22.50) 006 Literal V alues represented by URIs and Blank No des + (0.07) + (0.09) + (0.26) + (2.08) + (21.96) 007 Equal Classes + (0.07) + (0.08) + (0.26) + (2.08) + (22.23) 008 Inverse F unctional Data Prop erties + (0.06) + (0.09) + (0.27) + (2.08) + (21.97) 009 Existential Restriction Entailmen ts + (0.05) + (0.08) + (0.25) + (2.08) + (21.89) 010 Negative Prop ert y Assertions + (0.29) + (0.32) + (0.49) + (2.33) + (22.21) 011 Entity Types as Classes + (0.05) + (0.07) + (0.26) + (2.08) + (21.93) 012 T emplate Class + (0.27) + (0.24) + (0.41) + (2.25) + (22.13) 013 Cliques + (164.20) + (191.29) ? (256.09) ? (259.50) ? (300.00) 014 Harry belongs to some Species + (0.08) + (0.09) + (0.27) + (2.08) + (21.85) 015 Reflective T autologies I + (0.05) + (0.07) + (0.25) + (2.10) + (21.73) 016 Reflective T autologies I I + (0.11) + (0.13) + (0.30) + (2.12) + (21.76) 017 Builtin Based Definitions + (0.08) + (0.10) + (0.27) + (2.09) + (21.93) 018 Mo dified Logical V o cabulary Semantics + (0.05) + (0.07) + (0.24) + (2.07) + (22.50) 019 Disjoint Annotation Properties + (0.05) + (0.08) + (0.25) + (2.10) + (21.91) 020 Logical Complications + (40.67) + (45.04) + (47.79) + (42.16) + (62.69) 021 Comp osite Enumerations + (42.32) + (38.32) + (46.15) + (38.11) + (63.10) 022 List Member Access + (0.12) + (0.14) + (0.31) + (2.14) + (22.00) 023 Unique List Comp onen ts + (0.14) + (0.16) + (0.34) + (2.24) + (22.62) 024 Cardinality Restrictions on Complex Properties + (0.07) + (0.08) + (0.27) + (2.14) + (21.87) 025 Cyclic Dependencies between Complex Prop erties + (0.11) + (0.14) + (0.31) + (2.13) + (22.39) 026 Inferred Property Characteristics I + (0.12) + (0.15) + (0.32) + (2.14) + (22.04) 027 Inferred Property Characteristics II + (0.16) + (0.19) + (0.35) + (2.18) + (22.14) 028 Inferred Property Characteristics II I + (0.30) + (0.33) + (0.50) + (2.35) + (22.84) 029 Ex F also Quo dlibet + (0.09) + (0.11) + (0.28) + (2.11) + (22.51) 030 Bad Class + (0.07) + (0.08) + (0.26) + (2.09) + (22.07) 031 Large Universe + (0.34) + (0.36) + (0.54) + (2.39) + (22.29) 032 Datatype Relationships + (0.07) + (0.09) + (0.26) + (2.08) + (21.86) T able 14: Scalability results for the theorem prover V ampire (auto mode) using the small-sufficient OWL 2 F ull subaxiomatizations. T est Case 0 1200 10k 100k 1M 001 Subgraph Entailmen t + (0.00) + (0.01) + (0.20) + (2.41) + (48.09) 002 Existential Blank Nodes + (0.00) + (0.05) + (1.89) ? (300.00) ? (301.00) 003 Blank Nodes for Literals + (0.00) + (0.03) + (1.49) + (161.52) + (182.70) 004 Axiomatic T riples + (0.01) + (0.06) + (2.02) ? (300.00) ? (300.00) 005 Everything is a Resource + (0.00) + (0.08) + (3.17) ? (300.00) ? (300.00) 006 Literal V alues represented by URIs and Blank No des + (0.00) + (0.03) + (1.60) ? (300.00) ? (300.00) 007 Equal Classes + (0.00) + (0.05) + (1.81) ? (300.00) ? (300.00) 008 Inverse F unctional Data Prop erties + (0.00) + (0.05) + (1.84) ? (300.00) ? (300.00) 009 Existential Restriction Entailmen ts + (0.00) + (0.05) + (2.01) ? (300.00) ? (285.58) 010 Negative Prop ert y Assertions + (0.05) + (0.11) + (1.92) ? (300.00) ? (300.00) 011 Entity Types as Classes + (0.00) + (0.04) + (1.78) ? (285.25) ? (300.00) 012 T emplate Class + (0.01) + (0.17) + (6.56) ? (300.00) ? (274.53) 013 Cliques + (4.20) + (4.25) + (5.60) ? (300.00) ? (300.00) 014 Harry belongs to some Species + (0.00) + (0.05) + (1.52) ? (300.00) ? (300.00) 015 Reflective T autologies I + (0.00) + (0.18) + (5.07) ? (300.95) ? (300.00) 016 Reflective T autologies I I + (0.03) + (0.16) + (3.17) ? (300.00) ? (300.00) 017 Builtin Based Definitions + (0.00) + (0.05) + (1.46) ? (300.00) ? (300.00) 018 Mo dified Logical V o cabulary Semantics + (0.00) + (0.10) + (2.77) ? (300.00) ? (300.00) 019 Disjoint Annotation Properties + (0.00) + (0.03) + (1.45) ? (300.00) ? (300.00) 020 Logical Complications + (31.08) + (31.06) + (34.26) ? (300.00) ? (300.00) 021 Comp osite Enumerations + (3.79) + (3.86) + (5.21) ? (300.00) ? (300.00) 022 List Member Access + (0.02) + (0.20) + (7.69) ? (300.00) ? (300.00) 023 Unique List Comp onen ts + (0.00) + (0.05) + (1.42) ? (300.00) ? (300.00) 024 Cardinality Restrictions on Complex Properties + (0.01) + (0.34) + (9.76) ? (300.00) ? (300.00) 025 Cyclic Dependencies between Complex Prop erties + (0.01) + (0.16) + (7.36) ? (300.00) ? (300.00) 026 Inferred Property Characteristics I + (0.01) + (0.06) + (1.46) ? (300.00) ? (300.00) 027 Inferred Property Characteristics II + (0.01) + (0.01) + (0.18) + (1.83) ? (300.00) 028 Inferred Property Characteristics II I + (0.02) + (0.03) + (0.18) + (1.81) ? (300.00) 029 Ex F also Quo dlibet + (0.00) + (0.04) + (1.52) ? (300.00) ? (300.00) 030 Bad Class + (0.00) + (0.04) + (1.49) ? (300.00) ? (300.00) 031 Large Universe + (0.01) + (0.13) + (3.04) ? (300.00) ? (300.00) 032 Datatype Relationships + (0.00) + (0.05) + (1.49) ? (300.00) ? (300.00) Reasoning in OWL 2 F ull using First-Order A TP 25 T able 15: Scalability results for the theorem prov er V ampire-SInE using the small-sufficient OWL 2 F ull subaxiomatizations. T est Case 0 1200 10k 100k 1M 001 Subgraph Entailmen t + (0.00) + (0.01) + (0.16) + (1.80) + (19.53) 002 Existential Blank Nodes + (0.00) + (0.01) + (0.16) + (1.76) + (20.15) 003 Blank Nodes for Literals + (0.00) + (0.01) + (0.16) + (1.81) + (19.52) 004 Axiomatic T riples + (0.00) + (0.02) + (0.17) + (1.79) + (19.35) 005 Everything is a Resource + (0.00) + (0.01) + (0.16) + (1.78) + (19.54) 006 Literal V alues represented by URIs and Blank No des + (0.00) + (0.01) + (0.16) + (1.80) + (19.54) 007 Equal Classes ? (0.00) ? (0.01) ? (0.16) ? (1.79) ? (20.31) 008 Inverse F unctional Data Prop erties + (0.00) + (0.01) + (0.17) + (1.77) + (19.40) 009 Existential Restriction Entailmen ts + (0.00) + (0.01) + (0.17) + (1.85) + (19.45) 010 Negative Prop ert y Assertions + (0.01) + (0.03) + (0.18) + (1.82) + (20.14) 011 Entity Types as Classes + (0.00) + (0.01) + (0.17) + (1.76) + (19.43) 012 T emplate Class + (0.20) + (0.23) + (0.38) + (2.05) + (20.40) 013 Cliques ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 014 Harry belongs to some Species + (0.32) + (0.34) + (0.50) + (2.15) + (19.84) 015 Reflective T autologies I + (0.00) + (0.01) + (0.17) + (1.78) + (19.41) 016 Reflective T autologies I I + (0.00) + (0.02) + (0.17) + (1.79) + (19.52) 017 Builtin Based Definitions ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 018 Mo dified Logical V o cabulary Semantics + (0.00) + (0.01) + (0.16) + (1.84) + (19.44) 019 Disjoint Annotation Properties + (0.00) + (0.01) + (0.16) + (1.79) + (19.45) 020 Logical Complications ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 021 Comp osite Enumerations ? (300.00) ? (300.00) ? (300.00) ? (300.00) ? (300.00) 022 List Member Access + (0.00) + (0.02) + (0.18) + (1.82) + (19.34) 023 Unique List Comp onen ts + (0.00) + (0.02) + (0.17) + (1.78) + (19.77) 024 Cardinality Restrictions on Complex Properties + (0.00) + (0.02) + (0.17) + (1.80) + (19.53) 025 Cyclic Dependencies between Complex Prop erties + (0.01) + (0.03) + (0.18) + (1.79) + (19.43) 026 Inferred Property Characteristics I + (0.00) + (0.02) + (0.17) + (1.80) + (19.42) 027 Inferred Property Characteristics II + (0.03) + (0.05) + (0.19) + (1.81) + (19.36) 028 Inferred Property Characteristics II I ? (300.00) + (242.19) + (262.52) ? (300.00) ? (300.00) 029 Ex F also Quo dlibet + (0.00) + (0.02) + (0.17) + (1.78) + (19.47) 030 Bad Class + (0.00) + (0.01) + (0.16) + (1.84) + (19.33) 031 Large Universe ? (0.00) ? (0.01) ? (0.16) ? (1.80) ? (19.35) 032 Datatype Relationships + (0.00) + (0.02) + (0.17) + (1.80) + (19.63) A.3 Mo del Finding Results The following table pro vides the ra w result data that underlies the results for the mo del-ﬁnding exp erimen ts, as rep orted in Section 5.4. The only additional data here is the CPU time for each exp erimen t. All exp erimen ts were conducted using the test suite of c haracteristic OWL 2 F ull conclusions, as in tro duced in Section 4.2 (see App endix B for more detailed information ab out the test suite). T able 16: Mo del finding results for the mo del-finders Par adox and DarwinFM on the ALCO F ull and RDFS axiom sets. The black entries indicate p ositiv e entailmen ts or inconsistent ontologies. P A/A=Paradox/ALCO F ull, P A/R=Paradox/RDFS, DF/R=DarwinFM/RDFS. T est Case P A/A P A/R DF/R 001 Subgraph Entailmen t 002 Existential Blank Nodes 003 Blank Nodes for Literals 004 Axiomatic T riples + (13.60) + (0.73) + (0.45) 005 Everything is a Resource + (15.08) + (0.90) + (0.12) 006 Literal V alues represented by URIs and Blank No des + (20.95) + (0.81) + (0.04) 007 Equal Classes + (13.01) + (1.03) + (7.19) 008 Inverse F unctional Data Prop erties + (11.74) + (0.99) + (0.08) 009 Existential Restriction Entailmen ts + (1.17) + (0.05) 010 Negative Prop ert y Assertions + (1.61) + (0.07) 011 Entity Types as Classes + (14.15) + (0.86) + (0.01) 012 T emplate Class + (1.70) + (0.33) 013 Cliques ? (300.11) + (2.17) + (0.05) 014 Harry belongs to some Species + (1.16) + (0.56) 015 Reflective T autologies I + (10.68) + (0.75) + (0.04) 016 Reflective T autologies I I + (8.21) + (0.77) + (2.05) 017 Builtin Based Definitions + (14.61) + (0.99) + (0.06) 018 Mo dified Logical V o cabulary Semantics + (89.21) + (0.93) + (7.35) 019 Disjoint Annotation Properties + (14.55) + (0.89) + (0.01) 020 Logical Complications ? (300.28) + (1.80) + (0.85) 021 Comp osite Enumerations ? (300.15) + (2.21) + (0.11) 022 List Member Access ? (300.15) + (1.79) + (0.06) 26 Mic hael Schneider and Geoﬀ Sutcliﬀe 023 Unique List Comp onen ts ? (300.15) + (1.17) + (0.05) 024 Cardinality Restrictions on Complex Properties + (16.76) + (1.16) + (0.10) 025 Cyclic Dependencies between Complex Prop erties ? (300.17) + (1.65) + (0.06) 026 Inferred Property Characteristics I ? (301.78) + (1.20) + (0.07) 027 Inferred Property Characteristics II ? (300.12) + (1.04) + (0.07) 028 Inferred Property Characteristics II I + (17.62) + (1.10) + (0.07) 029 Ex F also Quo dlibet + (1.27) + (0.07) 030 Bad Class + (17.88) + (1.05) + (0.01) 031 Large Universe ? (300.55) + (0.93) + (0.01) 032 Datatype Relationships + (9.69) + (0.85) + (0.06) Reasoning in OWL 2 F ull using First-Order A TP 27 B O WL 2 F ull Characteristic Conclusions T est Suite This app endix presents the suite of O WL 2 F ull-characteristic conclusion te st cases that was used in the ev aluation and has b een in tro duced in Section 4.2. The app endix is divided into tw o parts: Section B.1 lists the test c ases , and Section B.2 provides c orr e ctness pr o ofs for them. The test suite is also av ail- able in electronic form as part of the supplementary material for this pap er (see the do wnload link at the b eginning of Section 4), and can alternatively b e obtained as a sep ar ate p ackage from http://www.fzi.de/downloads/ipe/ schneid/testsuite- fullish.zip . B.1 T est Cases Eac h test case is giv en by its name , its typ e (one of “En tailment” or “Inconsis- tency”), a textual description , and the testing data as one or t wo RDF graphs for an inconsistency test or entailmen t test, resp ectiv ely . The RDF graphs are represen ted in T urtle syntax 5 . The electronic form of the test suite additionally con tains serializations in RDF/XML syntax 6 and in the TPTP syn tax [14]. 001 Subgraph Entailmen t (En tailment) In OWL 2 F ull, a given RDF graph en tails any of its sub graphs, even sub graphs that app ear to enco de brok en language constructs of O WL. F or example, the enco ding of a class sub- sumption axiom that uses a prop ert y restriction as its sup erclass entails the single owl:onProperty triple of the serialization. This is a characteristic feature of the whole family of RDF-based languages, starting with RDF Simple Entail- men t, and it demonstrates the strictly triple-centered view that OWL 2 F ull adopts. This b eha vior is t ypically shown by RDF entailmen t-rule reasoners, but not by OWL DL reasoners. Premise Graph Conclusion Graph ex:c rdfs:subClassOf ex:r . ex:r rdf:type owl:Restriction . ex:r owl:onProperty ex:p . ex:r owl:someValuesFrom ex:d . ex:r rdf:type owl:Restriction . ex:r owl:onProperty ex:p . 002 Existential Blank No des (Entailmen t) In OWL 2 F ull, every blank no de in an RDF graph is interpreted as an existentially quantiﬁed v ariable. On the one hand, this means that triples with URIs entail corresp onding triples with blank no des substituting the URIs. On the other hand, this means that triples with blank no des entail corresponding triples with alternative blank no des. This feature stems from RDF Simple Entailmen t. Many reasoners, in particular most RDF en tailment-rule reasoners, do not pro vide the existential seman tics of blank no des. 5 T urtle RDF syntax: http://www.w3.org/TeamSubmission/turtle/ 6 RDF/XML syntax: http://www.w3.org/TR/rdf- syntax- grammar/ 28 Mic hael Schneider and Geoﬀ Sutcliﬀe Premise Graph Conclusion Graph ex:s ex:p _:o . _:o ex:q ex:s . _:x ex:p _:y . _:y ex:q _:x . 003 Blank No des for Literals (En tailmen t) In OWL 2 F ull, an RDF triple ha ving a data literal in ob ject position en tails a corresponding triple with a blank no de substituting the literal. This feature stems from RDF Simple Entailmen t. It cannot b e exp ected from OWL DL reasoners, since OWL 2 DL treats suc h blank no des as anon ymous individuals, while the domains of individuals and data v alues are deﬁned to b e disjoin t. Most RDF en tailment-rule reasoners do not show this b eha vior, since they typically do not implement the existen tial seman tics of blank no des. Premise Graph Conclusion Graph ex:s ex:p "foo" . ex:s ex:p _:x . 004 Axiomatic T riples (Entailmen t) OWL 2 F ull has man y tautologies, i.e. statemen ts that are entailed by the empt y premise graph. Some of these tau- tologies ha ve the form of “axiomatic triples”, as deﬁned by RDF and RDFS, but OWL 2 F ull go es b ey ond these sp eciﬁcations. An example is the triple “ owl:Class rdfs:subClassOf owl:Thing ”. RDF entailmen t-rule reasoners, such as OWL 2 RL/RDF rule reasoners, often prov e at least some of the tautologies that O WL 2 F ull pro vides, while for O WL 2 DL, man y of these tautologies are not v alid, neither syntactically nor semantically . Premise Graph Conclusion Graph owl:Class rdf:type owl:Thing . owl:Class rdf:type owl:Class . owl:Class rdfs:subClassOf owl:Thing . owl:Class owl:equivalentClass rdfs:Class . rdfs:Datatype rdfs:subClassOf owl:Class . 005 Everything is a Resource (Entailmen t) In OWL 2 F ull, following the seman tics of RDFS, all three no des of an RDF triple denote RDF resources ( rdfs:Resource ) and OWL individuals ( owl:Thing ). In addition, the predicate no de of an RDF triple denotes an RDF prop ert y ( rdf:Property ) and an OWL ob ject prop ert y ( owl:ObjectProperty ). RDF entailmen t-rule reasoners will of- ten supp ort this view to at least some extent. While OWL 2 DL oﬀers some supp ort for this view syn tactically in the form of “punning”, the strict separa- tion of individuals, classes and prop erties in the seman tics of O WL 2 DL prev en ts complian t OWL DL reasoners from pro ducing man y of the conclusions known from OWL 2 F ull. In addition, OWL DL has only v ery limited supp ort for RDF en tity types suc h as rdf:Property . Reasoning in OWL 2 F ull using First-Order A TP 29 Premise Graph Conclusion Graph ex:s ex:p ex:o . ex:s rdf:type rdfs:Resource . ex:s rdf:type owl:Thing . ex:p rdf:type rdfs:Resource . ex:p rdf:type owl:Thing . ex:p rdf:type rdf:Property . ex:p rdf:type owl:ObjectProperty . ex:o rdf:type rdfs:Resource . ex:o rdf:type owl:Thing . 006 Literal V alues represen ted b y URIs and Blank No des (Entailmen t) In OWL 2 F ull, literals can b e assigned URIs or blank no des via owl:sameAs statemen ts. One can then use these references to make further assertions ab out the literals and to dra w semantic conclusions from them. This is an often dis- cussed replacement for literals in the sub ject position of RDF triples, whic h is not supp orted b y the RDF syn tax. It is often supp orted by RDF en tailmen t-rule reasoners to some exten t, but is not allow ed in OWL 2 DL, where URIs and blank no des are used to refer to individuals but not to data v alues. Premise Graph Conclusion Graph ex:u owl:sameAs "abc" . _:x owl:sameAs "abc" . _:x owl:sameAs ex:w . ex:u owl:sameAs ex:w . 007 Equal Classes (Entailmen t) In OWL 2 F ull, asserting that tw o classes are equal makes them into equiv alent classes. This allows to substitute one class name for the other in all class-related axioms, such as class assertions, class subsumption axioms, and property range axioms. This can be observed in the Link ed Op en Data cloud, whic h con tains man y sameAs links b et w een en tities that are sometimes used as as classes in certain contexts. Many RDF en tailment- rule reasoners provide for the exp ected semantic results. While syn tactically allo wed in OWL 2 DL via “punning”, the semantic results are not av ailable due to the strict separation of individuals and classes. Premise Graph Conclusion Graph ex:c1 owl:sameAs ex:c2 . ex:w rdf:type ex:c1 . ex:c rdfs:subClassOf ex:c1 . ex:p rdfs:range ex:c1 . ex:w rdf:type ex:c2 . ex:c rdfs:subClassOf ex:c2 . ex:p rdfs:range ex:c2 . 008 Inv erse F unctional Data Prop erties (En tailment) In OWL 2 F ull, data properties can b e deﬁned as inv erse-functional prop erties. This option is, for example, frequently applied in the FO AF sp eciﬁcation. While man y RDF en tailment-rule reasoners support the semantic consequences from these deﬁni- tions, they are not supp orted b y OWL 2 DL, which only allows ob ject properties to b e inv erse-functional. 30 Mic hael Schneider and Geoﬀ Sutcliﬀe Premise Graph Conclusion Graph foaf:mbox_sha1sum rdf:type owl:DatatypeProperty ; rdf:type owl:InverseFunctionalProperty . ex:bob foaf:mbox_sha1sum "xyz" . ex:robert foaf:mbox_sha1sum "xyz" . ex:bob owl:sameAs ex:robert . 009 Existential Restriction En tailments (Entailmen t) In OWL 2 F ull, a class assertion using an existen tial property restriction en tails a prop ert y asser- tion with a corresp onding blank no de. This inference is generally b e pro vided by O WL DL reasoners, but in most cases is not prov able by RDF en tailment rule reasoners, whic h t ypically do not implement the existential semantics of blank no des and existential prop ert y restrictions. Premise Graph Conclusion Graph ex:p rdf:type owl:ObjectProperty . ex:c rdf:type owl:Class . ex:s rdf:type [ rdf:type owl:Restriction ; owl:onProperty ex:p ; owl:someValuesFrom ex:c ] . ex:s ex:p _:x . _:x rdf:type ex:c . 010 Negative Prop ert y Assertions (En tailment) OWL 2 has in tro duced explicit supp ort for negative property assertions (NP As). How ever, it was al- ready p ossible to encode NP As in O WL 1, in terms of O WL 1 axioms and class expressions. These deﬁnitions are rather complex and require strong semantic supp ort for sev eral of the O WL language features. O WL 2 F ull can infer that the new explicit encoding of NP As follows from the corresp onding old enco ding of OWL 1. The same holds for OWL 2 DL. In contrast, RDF entailmen t-rule reasoners typically do not allow for such inferences due to the high semantic requiremen ts. Premise Graph Conclusion Graph ex:p rdf:type owl:ObjectProperty . ex:s rdf:type [ owl:onProperty ex:p ; owl:allValuesFrom [ owl:complementOf [ owl:oneOf ( ex:o ) ] ] ] . _:z rdf:type owl:NegativePropertyAssertion . _:z owl:sourceIndividual ex:s . _:z owl:assertionProperty ex:p . _:z owl:targetIndividual ex:o . 011 Entit y T yp es as Classes (Inconsistency) In O WL 2 F ull, en tity types, suc h as owl:Class , are regular classes. This seman tic prop ert y is basically in- herited from RDFS. This makes it p ossible, for example, to state that the entit y Reasoning in OWL 2 F ull using First-Order A TP 31 t yp es of classes and prop erties are mutually disjoint, and to infer inconsistencies if an entit y is used as both a class and a property . Some RDF entailmen t rule reasoners, suc h as those implementing the O WL 2 RL/RDF rules, follow this seman tics. O WL 2 DL, on the other hand, do es not support it, since it sees en tity t yp es are purely syntactic information. Graph owl:Class owl:disjointWith owl:ObjectProperty . ex:x rdf:type owl:Class . ex:x rdf:type owl:ObjectProperty . 012 T emplate Class (Entailmen t) In O WL 2 F ull, instead of explicitly as- signing features to a property , such as an entit y t yp e, prop ert y c haracteristics, or a domain, it is p ossible to build a class represen ting all these features and then make the prop ert y an instance of this “template class”. Some RDF en tail- men t rule reasoners, such as those implemen ting the OWL 2 RL/RDF rules, will supp ort this approac h to a certain exten t, while in O WL 2 DL, in most cases it is be syn tactically illegal and generally do es not hav e the expected seman tic meaning. Premise Graph Conclusion Graph foaf:Person rdf:type owl:Class . ex:PersonAttribute owl:intersectionOf ( owl:DatatypeProperty owl:FunctionalProperty [ rdf:type owl:Restriction ; owl:onProperty rdfs:domain ; owl:hasValue foaf:Person ] ) . ex:name rdf:type ex:PersonAttribute . ex:alice ex:name "alice" . ex:name rdf:type owl:FunctionalProperty . ex:alice rdf:type foaf:Person . 013 Cliques (Entailemen t) OWL 2 F ull can deﬁne the metaclass of all cliques, for which each instance is a clique of p eople that kno w every one else in that clique. The enco ding is not supp orted by OWL 2 DL, since it uses built-in v o cabulary terms as regular entities. F or RDF entailmen t rule reasoners, the seman tic requiremen ts for pro ducing all exp ected results are typically to o high. 32 Mic hael Schneider and Geoﬀ Sutcliﬀe Premise Graph Conclusion Graph ex:Clique rdf:type owl:Class . ex:sameCliqueAs rdfs:subPropertyOf owl:sameAs ; rdfs:range ex:Clique . ex:Clique rdfs:subClassOf [ rdf:type owl:Restriction ; owl:onProperty ex:sameCliqueAs ; owl:someValuesFrom ex:Clique ] . foaf:knows rdf:type owl:ObjectProperty ; owl:propertyChainAxiom ( rdf:type ex:sameCliqueAs [owl:inverseOf rdf:type] ) . ex:JoesGang rdf:type ex:Clique . ex:alice rdf:type ex:JoesGang . ex:bob rdf:type ex:JoesGang . ex:alice foaf:knows ex:bob . 014 Harry b elongs to some Species (Entailmen t) OWL 2 F ull supp orts the com bination of metamo delling and class union. F or example, pro vided that the classes of eagles and falcons are b oth instances of the metaclass of sp ecies, if one do es not exactly know whether Harry is an eagle or a falcon, one can still conclude that Harry must b elong to some sp ecies. O WL 2 DL do es not supp ort seman tic conclusions from metamodeling, although it allo ws for some metamo deling syntactically via “punning”. While man y RDF en tailment-rule reasoners ha ve some restricted supp ort for semantic metamodeling, drawing said conclusion from the union of classes typically go es beyond the capabilities of these reasoners. Premise Graph Conclusion Graph ex:Eagle rdf:type ex:Species . ex:Falcon rdf:type ex:Species . ex:harry rdf:type [ owl:unionOf ( ex:Eagle ex:Falcon ) ] . ex:harry rdf:type _:x . _:x rdf:type ex:Species . 015 Reﬂective T autologies I (En tailment) In OWL 2 F ull, the statement “ owl:sameAs owl:sameAs owl:sameAs ” is a tautology . This is a classic example used to demonstrate the use of built-in v o cabulary terms as regular entities, sometimes referred to as “syn tax reﬂection”. It is not allo wed in O WL 2 DL. Some RDF entailmen t-rule reasoners, suc h as those implementing the O WL 2 RL/RDF rules, do provide this result. Premise Graph Conclusion Graph owl:sameAs owl:sameAs owl:sameAs . Reasoning in OWL 2 F ull using First-Order A TP 33 016 Reﬂective T autologies I I (Entailmen t) In O WL 2 F ull, the class equiv- alence prop ert y is a subproperty of the class subsumption prop ert y . This is an example of the use of built-in vocabulary terms as regular entities, o ccasion- ally referred to as “syn tax reﬂection”. It is not allow ed in O WL 2 DL. RDF en tailment-rule reasoners may contain this tautology as a sp ecial rule, but oth- erwise cannot b e exp ected to pro vide this result. F or example, the result do es not follow from the OWL 2 RL/RDF rules. Premise Graph Conclusion Graph owl:equivalentClass rdfs:subPropertyOf rdfs:subClassOf . 017 Builtin Based Deﬁnitions (En tailmen t) In OWL 2 F ull, custom prop- erties can b e deﬁned based on existing built-in prop erties. F or example, a prop- ert y ex:noInstanceOf that is disjoin t from rdf:type can b e deﬁned, and this new prop ert y can b e used to state non-membership, whic h has semantic rami- ﬁcations. OWL 2 DL do es not allo w this. En tailment-rule reasoners can mak e suc h assertions, and may pro vide some limited supp ort for seman tic conclusions. Premise Graph Conclusion Graph ex:notInstanceOf owl:propertyDisjointWith rdf:type . ex:w rdf:type ex:c . ex:u ex:notInstanceOf ex:c . ex:w owl:differentFrom ex:u . 018 Mo diﬁed Logical V o cabulary Seman tics (Entailmen t) The seman- tics of OWL built-in v o cabulary terms can b e enriched in a wa y suc h that their application leads to additional results that are not a v ailable from their original meaning. F or example, the domain and range of owl:sameAs can be restricted to the class of p ersons, whic h renders all things that are equal into persons. OWL 2 DL do es not allow this, while RDF entailmen t-rule reasoners often provide some limited supp ort. Premise Graph Conclusion Graph owl:sameAs rdfs:domain ex:Person . ex:w owl:sameAs ex:u . ex:u rdf:type ex:Person . 019 Disjoint Annotation Prop erties (Inconsistency) In OWL 2 F ull, an- notation prop erties are normal ob ject prop erties. Th us, tw o annotation prop- erties can b e speciﬁed to b e disjoint, and semantic conclusions can b e drawn from this disjoin tness. This feature is, for example, used in the SK OS speciﬁca- tion to deﬁne the meaning of lexical lab els. O WL 2 DL provides only limited 34 Mic hael Schneider and Geoﬀ Sutcliﬀe syn tactic support for putting axioms on annotation properties, and do es not pro- vide any seman tic conclusions. One can exp ect limited seman tic supp ort from some RDF entailmen t-rule reasoners, such as those implemen ting the O WL 2 RL/RDF rules. Graph skos:prefLabel rdf:type owl:AnnotationProperty . skos:prefLabel rdfs:subPropertyOf rdfs:label . skos:altLabel rdf:type owl:AnnotationProperty . skos:altLabel rdfs:subPropertyOf rdfs:label . skos:prefLabel owl:propertyDisjointWith skos:altLabel . ex:foo skos:prefLabel "foo" . ex:foo skos:altLabel "foo" . 020 Logical Complications (En tailment) O WL 2 F ull allows complex logi- cal reasoning to b e p erformed. F or example, non-ob vious subsumption relation- ships betw een t wo classes can be inferred based on the application of disjointness and diﬀerent Bo olean connectiv es. This kind of reasoning is generally p ossible in unrestricted form in OWL 2 DL, but typically not with RDF entailmen t-rule reasoners. Premise Graph Conclusion Graph ex:c owl:unionOf ( ex:c1 ex:c2 ex:c3 ) . ex:d owl:disjointWith ex:c1 . ex:d rdfs:subClassOf [ owl:intersectionOf ( ex:c [ owl:complementOf ex:c2 ] ) ] . ex:d rdfs:subClassOf ex:c3 . 021 Comp osite En umerations (Entailmen t) OWL 2 F ull allows for the comp osition of enumerations via b o olean connectiv es. F or example, the union of the classes { w 1 , w 2 } and { w 2 , w 3 } can b e inferred to b e equiv alent to the class { w 1 , w 2 , w 3 } . O WL 2 DL reasoners can b e exp ected to pro vide this result, while RDF entailmen t-rule reasoners are t ypically unable to produce the result. Premise Graph Conclusion Graph ex:c1 owl:oneOf ( ex:w1 ex:w2 ) . ex:c2 owl:oneOf ( ex:w2 ex:w3 ) . ex:c3 owl:oneOf ( ex:w1 ex:w2 ex:w3 ) . ex:c4 owl:unionOf ( ex:c1 ex:c2 ) . ex:c3 owl:equivalentClass ex:c4 . 022 List Member Access (Entailmen t) In OWL 2 F ull, one can refer to all items within an RDF list. F or example, Chapter 9 of the SKOS Reference de- ﬁnes ordered concept collections via the prop ert y skos:memberList applied to some RDF list consisting of items of t yp e skos:Concept . SK OS further deﬁnes Reasoning in OWL 2 F ull using First-Order A TP 35 non-ordered concept collections b y applying the prop ert y skos:member repeat- edly to single en tities of type skos:Concept . SKOS statement S36 sa ys that a non-ordered concept collection can be inferred from an ordered collection. An example is giv en in Section 9.6.1 of the SK OS Reference. OWL 2 F ull allows this statement to b e expressed semantically . Both the encoding of S36 and the example inference is giv en here. RDF entailmen t-rule reasoners implementing the O WL 2 RL/RDF rules also pro duce the result. O WL 2 DL cannot make assertions ab out RDF lists. Premise Graph Conclusion Graph skos:memberList rdfs:subPropertyOf _:pL . skos:member owl:propertyChainAxiom ( _:pL rdf:first ) . _:pL owl:propertyChainAxiom ( _:pL rdf:rest ) . ex:MyOrderedCollection rdf:type skos:OrderedCollection ; skos:memberList ( ex:X ex:Y ex:Z ) . ex:MyOrderedCollection skos:member ex:X . ex:MyOrderedCollection skos:member ex:Y . ex:MyOrderedCollection skos:member ex:Z . 023 Unique List Comp onen ts (Entailmen t) In principle, it is p ossible to create argumen t lists of OWL constructs that are non-linear. Section 3.3.3 of the RDF Semantics sp eciﬁcation allo ws semantic extensions to place extra syntactic w ellformedness restrictions on the use of the RDF Collections v o cabulary in order to rule out graphs containing non-linear lists. While OWL 2 F ull do es not pro vide this directly , it can state that the List v o cabulary property rdf:first is a functional property . This has seman tic consequences even if the argument list of an OWL construct is giv en in a non-linear form. RDF entailmen t-rule reasoners often hav e some limited supp ort for these kinds of results. OWL 2 DL cannot make assertions about RDF lists. Premise Graph Conclusion Graph rdf:first rdf:type owl:FunctionalProperty . ex:w rdf:type [ rdf:type owl:Class ; owl:oneOf _:l ] . _:l rdf:first ex:u . _:l rdf:first ex:v . _:l rdf:rest rdf:nil . ex:w owl:sameAs ex:u . ex:w owl:sameAs ex:v . 024 Cardinality Restrictions on Complex Prop erties (Entailmen t) OWL 2 DL do es cannot place cardinality restrictions on transitive properties. OWL 2 F ull allows this. This can, for example, b e used to state that ev ery person has at least one ancestor. The existence of an ancestor can then b e inferred for any giv en p erson. RDF en tailment-rule reasoners may provide some limited supp ort but typically are unable to produce the result of this particular example. 36 Mic hael Schneider and Geoﬀ Sutcliﬀe Premise Graph Conclusion Graph ex:hasAncestor rdf:type owl:TransitiveProperty . ex:Person rdfs:subClassOf [ rdf:type owl:Restriction ; owl:onProperty ex:hasAncestor ; owl:minCardinality "1"^^xsd:nonNegativeInteger ] . ex:alice rdf:type ex:Person . ex:bob rdf:type ex:Person . ex:alice ex:hasAncestor ex:bob . ex:bob ex:hasAncestor _:x . ex:alice ex:hasAncestor _:x . 025 Cyclic Dep endencies b et ween Complex Properties (Entailmen t) OWL 2 DL do es not allow cyclic dep endencies b et w een complex prop erties that are de- ﬁned via subprop ert y chain axioms. O WL 2 F ull allo ws this. F or example, the uncle relation and the cousin relation can b e expressed mutually in terms of the other relation using t wo subproperty chain axioms. This pro vides for more precise characterizations of prop erties than it is possible in OWL 2 DL. RDF en tailment rule reasoners that implement the OWL 2 RL/RDF rules provide limited supp ort for reasoning in suc h scenarios. Premise Graph Conclusion Graph ex:hasUncle owl:propertyChainAxiom ( ex:hasCousin ex:hasFather ) . ex:hasCousin owl:propertyChainAxiom ( ex:hasUncle [ owl:inverseOf ex:hasFather ] ) . ex:alice ex:hasFather ex:dave . ex:alice ex:hasCousin ex:bob . ex:bob ex:hasFather ex:charly . ex:bob ex:hasUncle ex:dave . ex:alice ex:hasUncle ex:charly . ex:bob ex:hasCousin ex:alice . 026 Inferred Prop ert y Characteristics I (En tailment) In OWL 2 F ull, as in O WL 2 DL, a prop ert y that has a domain and a range being singleton classes is entailed to b e an inv erse-functional prop ert y . RDF en tailment-rule reasoners cannot b e exp ected to pro vide this result, since it requires sophisticated reasoning. Premise Graph Conclusion Graph ex:p rdfs:domain [ owl:oneOf ( ex:w ) ] . ex:p rdfs:range [ owl:oneOf ( ex:u ) ] . ex:p rdf:type owl:InverseFunctionalProperty . 027 Inferred Prop ert y Characteristics I I (En tailment) In O WL 2 F ull, if the c hain of a prop ert y and its in verse prop ert y builds a subproperty chain of owl:sameAs , then that property is in verse-functional. The application of the Reasoning in OWL 2 F ull using First-Order A TP 37 built-in v o cabulary term owl:sameAs is not allow ed in OWL 2 DL. Newer RDF en tailment-rule reasoners, such as those implemen ting the OWL 2 RL/RDF rules, may provide some limited semantic supp ort. Premise Graph Conclusion Graph owl:sameAs owl:propertyChainAxiom ( ex:p [owl:inverseOf ex:p] ) . ex:p rdf:type owl:InverseFunctionalProperty . 028 Inferred Prop ert y Characteristics I I I (En tailmen t) In OWL 2 F ull, instead of using the built-in prop ert y c haracteristics of in verse-functional prop- erties, prop erties can be made into instances of the custom class of the inv erses of all functional prop erties. OWL 2 DL do es not allow the use of built-in v o cab- ulary terms as regular entities. F or RDF en tailment-rule reasoners, the semantic result given in this example is typically to o demanding. Premise Graph Conclusion Graph ex:InversesOfFunctionalProperties owl:equivalentClass [ rdf:type owl:Restriction ; owl:onProperty owl:inverseOf ; owl:someValuesFrom owl:FunctionalProperty ] . ex:InversesOfFunctionalProperties rdfs:subClassOf owl:InverseFunctionalProperty . 029 Ex F also Quo dlibet (Entailmen t) In O WL 2 F ull, an inconsistent premise on tology entails arbitrary conclusion ontologies (“principle of explosion”, “ex falso sequitur quo dlibet”). OWL 2 DL has the same seman tic prop ert y , but man y existing OWL 2 DL reasoners signal an error when given an inconsistent premise ontology , and do not pro duce the exp ected result (how ever, it would b e trivial to extend an OWL 2 DL reasoner to give the result as a reaction to an inconsistency error). RDF entailmen t-rule reasoners cannot b e exp ected to pro duce tyhis result, since it requires full seman tic support for classical negation. Premise Graph Conclusion Graph ex:A rdf:type owl:Class . ex:B rdf:type owl:Class . ex:w rdf:type [ owl:intersectionOf ( ex:A [owl:complementOf ex:A] ) ] . ex:w rdf:type ex:B . 030 Bad Class (Inconsistency) If an OWL 2 F ull ontology contains a class that has the Russell Set as its class extension, then the ontology is inconsistent. 38 Mic hael Schneider and Geoﬀ Sutcliﬀe This situation would o ccur for ev en the empty ontology if the so-called OWL 2 F ull comprehension conditions, as non-normatively deﬁned in Chapter 8 of the O WL 2 RDF-Based Semantics, were a normative part of OWL 2 F ull, as ex- plained in Chapter 9 of the sp eciﬁcation document. O WL 2 DL do es not kno w ab out this issue, and RDF en tailment-rule reasoners cannot b e expected to know ab out it due to their relatively weak semantics. Graph ex:c rdf:type owl:Class . ex:c owl:complementOf [ rdf:type owl:Restriction ; owl:onProperty rdf:type ; owl:hasSelf "true"^^xsd:boolean ] . 031 Large Universe (Inconsistency) The univ erse of an OWL 2 F ull in ter- pretation cannot consist of only a single individual. This means that owl:Thing cannot b e equiv alen t to a singleton enumeration class, without leading to an inconsisten t ontology . This is diﬀeren t from O WL 2 DL, for whic h the only re- striction on the universe is that it has to b e non-empt y . RDF entailmen t-rule reasoners cannot b e exp ected to provide the inconsistency result, since this re- quires strong logic-based reasoning. Graph owl:Thing owl:equivalentClass [ owl:oneOf ( ex:w ) ] . 032 Datatype Relationships (En tailment) According to the XSD Datat yp es sp eciﬁcation, the v alue spaces of the datatypes xsd:decimal and xsd:string are disjoint, while the v alue space of xsd:integer is a subset of the v alue space of xsd:decimal . In OWL 2 F ull, these relationships b et ween the data v alues of datat yp es can b e observed as corresponding relationships b et w een the classes represen ting these datatypes. OWL 2 DL also follows the XSD semantics, but it do es not supp ort to explicitly query for subsumption or disjointness relationships b et ween datat yp es. Some RDF entailmen t-rule reasoners may decide to provide the diﬀeren t relationships b et ween XSD datatypes as explicit facts or rules, but cannot, in general, b e exp ected to do so. Premise Graph Conclusion Graph xsd:decimal owl:disjointWith xsd:string . xsd:integer rdfs:subClassOf xsd:decimal . Reasoning in OWL 2 F ull using First-Order A TP 39 B.2 Correctness Proofs This section provides c orr e ctness pr o ofs for all test cases listed in Section B.1. The pro ofs hav e b een constructed with resp ect to the OWL 2 RDF Based Seman- tics [12] and the RDF Seman tics [5], whic h conjoin tly specify the mo del-theoretic seman tics of OWL 2 F ull. 001 Subgraph Entailmen t (Pro of ) Let I b e an O WL 2 RDF-Based inter- pretation that satisﬁes the premise graph, so the follo wing b ecomes true: h I ( ex:c ) , I ( ex:r ) i ∈ IEXT( I ( rdfs:subClassOf )) h I ( ex:r ) , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) h I ( ex:r ) , I ( ex:p ) i ∈ IEXT( I ( owl:onProperty )) h I ( ex:r ) , I ( ex:d ) i ∈ IEXT( I ( owl:someValuesFrom )) Then, in particular, the conjunction of the subset of atoms h I ( ex:r ) , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) h I ( ex:r ) , I ( ex:p ) i ∈ IEXT( I ( owl:onProperty )) is also satisﬁed. 002 Existential Blank No des (Pro of ) Let I b e an O WL 2 RDF-Based in- terpretation interpretation and B b e a blank no de mapping for the blank no des in the premise graph, suc h that I + B satisﬁes the premise graph. This gives ∃ o : h I ( ex:s ) , o i ∈ IEXT( I ( ex:p )) ∧ h o, I ( ex:s ) i ∈ IEXT( I ( ex:q )) W eakening this statement by introducing an existentially quantiﬁed v ariable for I ( ex:s ) logically implies ∃ x, y : h x, y i ∈ IEXT( I ( ex:p )) ∧ h y , x i ∈ IEXT( I ( ex:q )) Th us, there is a blank no de mapping B 0 , suc h that I + B 0 satisﬁes the conclusion graph. 003 Blank No des for Literals (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation that satisﬁes the premise graph. Then from h I ( ex:s ) , I ( "foo" ) i ∈ IEXT( I ( ex:p )) and taking in to account that literals denote individuals in the universe, we receiv e the formally w eaker assertion ∃ x : h I ( ex:s ) , x i ∈ IEXT( I ( ex:p )) Th us, there is a blank no de mapping B , such that I + B satisﬁes the conclusion graph. 40 Mic hael Schneider and Geoﬀ Sutcliﬀe 004 Axiomatic T riples (Pro of ) Given a satisfying O WL 2 RDF-Based in- terpretation I for the empt y graph. 1) Claim: h I ( owl:Class ) , I ( owl:Thing ) i ∈ IEXT( I ( rdf:type )). Pr o of: The denotation of owl:Class is in the universe, i.e., I(owl:Class) ∈ IR. The claim follows from ICEXT( I ( owl:Thing )) = IR (OWL2/T ab5.2) and from the RDFS seman tic condition deﬁning “ICEXT”. 2) Claim: h I ( owl:Class ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )). Pr o of: I ( owl:Class ) ∈ IC and ICEXT( I ( owl:Class )) = IC (OWL2/T ab5.2). The claim follo ws from the RDFS semantic condition deﬁning “ICEXT”. 3) Claim: h I ( owl:Class ) , I ( owl:Thing ) i ∈ IEXT( I ( rdfs:subClassOf )). Pr o of: According to 2), I ( owl:Class ) ∈ IC. F urther, I ( owl:Thing ) ∈ IC accord- ing to OWL2/T ab5.2. Given arbitrary x ∈ ICEXT( I ( owl:Class )), then x ∈ IR, and thus x ∈ ICEXT( I ( owl:Thing )) according to O WL2/T ab5.2. The claim follo ws from using the “ ← ” direction of the OWL 2 semantic condition for class subsumption (OWL2/T ab5.8). 4) Claim: h I ( owl:Class ) , I ( rdfs:Class ) i ∈ IEXT( I ( owl:equivalentClass )). Pr o of: According to 2), we get I ( owl:Class ) ∈ IC. According to O WL2/T ab5.2, w e get I ( rdfs:Class ) ∈ IC. F rom O WL2/T ab5.2 we get ICEXT( I ( owl:Class )) = IC = ICEXT( I ( rdfs:Class )). The claim follo ws from using the “ ← ” direction of the O WL 2 semantic condition for class equiv alence (O WL2/T ab5.9). 5) Claim: h I ( rdfs:Datatype ) , I ( owl:Class ) i ∈ IEXT( I ( rdfs:subClassOf )). Pr o of: According to 2), we get I ( owl:Class ) ∈ IC. According to O WL2/T ab5.2, w e get I ( rdfs:Datatype ) ∈ IC. Given arbitrary x ∈ ICEXT( I ( rdfs:Datatype )). By OWL2/T ab5.2 we get x ∈ IDC. Then, OWL2/T ab5.1 giv es x ∈ IC. Fi- nally , O WL2/T ab5.2 gives x ∈ ICEXT( I ( owl:Class )). The claim now follo ws from the “ ← ” direction of the OWL 2 semantic condition for class subsumption (O WL2/T ab5.8). 005 Everything is a Resource (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation that satisﬁes the premise graph. 1a) Claim: I ( ex:s ) ∈ IR, I ( ex:p ) ∈ IR, I ( ex:o ) ∈ IR. Pr o of: Since I is a simple-interpretation, I ( ex:s ) and I ( ex:o ) are in IR, and I ( ex:p ) is in IP. According to the RDF seman tic condition that deﬁnes “IP”, h I ( ex:p ) , I ( rdf:Property ) i ∈ IEXT( I ( rdf:type ), and thus I ( ex:p ) ∈ IR. 1b) Claim: h I ( ex:s ) , I ( rdfs:Resource ) i ∈ IEXT( I ( rdf:type )), and ditto for ex:p and ex:o . Pr o of: 1a) show ed I ( ex:s ) ∈ I R , and from the RDFS seman tic condition that deﬁnes the class extension of rdfs:Resource to b e the set IR follows I ( ex:s ) ∈ ICEXT( I ( rdfs:Resource )). The claim follo ws from the RDFS semantic condi- tion that deﬁnes “ICEXT”. Analog pro ofs apply to ex:p and ex:o . 1c) Claim: h I ( ex:s ) , I ( owl:Thing ) i ∈ IEXT( I ( rdf:type )), and ditto for ex:p and ex:o . Pr o of: As for 1b), but applying the OWL 2 semantic condition that deﬁnes the extension of owl:Thing (OWL2/T ab5.2) instead of rdfs:Resource . 2a) Claim: I ( ex:p ) ∈ IP. Reasoning in OWL 2 F ull using First-Order A TP 41 Pr o of: This follows directly from I b eing a simple-interpretation that satisﬁes the premise graph. 2b) Claim: h I ( ex:p ) , I ( rdf:Property ) i ∈ IEXT( I ( rdf:type )). Pr o of: According to 2a), I ( ex:p ) ∈ IP, and the RDF semantic condition that deﬁnes “IP” pro vides the claim. 2c) Claim: h I ( ex:p ) , I ( owl:ObjectProperty ) i ∈ IEXT( I ( rdf:type )). Pr o of: According to 2a), I ( ex:p ) ∈ IP, and according to O WL2/T ab5.2 the class extension of I ( owl:ObjectProperty ) is IP. The claim follo ws from the RDFS seman tic extension that deﬁnes “ICEXT”. 006 Literal V alues represen ted b y URIs and Blank No des (Pro of ) Let I b e an O WL 2 RDF-Based in terpretation and B b e a blank no de mapping for the blank nodes in the premise graph suc h that I + B satisﬁes the premise graph. Giv en an x , suc h that (1) h I ( ex:u ) , I ( "abc" ) i ∈ IEXT( I ( owl:sameAs )), and (2) h x, I ( "abc" ) i ∈ IEXT( I ( owl:sameAs )), and (3) h x, I ( ex:w ) i ∈ IEXT( I ( owl:sameAs )). By the “ → ” direction of the semantic condition for owl:sameAs (OWL2/T ab5.9), w e receiv e that (1 0 ) I ( ex:u ) = I ( "abc" ), and (2 0 ) x = I ( "abc" ), and (3 0 ) x = I ( ex:w ). F rom (2’) and (3’) we conclude that (4) I ( "abc" ) = I ( ex:w ). F rom (1’) and (4) we conclude (5) I ( ex:u ) = I ( ex:w ). F rom the “ ← ” direction of the semantic condition for owl:sameAs (OWL2/T ab5.9), w e conclude (6) h I ( ex:u ) , I ( ex:w ) i ∈ IEXT( I ( owl:sameAs )). 007 Equal Classes (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation that satisﬁes the premise graph. F rom the fact h I ( ex:c1 ) , I ( ex:c2 ) i ∈ IEXT( I ( owl:sameAs )) the “ → ” direction of the semantic condition for owl:sameAs (OWL2/T ab5.9) pro vides I ( ex:c1 ) = I ( ex:c2 ). 42 Mic hael Schneider and Geoﬀ Sutcliﬀe So w e can substitute an y o ccurrence of I ( ex:c1 ) by I ( ex:c2 ). Hence, from the premises h I ( ex:w ) , I ( ex:c1 ) i ∈ IEXT( I ( rdf:type )) h I ( ex:c ) , I ( ex:c1 ) i ∈ IEXT( I ( rdfs:subClassOf )) h I ( ex:p ) , I ( ex:c1 ) i ∈ IEXT( I ( rdfs:range )) w e receiv e the corresp onding conclusions h I ( ex:w ) , I ( ex:c2 ) i ∈ IEXT( I ( rdf:type )) h I ( ex:c ) , I ( ex:c2 ) i ∈ IEXT( I ( rdfs:subClassOf )) h I ( ex:p ) , I ( ex:c2 ) i ∈ IEXT( I ( rdfs:range )) 008 Inv erse F unctional Data Prop erties (Pro of ) Let I b e an OWL 2 RDF-Based interpretation that satisﬁes the premise graph. W e start from (1 a ) h I ( ex:bob ) , I ( "xyz" ) i ∈ IEXT( I ( foaf:mbox sha1sum )), and (1 b ) h I ( ex:robert ) , I ( "xyz" ) i ∈ IEXT( I ( foaf:mbox sha1sum )), as well as from (2) h I ( foaf:mbox sha1sum ) , I ( owl:InverseFunctionalProperty ) i ∈ IEXT( I ( rdf:type )) . F rom (2) and the “ → ” direction of the RDFS semantic condition for ICEXT w e receiv e (2 0 ) I ( foaf:mbox sha1sum ) ∈ ICEXT( I ( owl:InverseFunctionalProperty )) . This allows to apply the “ → ” direction of semantic condition for in verse-functional prop erties (OWL2/T ab5.13), whic h pro vides (3) ∀ x 1 , x 2 , y : h x 1 , y i ∈ IEXT( I ( foaf:mbox sha1sum )) ∧ h x 2 , y i ∈ IEXT( I ( foaf:mbox sha1sum )) ⇒ x 1 = x 2 Applying (3) to (1a) and (1b) with x 1 := I ( ex:bob ), x 2 := I ( ex:robert ), and y := I ( "xyz" ) results in (4) I ( ex:bob ) = I ( ex:robert ) . Using the “ ← ” direction of the semantic condition for owl:sameAs (O WL2/T ab5.9) results in (5) h I ( ex:bob ) , I ( ex:robert ) i ∈ IEXT( I ( owl:sameAs )). Reasoning in OWL 2 F ull using First-Order A TP 43 009 Existential Restriction En tailments (Pro of ) Let I b e an OWL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Given a z such that the follo wing holds: (1 a ) h I ( ex:p ) , I ( owl:ObjectProperty ) i ∈ IEXT( I ( rdf:type )) , (1 b ) h I ( ex:c ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 c ) h I ( ex:s ) , z i ∈ IEXT( I ( rdf:type )) , (1 d ) h z , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) , (1 e ) h z , I ( ex:p ) i ∈ IEXT( I ( owl:onProperty )) , (1 f ) h z , I ( ex:c ) i ∈ IEXT( I ( owl:someValuesFrom )) . F rom the RDFS semantic condition for ICEXT (“ → ” direction) and (1c) follows (2) I ( ex:s ) ∈ ICEXT( z ) . F rom the seman tic condition of owl:someValuesFrom (O WL2/T ab5.6), (1e) and (1f ) follo ws (3) ∀ y : y ∈ ICEXT( z ) ⇔ ∃ x : [ h y , x i ∈ IEXT( I ( ex:p )) ∧ x ∈ ICEXT( I ( ex:c )) ] . F rom (2) and (3) follows (4) ∃ x : h I ( ex:s ) , x i ∈ IEXT( I ( ex:p )) ∧ x ∈ ICEXT( I ( ex:c )) . By the RDFS semantic condition for ICEXT (“ ← ” direction) and (4) we receive (5) ∃ x : h I ( ex:s ) , x i ∈ IEXT( I ( ex:p )) ∧ h x, I ( ex:c ) i ∈ IEXT( I ( rdf:type )) . 010 Negative Prop ert y Assertions (Pro of ) Let I be an O WL 2 RDF- Based in terpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let x 1 , x 2 , x 3 and x 4 b e individuals, suc h that the follo wing holds (1 a ) h I ( ex:p ) , I ( owl:ObjectProperty ) i ∈ IEXT( I ( rdf:type )) , (1 b ) h I ( ex:s ) , x 1 i ∈ IEXT( I ( rdf:type )) , (1 c ) h x 1 , I ( ex:p ) i ∈ IEXT( I ( owl:onProperty )) , (1 d ) h x 1 , x 2 i ∈ IEXT( I ( owl:allValuesFrom )) , (1 e ) h x 2 , x 3 i ∈ IEXT( I ( owl:complementOf )) , (1 f ) h x 3 , x 4 i ∈ IEXT( I ( owl:oneOf )) , (1 g ) h x 4 , I ( ex:o ) i ∈ IEXT( I ( rdf:first )) , (1 h ) h x 4 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . F rom the RDFS semantic condition for ICEXT (“ → ” direction) and (1b) follo ws (2 b ) I ( ex:s ) ∈ ICEXT( x 1 ) . F rom the semantic condition for owl:allValuesFrom (OWL2/T ab5.6), (1c) and (1d) follows (2 c ) ∀ y : y ∈ ICEXT( x 1 ) ⇔ ∀ z : [ h y , z i ∈ IEXT( I ( ex:p )) ⇒ z ∈ ICEXT( x 2 )] . 44 Mic hael Schneider and Geoﬀ Sutcliﬀe F rom the “ → ” direction of the semantic condition for class complement (OWL2/T ab5.4) and (1e) follo ws (2 e ) ∀ y : y ∈ ICEXT( x 2 ) ⇔ y / ∈ ICEXT( x 3 ) . F rom the “ → ” direction of the seman tic condition for singleton en umerations (O WL2/T ab5.5) and (1f ), (1g) and (1h) follows (2 f ) ∀ y : y ∈ ICEXT( x 3 ) ⇔ y = I ( ex:o ) . Assume that (3) h I ( ex:s ) , I ( ex:o ) i ∈ I E X T ( I ( ex:p )) . By (2b), (2c) and (3) we receive (4) I ( ex:o ) ∈ ICEXT( x 2 ) . By (4) and (2e) follo ws (5) I ( ex:o ) / ∈ ICEXT( x 3 ) . By (5) and (2f ) follo ws (6) I ( ex:o ) 6 = I ( ex:o ) . This is a contradiction, hence assumption (3) w as wrong. So w e ha ve: (3 0 ) h I ( ex:s ) , I ( ex:o ) i / ∈ IEXT( I ( ex:p )) . Since I is a simple-in terpretation, w e receiv e (7 a ) I ( ex:s ) ∈ IR , (7 b ) I ( ex:o ) ∈ IR . By the prop ert y extension of owl:onProperty (OWL2/T ab5.3) follo ws (7 c ) I ( ex:p ) ∈ IP . F rom the second semantic condition for NP As in O WL2/T ab5.15, (3’), (7a), (7b) and (7c) follo ws that there exists some z , such that (8 a ) h z , I ( ex:s ) i ∈ IEXT( I ( owl:sourceIndividual )) , (8 b ) h z , I ( ex:p ) i ∈ IEXT( I ( owl:assertionProperty )) , (8 c ) h z , I ( ex:o ) i ∈ IEXT( I ( owl:targetIndividual )) . Finally , from the prop ert y extension of owl:sourceIndividual and (8a) follows (9) z ∈ ICEXT( I ( owl:NegativePropertyAssertion )) , whic h by the “ ← ” direction of the RDFS seman tic condition for ICEXT results in (9 0 ) h z , I ( owl:NegativePropertyAssertion ) i ∈ IEXT( I ( rdf:type )) . The result consists of (9), (8a), (8b) and (8c). Reasoning in OWL 2 F ull using First-Order A TP 45 011 Entit y T yp es as Classes (Proof ) Let I b e an OWL 2 RDF-Based in- terpretation that satisﬁes the premise graph. W e start from the facts: (1 a ) h I ( owl:Class ) , I ( owl:ObjectProperty )) ∈ IEXT( I ( owl:disjointWith )) , (1 b ) h I ( ex:x ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 c ) h I ( ex:x ) , I ( owl:ObjectProperty ) i ∈ IEXT( I ( rdf:type )) . First, b y the “ → ” direction of RDFS semantic condition of ICEXT, w e rewrite (1b) and (1c) to (1 b 0 ) I ( ex:x ) ∈ ICEXT( I ( owl:Class )) , (1 c 0 ) I ( ex:x ) ∈ ICEXT( I ( owl:ObjectProperty )) . By the “ → ” direction of the semantic condition for class disjoin tness (OWL2/T ab5.9) and (1a) follo ws (2) ∀ z : ¬ [ z ∈ ICEXT( I ( owl:Class )) ∧ z ∈ ICEXT( I ( owl:ObjectProperty ))] . Sp ecialization (2) to z := I ( ex:x ) implies (3) ¬ [ I ( ex:x ) ∈ ICEXT( I ( owl:Class )) ∧ I ( ex:x ) ∈ ICEXT( owl:ObjectProperty ))] . No w (3) is in contradiction with (1b’) and (1c’). 012 T emplate Class (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Given the existence of individuals l 1 , l 2 , l 3 and r , such that (1 a 1) h I ( foaf:Person ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 b 1) h I ( ex:PersonAttribute ) , l 1 i ∈ IEXT( I ( owl:intersectionOf )) , (1 c 1) h l 1 , I ( owl:DatatypeProperty ) i ∈ IEXT( I ( rdf:first )) , (1 c 2) h l 1 , l 2 i ∈ IEXT( I ( rdf:rest )) , (1 c 3) h l 2 , I ( owl:FunctionalProperty ) i ∈ IEXT( I ( rdf:first )) , (1 c 4) h l 2 , l 3 i ∈ IEXT( I ( rdf:rest )) , (1 c 5) h l 3 , r i ∈ IEXT( I ( rdf:first )) , (1 c 6) h l 3 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 d 1) h r , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) , (1 d 2) h r , I ( rdfs:domain ) i ∈ IEXT( I ( owl:onProperty )) , (1 d 3) h r , I ( foaf:Person ) i ∈ IEXT( I ( owl:hasValue )) , (1 e 1) h I ( ex:name ) , I ( ex:PersonAttribute ) i ∈ IEXT( I ( rdf:type )) , (1 f 1) h I ( ex:alice ) , I ( "alice" ) i ∈ IEXT( I ( ex:name )) . F rom (1b1), (1c1) – (1c6), and the semantic condition for class intersection (O WL2/T ab5.4, “ → ”, ternary) follo ws (2) ∀ x : x ∈ ICEXT( I ( ex:PersonAttribute )) ⇔ x ∈ ICEXT( I ( owl:DatatypeProperty )) ∧ x ∈ ICEXT( I ( owl:FunctionalProperty )) ∧ x ∈ ICEXT( r ) . 46 Mic hael Schneider and Geoﬀ Sutcliﬀe F rom (1e1) and the RDFS seman tic condition for ICEXT (“ → ”) follo ws (3) I ( ex:name ) ∈ ICEXT( I ( ex:PersonAttribute )) . F rom (3) and (2) follows (4) I ( ex:name ) ∈ ICEXT( I ( owl:FunctionalProperty )) . F rom (4) and the RDFS seman tic condition for ICEXT (“ ← ”) follo ws (5) h I ( ex:name ) , I ( owl:FunctionalProperty ) i ∈ IEXT( I ( rdf:type )) . F rom (1d2), (1d3) and the semantic condition for has-v alue restrictions (OWL2/T ab5.6) follo ws (6) ∀ x : x ∈ I C E X T ( r ) ⇔ h x, I ( foaf:Person ) i ∈ IEXT( I ( rdfs:domain )) . F rom (3) and (2) follows (7) I ( ex:name ) ∈ ICEXT( r ) . F rom (7) and (6) follows (8) h I ( ex:name ) , I ( foaf:Person ) i ∈ IEXT( I ( rdfs:domain )) . F rom (1f1), (8) and the RDFS semantic condition for prop ert y domains follows (9) I ( ex:alice ) ∈ ICEXT( I ( foaf:Person )) . F rom (9) and the RDFS seman tic condition for ICEXT (“ ← ”) follo ws (10) h I ( ex:alice ) , I ( foaf:Person ) i ∈ IEXT( I ( rdf:type )) . The conjecture follo ws from (5) and (10). 013 Cliques (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation and B b e a blank no de mapping for the blank nodes in the premise graph suc h that Reasoning in OWL 2 F ull using First-Order A TP 47 I + B satisﬁes the premise graph. Let there be r , i , l 1 , l 2 and l 3 suc h that (1 a ) h I ( ex:Clique ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 b ) h I ( ex:sameCliqueAs ) , I ( owl:sameAs ) i ∈ IEXT( I ( rdfs:subPropertyOf )) , (1 c ) h I ( ex:sameCliqueAs ) , I ( ex:Clique ) i ∈ IEXT( I ( rdfs:range )) , (1 d ) h I ( ex:Clique ) , r i ∈ IEXT( I ( rdfs:subClassOf )) , (1 e ) h r , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) , (1 f ) h r , I ( ex:sameCliqueAs ) i ∈ IEXT( I ( owl:onProperty )) , (1 g ) h r , I ( ex:Clique ) i ∈ IEXT( I ( owl:someValuesFrom )) , (1 h ) h I ( foaf:knows ) , I ( owl:ObjectProperty ) i ∈ IEXT( I ( rdf:type )) , (1 j ) h I ( foaf:knows ) , l 1 i ∈ IEXT( I ( owl:propertyChainAxiom )) , (1 k ) h l 1 , I ( rdf:type ) i ∈ IEXT( I ( rdf:first )) , (1 m ) h l 1 , l 2 i ∈ IEXT( I ( rdf:rest )) , (1 n ) h l 2 , I ( ex:sameCliqueAs ) i ∈ IEXT( I ( rdf:first )) , (1 o ) h l 2 , l 3 i ∈ IEXT( I ( rdf:rest )) , (1 p ) h l 3 , i i ∈ IEXT( I ( rdf:first )) , (1 q ) h l 3 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 r ) h i, I ( rdf:type ) i ∈ IEXT( I ( owl:inverseOf )) , (1 s ) h I ( ex:JoesGang ) , I ( ex:Clique ) i ∈ IEXT( I ( rdf:type )) , (1 t ) h I ( ex:alice ) , I ( ex:JoesGang ) i ∈ IEXT( I ( rdf:type )) , (1 u ) h I ( ex:bob ) , I ( ex:JoesGang ) i ∈ IEXT( I ( rdf:type )) . F rom (1s) and the RDFS seman tic condition of “ICEXT” (“ → ”) follo ws (1 s 0 ) I ( ex:JoesGang ) ∈ ICEXT( I ( ex:Clique )) . F rom (1d) and the OWL 2 semantic condition of class subsumption (OWL2/T ab5.8, “ → ”) follows (2) ∀ x : x ∈ ICEXT( I ( ex:Clique )) ⇒ x ∈ ICEXT( r ) . F rom (1f ) – (1g) and the semantic condition for existential prop ert y restrictions (O WL2/T ab5.6) follo ws (3) ∀ x : x ∈ ICEXT( r ) ⇔ ∃ y : h x, y i ∈ IEXT( I ( ex:sameCliqueAs )) ∧ y ∈ ICEXT( I ( ex:Clique )) . F rom (1s’), (2) and (3) follo ws (4) ∃ y : h I ( ex:JoesGang ) , y i ∈ IEXT( I ( ex:sameCliqueAs )) . According to (4), we can ﬁnd some y such that (5) h I ( ex:JoesGang ) , y i ∈ IEXT( I ( ex:sameCliqueAs )) . By applying the OWL 2 seman tic condition for prop ert y subsumption OWL2/T ab5.8, “ ↔ ”) and (1b) to (5), we receive (6) h I ( ex:JoesGang ) , y i ∈ IEXT( I ( ex:sameAs )) . 48 Mic hael Schneider and Geoﬀ Sutcliﬀe No w, the semantic condition for owl:sameAs (O WL2/T ab5.9, “ → ”) applied to (6) which yields (6) y = I ( ex:JoesGang ) . By (4) and (6) w e receiv e (7) h I ( ex:JoesGang ) , I ( ex:JoesGang ) i ∈ IEXT( I ( ex:sameCliqueAs )) . F rom (1r) and the seman tic condition for in verse prop erties (OWL2/T ab5.12, “ → ”) follows (8) ∀ x y : h x, y i ∈ IEXT( i ) ⇔ h y , x i ∈ IEXT( I ( rdf:type )) . F rom (1u) and (8) follows: (9) h I ( ex:JoesGang ) , I ( ex:bob ) i ∈ IEXT( i ) . F rom (1j) – (1q) and the semantic condition for sub prop ert y chains (OWL2/T ab5.11, “ → ”, ternary) w e receiv e (10) ∀ y 0 , y 1 , y 2 , y 3 : h y 0 , y 1 i ∈ IEXT( I ( rdf:type )) ∧ h y 1 , y 2 i ∈ IEXT( I ( ex:sameCliqueAs )) ∧ h y 2 , y 3 i ∈ IEXT( i ) ⇒ h y 0 , y 3 i ∈ IEXT( I ( foaf:knows )) . Finally , from (10), (1t), (7) and (9) follo ws (11) h I ( ex:alice ) , I ( ex:bob ) i ∈ IEXT( I ( foaf:knows )) . 014 Harry b elongs to some Sp ecies (Pro of ) Let I b e an OWL 2 RDF- Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph suc h that I + B satisﬁes the premise graph. Let u , l 1 and l 2 b e individuals such that the following holds: (1 a ) h I ( ex:Eagle ) , I ( ex:Species ) i ∈ IEXT( I ( rdf:type )) , (1 b ) h I ( ex:Falcon ) , I ( ex:Species ) i ∈ IEXT( I ( rdf:type )) , (1 c ) h I ( ex:harry ) , u i ∈ IEXT( I ( rdf:type )) , (1 d 1) h u, l 1 i ∈ IEXT( I ( owl:unionOf )) , (1 d 2) h l 1 , I ( ex:Eagle ) i ∈ IEXT( I ( rdf:first )) , (1 d 3) h l 1 , l 2 i ∈ IEXT( I ( rdf:rest )) , (1 d 4) h l 2 , I ( ex:Falcon ) i ∈ IEXT( I ( rdf:first )) , (1 d 5) h l 2 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . W e pro ve the claim by classical dilemma. Case 1: h I ( ex:harry ) , I ( ex:Eagle ) i ∈ IEXT( I ( rdf:type )) . Then, together with (1a) follo ws: (2) ∃ x : h I ( ex:harry ) , x i ∈ IEXT( I ( rdf:type )) ∧ h x, I ( ex:Species ) i ∈ IEXT( I ( rdf:type )) . Reasoning in OWL 2 F ull using First-Order A TP 49 Case 2: h I ( ex:harry ) , I ( ex:Eagle ) i / ∈ IEXT( I ( rdf:type )) . By the RDFS semantic condition of ICEXT (“ → ”) we get (3) I ( ex:harry ) / ∈ ICEXT( I ( ex:Eagle )) . Lik ewise, from (1c) and the RDFS semantic condition of ICEXT (“ → ”) we get (4) I ( ex:harry ) ∈ ICEXT( u ) . F rom (1d1) – (1d5) and the semantic condition of class union (OWL2/T ab5.4, “ → ”, binary) follo ws (5) ∀ x : x ∈ ICEXT( u ) ⇔ x ∈ ICEXT( I ( ex:Eagle )) ∨ x ∈ ICEXT( I ( ex:Falcon )) . Sp ecializing (4) to x := I ( ex:harry ) implies (6) I ( ex:harry ) ∈ ICEXT( u ) ⇔ I ( ex:harry ) ∈ ICEXT( ex:Eagle ) ∨ I ( ex:harry ) ∈ ICEXT( ex:Falcon ) . By (4), (6) and (3) we get (7) I ( ex:harry ) ∈ ICEXT( I ( ex:Falcon )) . Using the RDFS semantic condition of ICEXT (“ ← ”) results in (8) h I ( ex:harry ) , I ( ex:Falcon ) i ∈ IEXT( I ( rdf:type )) . F rom (8) and (1b) follows: (9) ∃ x : h I ( ex:harry ) , x i ∈ IEXT( I ( rdf:type )) ∧ h x, I ( ex:Species ) i ∈ IEXT( I ( rdf:type )) . Since (2) and (9) are the same result from the contrary assumed cases, w e get the claimed result. 015 Reﬂective T autologies I (Pro of ) Let I b e a satisfying O WL 2 RDF- Based interpretation for the empty graph. It is true that I ( owl:sameAs ) = I ( owl:sameAs ) . By the “ ← ” direction of semantic condition for owl:sameAs (OWL2/T ab5.9) we receiv e h I ( owl:sameAs ) , I ( owl:sameAs ) i ∈ IEXT( I ( owl:sameAs )) . 50 Mic hael Schneider and Geoﬀ Sutcliﬀe 016 Reﬂective T autologies I I (Pro of ) Let I b e a satisfying O WL 2 RDF- Based interpretation for the empt y graph. Giv en arbitrary c 1 , c 2 , and assume the following to hold: (1) h c 1 , c 2 i ∈ IEXT( I ( owl:equivalentClass )) . F rom the “ → ” direction of the semantic condition for class equiv alence (O WL2/T ab5.9) and from the prop ert y extension of owl:equivalentClass (OWL2/T ab5.3) fol- lo ws (2 a ) c 1 ∈ IC , (2 b ) c 2 ∈ IC , (2 c ) ∀ x : x ∈ ICEXT( c 1 ) ⇔ x ∈ ICEXT( c 2 ) . F rom (2c) follows the w eaker result (3) ∀ x : x ∈ ICEXT( c 1 ) ⇒ x ∈ ICEXT( c 2 ) . F rom (2a), (2b) and (3) and the “ ← ” direction of the O WL 2 seman tic condition of class subsumption (OWL2/T ab5.8) follo ws (4) h c 1 , c 2 i ∈ IEXT( I ( rdfs:subClassOf )) . Since (4) follo ws from (1) and since c 1 and c 2 w ere c hosen arbitrarily , w e get (5) ∀ x, y : h x, y i ∈ IEXT( I ( owl:equivalentClass )) ⇒ h x, y i ∈ IEXT( I ( rdfs:subClassOf )) . F or prop ert y owl:equivalentClass w e receiv e from OWL2/T ab5.3 (6 a ) I ( owl:equivalentClass ) ∈ IP . and for prop ert y rdfs:subClassOf we receive from the RDFS axiomatic triples (6 b ) h I ( rdfs:subClassOf ) , I ( rdf:Property ) i ∈ IEXT( I ( rdf:type )) . By the “ ← ” direction of the RDF semantic condition of IP and IEXT we receiv e from (6b): (6 b 0 ) I ( rdfs:subClassOf ) ∈ IP . F rom (5), (6a) and (6b’) and from the “ ← ” direction of the OWL 2 seman tic condition for prop ert y subsumption (OWL2/T ab5.8) w e ﬁnally receive (7) h I ( owl:equivalentClass ) , I ( rdfs:subClassOf ) i ∈ IEXT( I ( rdfs:subPropertyOf )) . 017 Builtin Based Deﬁnitions (Pro of ) Let I b e an O WL 2 RDF-Based in terpretation that satisﬁes the premise graph. Let the following assertions hold: (1 a ) h I ( ex:notInstanceOf ) , I ( rdf:type ) i ∈ IEXT( I ( owl:propertyDisjointWith )) , (1 b ) h I ( ex:w ) , I ( ex:c ) i ∈ IEXT( I ( rdf:type )) , (1 c ) h I ( ex:u ) , I ( ex:c ) i ∈ IEXT( I ( ex:notInstanceOf )) . Reasoning in OWL 2 F ull using First-Order A TP 51 F rom (1a) and the semantic condition for disjoint prop erties (OWL2/5.9, “ → ”) follo ws: (1 a 0 ) ∀ x, y : ¬ [ h x, y i ∈ IEXT( I ( ex:notInstanceOf )) ∧ h x, y i ∈ IEXT( I ( rdf:type )) ] . Sp ecializing (1a’) to x := I ( ex:w ) and y := I ( ex:c ) implies (2) ¬ [ h I ( ex:w ) , I ( ex:c ) i ∈ IEXT( I ( ex:notInstanceOf )) ∧ h I ( ex:w ) , I ( ex:c ) i ∈ IEXT( I ( rdf:type )) ] . F rom (1b) and (2) follows (3) h I ( ex:w ) , I ( ex:c ) i / ∈ IEXT( I ( ex:notInstanceOf )) . F rom (1c) and (3) follows (4) I ( ex:w ) 6 = I ( ex:u ) . F rom (4) and the semantic condition for owl:differentFrom (OWL2/T ab5.9, “ ← ”) follows: (5) h I ( ex:w ) , I ( ex:u ) i ∈ IEXT( I ( owl:differentFrom )) . 018 Mo diﬁed Logical V o cabulary Seman tics (Pro of ) Let I b e an O WL 2 RDF-Based interpretation that satisﬁes the premise graph. W e start from: (1 a ) h I ( owl:sameAs ) , I ( ex:Person ) i ∈ IEXT( I ( rdfs:domain )) , (1 b ) h I ( ex:w ) , I ( ex:u ) i ∈ IEXT( I ( owl:sameAs )) . F rom this we get via the RDFS semantic condition for prop ert y domains: (2) I ( ex:w ) ∈ ICEXT( I ( ex:Person )) . F urther, from (1b) and the seman tic condition for owl:sameAs (OWL2/T ab5.9, “ → ”) we get (3) I ( ex:w ) = I ( ex:u ) . This allows for substitution in (2), providing (4) I ( ex:u ) ∈ ICEXT( I ( ex:Person )) . Finally , by applying the RDFS seman tic extension for ICEXT (“ ← ”) to (4) w e get (5) h I ( ex:u ) , I ( ex:Person ) i ∈ IEXT( I ( rdf:type )) . 52 Mic hael Schneider and Geoﬀ Sutcliﬀe 019 Disjoint Annotation Prop erties (Pro of ) Let I b e an OWL 2 RDF- Based interpretation that satisﬁes the premise graph. Starting from: (1 a ) h I ( skos:prefLabel ) , I ( skos:altLabel ) i ∈ IEXT( I ( owl:propertyDisjointWith )) , (1 b ) h I ( ex:foo ) , I ( "foo" ) i ∈ IEXT( I ( skos:prefLabel )) , (1 c ) h I ( ex:foo ) , I ( "foo" ) i ∈ IEXT( I ( skos:altLabel )) . F rom (1a) and the semantic condition of prop ert y disjointness (OWL2/T ab5.9, “ → ”) we receive (2) ∀ x, y : ¬ [ h x, y i ∈ IEXT( I ( skos:prefLabel )) ∧ h x, y i ∈ IEXT( I ( skos:altLabel )) ] . Sp eciﬁcally , we receive: (3) ¬ [ h I ( ex:foo ) , I ( "foo" ) i ∈ IEXT( I ( skos:prefLabel )) ∧ h I ( ex:foo ) , I ( "foo" ) i ∈ IEXT( I ( skos:altLabel )) ] . W e no w ha ve a con tradiction betw een (1b), (1c) and (3). 020 Logical Complications (Pro of ) Let I b e an OWL 2 RDF-Based inter- pretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e xs , xc , l u 1 , l u 2 , lu 3 , l i 1 , l i 2 , l i 3 , such that (1 a ) h I ( ex:c ) , lu 1 i ∈ IEXT( I ( owl:unionOf )) , (1 b ) h lu 1 , I ( ex:c1 ) i ∈ IEXT( I ( rdf:first )) , (1 c ) h lu 1 , lu 2 i ∈ IEXT( I ( rdf:rest )) , (1 d ) h lu 2 , I ( ex:c2 ) i ∈ IEXT( I ( rdf:first )) , (1 e ) h lu 2 , lu 3 i ∈ IEXT( I ( rdf:rest )) , (1 f ) h lu 3 , I ( ex:c3 ) i ∈ IEXT( I ( rdf:first )) , (1 g ) h lu 3 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 h ) h I ( ex:d ) , I ( ex:c1 ) i ∈ IEXT( I ( owl:disjointWith )) , (1 j ) h I ( ex:d ) , xs i ∈ IEXT( I ( rdfs:subClassOf )) , (1 k ) h xs, li 1 i ∈ IEXT( I ( owl:intersectionOf )) , (1 m ) h li 1 , I ( ex:c ) i ∈ IEXT( I ( rdf:first )) , (1 n ) h li 1 , li 2 i ∈ IEXT( I ( rdf:rest )) , (1 o ) h li 2 , xc i ∈ IEXT( I ( rdf:first )) , (1 p ) h li 2 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 q ) h xc, I ( ex:c2 ) i ∈ IEXT( I ( owl:complementOf )) . F rom (1a), (1b) – (1g) and the semantic condition for class union (OWL2/T ab.5.4, “ → ”, ternary) follo ws (2) I ( ex:c3 ) ∈ IC . and (3) ∀ x : x ∈ ICEXT( I ( ex:c )) ⇔ x ∈ ICEXT( I ( ex:c1 )) ∨ x ∈ ICEXT( I ( ex:c2 )) ∨ x ∈ ICEXT( I ( ex:c3 )) . Reasoning in OWL 2 F ull using First-Order A TP 53 F rom (1h) and the semantic condition for class disjoin tness follo ws (4) I ( ex:d ) ∈ IC and from (1h) and the semantic condition for class disjointness (O WL2/T ab.5.9, “ → ”) follows (5) ∀ x : ¬ [ x ∈ ICEXT( I ( ex:d )) ∧ x ∈ ICEXT( I ( ex:c1 )) ] . F rom (1j) and the OWL 2 semantic condition of class subsumption (OWL2/T ab5.9, “ → ”) follows (6) ∀ x : x ∈ ICEXT( I ( ex:d )) ⇒ x ∈ ICEXT( xs ) . F rom (1k), (1m) – (1p) and the semantic condition for class intersection (OWL2/T ab5.4, “ → ”, binary) follo ws (7) ∀ x : x ∈ ICEXT( xs ) ⇔ x ∈ ICEXT( I ( ex:c )) ∧ x ∈ ICEXT( xc ) . F rom (1q) and the seman tic condition for class complement (OWL2/T ab5.4, “ → ”) follows (8) ∀ x : x ∈ I C E X T ( xc ) ⇔ x / ∈ ICEXT( I ( ex:c2 )) . F rom (6), (7) and (8) follo ws (9) ∀ x : x ∈ ICEXT( I ( ex:d )) ⇒ x ∈ ICEXT( I ( ex:c )) ∧ x / ∈ ICEXT( I ( ex:c2 )) . F rom (9) and (3) follows (10) ∀ x : x ∈ ICEXT( I ( ex:d )) ⇒ x ∈ ICEXT( I ( ex:c1 )) ∨ x ∈ ICEXT( I ( ex:c3 )) . F rom (10) and (5) follows (11) ∀ x : x ∈ ICEXT( I ( ex:d )) ⇒ x ∈ ICEXT( I ( ex:c3 )) . Finally , from (4), (2), (11) and the OWL 2 semantic extension of class subsump- tion (OWL2/T ab5.8, “ ← ”) follows (12) h I ( ex:d ) , I ( ex:c3 ) i ∈ IEXT( I ( rdfs:subClassOf )) . 021 Comp osite En umerations (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e l 11 , l 12 , l 21 , l 22 , 54 Mic hael Schneider and Geoﬀ Sutcliﬀe l 31 , l 32 , l 41 , l 42 , such that the following statements hold: (1 a 1) h I ( ex:c1 ) , l 11 i ∈ IEXT( I ( owl:oneOf )) , (1 a 2) h l 11 , I ( ex:w1 ) i ∈ IEXT( I ( rdf:first )) , (1 a 3) h l 11 , l 12 ) i ∈ IEXT( I ( rdf:rest )) , (1 a 4) h l 12 , I ( ex:w2 ) i ∈ IEXT( I ( rdf:first )) , (1 a 5) h l 12 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 b 1) h I ( ex:c2 ) , l 21 ) i ∈ IEXT( I ( owl:oneOf )) , (1 b 2) h l 21 , I ( ex:w2 ) i ∈ IEXT( I ( rdf:first )) , (1 b 3) h l 21 , l 22 ) i ∈ IEXT( I ( rdf:rest )) , (1 b 4) h l 22 , I ( ex:w3 ) i ∈ IEXT( I ( rdf:first )) , (1 b 5) h l 22 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 c 1) h I ( ex:c3 ) , l 31 ) i ∈ IEXT( I ( owl:oneOf )) , (1 c 2) h l 31 , I ( ex:w1 ) i ∈ IEXT( I ( rdf:first )) , (1 c 3) h l 31 , l 32 i ∈ IEXT( I ( rdf:rest )) , (1 c 4) h l 32 , I ( ex:w2 ) i ∈ IEXT( I ( rdf:first )) , (1 c 5) h l 32 , l 33 i ∈ IEXT( I ( rdf:rest )) , (1 c 6) h l 33 , I ( ex:w3 ) i ∈ IEXT( I ( rdf:first )) , (1 c 7) h l 33 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 d 1) h I ( ex:c4 ) , l 41 i ∈ IEXT( I ( owl:unionOf )) , (1 d 2) h l 41 , I ( ex:c1 ) i ∈ IEXT( I ( rdf:first )) , (1 d 3) h l 41 , l 42 ) i ∈ IEXT( I ( rdf:rest )) , (1 d 4) h l 42 , I ( ex:c2 ) i ∈ IEXT( I ( rdf:first )) , (1 d 5) h l 42 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . By applying the seman tic condition for enumeration classes (OWL2/T ab5.5, “ → ”, binary and ternary) to (1a1)–(1a5), (1b1)–(1b5) and (1c1)–(1c7), resp ec- tiv ely , w e receiv e: (2 a ) ∀ x : x ∈ ICEXT( I ( ex:c1 )) ⇔ x = I ( ex:w1 ) ∨ x = I ( ex:w2 ) , (2 b ) ∀ x : x ∈ ICEXT( I ( ex:c2 )) ⇔ x = I ( ex:w2 ) ∨ x = I ( ex:w3 ) , (2 c ) ∀ x : x ∈ ICEXT( I ( ex:c3 )) ⇔ x = I ( ex:w1 ) ∨ x = I ( ex:w2 ) ∨ x = I ( ex:w3 ) . By applying the seman tic condition for class union (OWL2/T ab5.4, “ → ”, binary) to (1d1)–(1d5), w e receiv e: (2 d ) ∀ x : x ∈ ICEXT( ex:c4 ) ⇔ x ∈ ICEXT( I ( ex:c1 )) ∨ x ∈ ICEXT( I ( ex:c2 )) . F rom the prop ert y extension of owl:oneOf (OWL2/T ab5.3) and (1c1) follo ws (3 a ) I ( ex:c3 ) ∈ IC . F rom the prop ert y extension of owl:unionOf (OWL2/T ab5.3) and (1d1) follows (3 b ) I ( ex:c4 ) ∈ IC . F rom (2a), (2b) and (2c) follo ws (4) ∀ x : x ∈ ICEXT( I ( ex:c3 )) ⇔ x ∈ ICEXT( I ( ex:c1 )) ∨ x ∈ ICEXT( I ( ex:c2 )) . Reasoning in OWL 2 F ull using First-Order A TP 55 F rom (2d) and (4) follows (5) ∀ x : x ∈ ICEXT( I ( ex:c3 )) ⇔ x ∈ ICEXT( I ( ex:c4 )) . F rom the seman tic condition for class equiv alence (O WL2/T ab5.9, ← ), (3a), (3b) and (5) follo ws (6) h I ( ex:c3 ) , I ( ex:c4 ) i ∈ IEXT( I ( owl:equivalentClass )) . 022 List Member Access (Proof ) Let I b e an O WL 2 RDF-Based interpre- tation and B b e a blank no de mapping for the blank no des in the premise graph suc h that I + B satisﬁes the premise graph. Given an individual pL as well as list individuals l 11 , l 12 , l 21 , l 22 , l 31 , l 32 and l 33 , such that the following assertions hold: (1 a 1) h I ( skos:memberList ) , pL i ∈ IEXT( I ( rdfs:subPropertyOf )) , (1 b 1) h I ( skos:member ) , l 11 i ∈ IEXT( I ( owl:propertyChainAxiom )) , (1 b 2) h l 11 , pL i ∈ IEXT( I ( rdf:first )) , (1 b 3) h l 11 , l 12 i ∈ IEXT( I ( rdf:rest )) , (1 b 4) h l 12 , I ( rdf:first ) i ∈ IEXT( I ( rdf:first )) , (1 b 5) h l 12 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 c 1) h pL, l 21 i ∈ IEXT( I ( owl:propertyChainAxiom )) , (1 c 2) h l 21 , pL i ∈ IEXT( I ( rdf:first )) , (1 c 3) h l 21 , l 22 i ∈ IEXT( I ( rdf:rest )) , (1 c 4) h l 22 , I ( rdf:rest ) i ∈ IEXT( I ( rdf:first )) , (1 c 5) h l 22 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:first )) , (1 d 1) h I ( ex:MyOrderedCollection ) , I ( skos:OrderedCollection ) i ∈ IEXT( I ( rdf:type )) , (1 e 1) h I ( ex:MyOrderedCollection ) , l 31 i ∈ IEXT( I ( skos:memberList )) , (1 e 2) h l 31 , I ( ex:X ) i ∈ IEXT( I ( rdf:first )) , (1 e 3) h l 31 , l 32 ) i ∈ IEXT( I ( rdf:rest )) , (1 e 4) h l 32 , I ( ex:Y ) i ∈ IEXT( I ( rdf:first )) , (1 e 5) h l 32 , l 33 ) i ∈ IEXT( I ( rdf:rest )) , (1 e 6) h l 33 , I ( ex:Z ) i ∈ IEXT( I ( rdf:first )) , (1 e 7) h l 33 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . By the RDFS seman tic condition for prop ert y subsumption, (1a1) and (1e1) we receiv e (1 e 1 0 ) h I ( ex:MyOrderedCollection ) , l 31 i ∈ IEXT( pL ) . F rom the seman tic condition of sub prop ert y c hains (O WL2/T ab5.11, “ → ”, bi- nary version) we get from (1b1) to (1b5): (2 b ) ∀ y 0 , y 1 , y 2 : h y 0 , y 1 i ∈ IEXT( pL ) ∧ h y 1 , y 2 i ∈ IEXT( I ( rdf:first )) ⇒ h y 0 , y 2 i ∈ IEXT( I ( skos:member )) . and from (1c1) to (1c5) we get (2 c ) ∀ y 0 , y 1 , y 2 : h y 0 , y 1 i ∈ IEXT( pL ) ∧ h y 1 , y 2 i ∈ IEXT( I ( rdf:rest )) ⇒ h y 0 , y 2 i ∈ IEXT( pL ) . 56 Mic hael Schneider and Geoﬀ Sutcliﬀe W e receiv e the ﬁrst result triple (3 a ) h I ( ex:MyOrderedCollection ) , I ( ex:X ) i ∈ IEXT( I ( skos:member )) . from (2b), (1e1’) and (1e2). F urther, b y (2c), (1e1’) and (1e3) w e receiv e (1 e 1 00 ) h I ( ex:MyOrderedCollection ) , l 32 i ∈ IEXT( pL ) . In the same wa y , from (2b), (1e1”), (1e4) and (1e5) we receive (3 b ) h I ( ex:MyOrderedCollection ) , I ( ex:Y ) i ∈ IEXT( I ( skos:member )) . and (1 e 1 000 ) h I ( ex:MyOrderedCollection ) , l 33 i ∈ IEXT( pL ) . And likewise, we rec eiv e (3 c ) h I ( ex:MyOrderedCollection ) , I ( ex:Z ) i ∈ IEXT( I ( skos:member )) . 023 Unique List Comp onen ts (Pro of ) Let I b e an OWL 2 RDF-Based in terpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e individuals o and l such that (1 a ) h I ( rdf:first ) , I ( owl:FunctionalProperty ) i ∈ IEXT( I ( rdf:type )) , (1 b ) h I ( ex:w ) , o i ∈ IEXT( I ( rdf:type )) , (1 c ) h o, I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 d ) h o, l i ∈ IEXT( I ( owl:oneOf )) , (1 e ) h l, I ( ex:u ) i ∈ IEXT( I ( rdf:first )) , (1 f ) h l, I ( ex:v ) i ∈ IEXT( I ( rdf:first )) , (1 g ) h l, I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . By the RDFS semantic condition for ICEXT and (1b) follo ws (1 b 0 ) I ( ex:w ) ∈ ICEXT( o ) . By the semantic condition for enumeration classes (OWL2/T ab5.5, “ → ”, single- ton) and (1d), (1e) and (1g) follo ws (2) ∀ x : x ∈ ICEXT( o ) ⇔ x = I ( ex:u ) . F rom (1b’) and sp ecializing (2) to x := I ( ex:w ) follo ws (3) I ( ex:w ) = I ( ex:u ) . By the semantic condition for functional prop erties (OWL2/T ab5.13, “ → ”) and (1a) follows (4) ∀ x, y 1 , y 2 : h x, y 1 i ∈ IEXT( I ( rdf:first )) ∧ h x, y 2 i ∈ IEXT( I ( rdf:first )) ⇒ y 1 = y 2 . Reasoning in OWL 2 F ull using First-Order A TP 57 F rom (1e), (1f ) and from sp ecializing (4) to x := l follows (5) I ( ex:u ) = I ( ex:v ) . F rom the seman tic condition of owl:sameAs (OWL2/T ab5.9, “ ← ”) and (3) and (5) follows (6 a ) h I ( ex:w ) , I ( ex:u ) i ∈ IEXT( I ( owl:sameAs )) , (6 b ) h I ( ex:w ) , I ( ex:v ) i ∈ IEXT( I ( owl:sameAs )) . 024 Cardinality Restrictions on Complex Prop erties (Pro of ) Let I be an OWL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e z suc h that (1 a ) h I ( ex:hasAncestor ) , I ( owl:TransitiveProperty ) i ∈ IEXT( I ( rdf:type )) , (1 b ) h I ( ex:Person ) , z i ∈ IEXT( I ( rdfs:subClassOf )) , (1 c ) h z , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) , (1 d ) h z , I ( ex:hasAncestor ) i ∈ IEXT( I ( owl:onProperty )) , (1 e ) h z , I ( "1"^^xsd:nonNegativeInteger ) i ∈ IEXT( I ( owl:minCardinality )) , (1 f ) h I ( ex:alice ) , I ( ex:Person ) i ∈ IEXT( I ( rdf:type )) , (1 g ) h I ( ex:bob ) , I ( ex:Person ) i ∈ IEXT( I ( rdf:type )) , (1 h ) h I ( ex:alice ) , I ( ex:bob ) i ∈ IEXT( I ( ex:hasAncestor )) . F rom (1a) and the seman tic condition of transitive prop erties (OWL2/T ab5.13, “ → ”) follows (2) ∀ y 1 , y 2 , y 3 : h y 1 , y 2 i ∈ IEXT( I ( ex:hasAncestor )) ∧ h y 2 , y 3 i ∈ IEXT( I ( ex:hasAncestor )) ⇒ h y 1 , y 3 i ∈ IEXT( I ( ex:hasAncestor )) . F rom (1b) and the OWL 2 semantic condition for class subsumption (OWL2/T ab5.8, “ → ”) follows (3) ∀ y : y ∈ ICEXT( I ( ex:Person )) ⇒ y ∈ ICEXT( z ) . F rom (1c)–(1e) and the seman tic condition for 1-min cardinalit y restrictions (O WL2/T ab5.6) follo ws (4) ∀ y : y ∈ ICEXT( z ) ⇔ ∃ x : h y , x i ∈ ICEXT( I ( ex:hasAncestor )) . F rom (3) and (4) follows (5) ∀ y : ICEXT( ex:Person , y ) ⇒ ∃ x : h y , x i ∈ IEXT( ex:hasAncestor ) . Applying the RDFS semantic condition for ICEXT (“ → ”) to (1g) yields (6) I ( ex:bob ) ∈ ICEXT( I ( ex:Person )) . 58 Mic hael Schneider and Geoﬀ Sutcliﬀe F rom (6) and (5) follows the existence of some x suc h that (7) h I ( ex:bob ) , x i ∈ IEXT( I ( ex:hasAncestor )) . F rom (1h), (7) and (2) follo ws, for the same x , (8) h I ( ex:alice ) , x i ∈ IEXT( I ( ex:hasAncestor )) . Hence we hav e shown in (7) and (8) that (9) ∃ x : h I ( ex:bob ) , x i ∈ IEXT( I ( ex:hasAncestor )) ∧ h I ( ex:alice ) , x i ∈ IEXT( I ( ex:hasAncestor )) . Therefore, there is some blank no de mapping B 0 for the blank no des in the conclusion graph suc h that I + B 0 satisﬁes the conclusion graph. 025 Cyclic Dep endencies b et ween Complex Properties (Pro of ) Let I b e an OWL 2 RDF-Based interpretation and B be a blank node mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e individuals l 11 , l 12 , l 21 , l 22 , l 3 , such that: (1 a 1) h I ( ex:hasUncle ) , l 11 i ∈ IEXT( I ( owl:propertyChainAxiom )) , (1 a 2) h l 11 , I ( ex:hasCousin ) i ∈ IEXT( I ( rdf:first )) , (1 a 3) h l 11 , l 12 i ∈ IEXT( I ( rdf:rest )) , (1 a 4) h l 12 , I ( ex:hasFather ) i ∈ IEXT( I ( rdf:first )) , (1 a 5) h l 12 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 b 1) h I ( ex:hasCousin ) , l 21 i ∈ IEXT( I ( owl:propertyChainAxiom )) , (1 b 2) h l 21 , I ( ex:hasUncle ) i ∈ IEXT( I ( rdf:first )) , (1 b 3) h l 21 , l 22 i ∈ IEXT( I ( rdf:rest )) , (1 b 4) h l 22 , l 3 i ∈ IEXT( I ( rdf:first )) , (1 b 5) h l 22 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 c ) h l 3 , I ( ex:hasFather ) i ∈ IEXT( I ( owl:inverseOf )) , (1 d ) h I ( ex:alice ) , I ( ex:dave ) i ∈ IEXT( I ( ex:hasFather )) , (1 e ) h I ( ex:alice ) , I ( ex:bob ) i ∈ IEXT( I ( ex:hasCousin )) , (1 f ) h I ( ex:bob ) , I ( ex:charly ) i ∈ IEXT( I ( ex:hasFather )) , (1 g ) h I ( ex:bob ) , I ( ex:dave ) i ∈ IEXT( I ( ex:hasUncle )) . F rom the semantic condition for sub prop ert y c hains (OWL2/T ab5.11, “ → ”, binary) and (1a1) to (1a5) follows (2 a ) ∀ y 0 , y 1 , y 2 : h y 0 , y 1 i ∈ IEXT( I ( ex:hasCousin )) ∧ h y 1 , y 2 i ∈ IEXT( I ( ex:hasFather )) ⇒ h y 0 , y 2 i ∈ IEXT( I ( ex:hasUncle )) . F rom the semantic condition for sub prop ert y c hains (OWL2/T ab5.11, “ → ”, binary) and (1b1) to (1b5) follows (2 b ) ∀ y 0 , y 1 , y 2 : h y 0 , y 1 i ∈ IEXT( I ( ex:hasUncle )) ∧ h y 1 , y 2 i ∈ IEXT( l 3 ) ⇒ h y 0 , y 2 i ∈ IEXT( I ( ex:hasCousin )) . Reasoning in OWL 2 F ull using First-Order A TP 59 F rom the seman tic condition for inv erse prop erties (OWL2/T ab5.12, “ → ”) and (1c) follows (2 c ) ∀ x, y : h x, y i ∈ IEXT( l 3 ) ⇔ h y , x i ∈ IEXT( I ( ex:hasFather )) . F rom (2c) and (1d) follows (1 d 0 ) h I ( ex:dave ) , I ( ex:alice ) i ∈ IEXT( l 3 ) . F rom (2a), (1e) and (1f ) follo ws (3 a ) h I ( ex:alice ) , I ( ex:charly ) i ∈ IEXT( I ( ex:hasUncle )) . F rom (2b), (1g) and (1d’) (3 b ) h I ( ex:bob ) , I ( ex:alice ) i ∈ IEXT( I ( ex:hasCousin )) . The resulting triples are (3a) and (3b). 026 Inferred Prop ert y Characteristics I (Pro of ) Let I b e an OWL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let x 1 , x 2 , l 1 and l 2 b e individuals such that the following holds: (1 a ) h I ( ex:p ) , x 1 i ∈ IEXT( I ( rdfs:domain )) , (1 b ) h x 1 , l 1 i ∈ IEXT( I ( owl:oneOf )) , (1 c ) h l 1 , I ( ex:w ) i ∈ IEXT( I ( rdf:first )) , (1 d ) h l 1 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 e ) h I ( ex:p ) , x 2 i ∈ IEXT( I ( rdfs:range )) , (1 f ) h x 2 , l 2 i ∈ IEXT( I ( owl:oneOf )) , (1 g ) h l 2 , I ( ex:u ) i ∈ IEXT( I ( rdf:first )) , (1 h ) h l 2 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . F rom the semantic condition for enumeration classes (OWL2/T ab5.5, “ → ”, sin- gleton) and (1b) to (1d) follows: (2) ∀ z : z ∈ ICEXT( x 1 ) ⇔ z = I ( ex:w ) . No w let us assume that there are s 1 , s 2 and o suc h that (3 a ) h s 1 , o i ∈ IEXT( I ( ex:p )) , and (3 b ) h s 2 , o i ∈ IEXT( I ( ex:p )) . F rom the RDFS semantic condition for prop ert y domains, (1a) and (3a) follows (4 a ) s 1 ∈ ICEXT( x 1 ) . F rom the RDFS semantic condition for prop ert y domains, (1a) and (3b) follows (4 b ) s 2 ∈ ICEXT( x 1 ) . 60 Mic hael Schneider and Geoﬀ Sutcliﬀe F rom (2) and (4a) follows (5 a ) s 1 = I ( ex:w ) . F rom (2) and (4b) follows (5 b ) s 2 = I ( ex:w ) . Finally , from (5a) and (5b) follo ws (6) s 1 = s 2 . Since s 1 , s 2 and o w ere c hosen arbitrarily , w e can generalize (7) ∀ s 1 , s 2 , o : h s 1 , o i ∈ IEXT( I ( ex:p )) ∧ h s 2 , o i ∈ IEXT( I ( ex:p )) ⇒ s 1 = s 2 . F rom the RDFS axiomatic triple for the domain of rdfs:domain , we receive (8) h I ( rdfs:domain ) , I ( rdf:Property ) i ∈ IEXT( I ( rdfs:domain )) . F rom the RDFS seman tic condition for prop ert y domains, (8) and (1a) follows (9) I ( ex:p ) ∈ ICEXT( I ( rdf:Property )) . With the RDFS seman tic condition for ICEXT and the RDF semantic condition for IP and IEXT follo ws (9 0 ) I ( ex:p ) ∈ IP . By the semantic condition for in verse-functional properties (O WL2/T ab5.13, “ ← ”), (9’) and (7) follo ws (10) I ( ex:p ) ∈ ICEXT( I ( owl:InverseFunctionalProperty )) . And by the RDFS semantic condition for ICEXT and (10) follows (10 0 ) h I ( ex:p ) , I ( owl:InverseFunctionalProperty ) i ∈ IEXT( I ( rdf:type )) . 027 Inferred Prop ert y Characteristics I I (Pro of ) Let I b e an O WL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e l 1 , l 2 and v , such that (1 a ) h I ( owl:sameAs ) , l 1 i ∈ IEXT( I ( owl:propertyChainAxiom )) , (1 b ) h l 1 , I ( ex:p ) i ∈ IEXT( I ( rdf:first )) , (1 c ) h l 1 , l 2 i ∈ IEXT( I ( rdf:rest )) , (1 d ) h l 2 , v i ∈ IEXT( I ( rdf:first )) , (1 e ) h l 2 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 f ) h v , I ( ex:p ) i ∈ IEXT( I ( owl:inverseOf )) . Reasoning in OWL 2 F ull using First-Order A TP 61 F rom (1a), (1b) – (1e) and the semantic condition for sub prop ert y c hains (O WL2/T ab5.11, “ → ”, binary) follo ws (2) I ( ex:p ) ∈ IP and (3) ∀ y 0 , y 1 , y 2 : h y 0 , y 1 i ∈ IEXT( I ( ex:p )) ∧ h y 1 , y 2 i ∈ IEXT( v ) ⇒ h y 0 , y 2 i ∈ IEXT( I ( owl:sameAs )) . F rom (1f ) and the seman tic condition for in verse prop erties (O WL2/T ab5.12, “ → ”) follows: (4) ∀ x, y : h x, y i ∈ IEXT( v ) ⇔ h y , x i ∈ IEXT( I ( ex:p )) . F rom the semantic condition for owl:sameAs (O WL2/T ab5.9, “ → ”) follo ws: (5) ∀ x, y : h x, y i ∈ IEXT( I ( owl:sameAs )) ⇒ x = y . F rom (3), (4) and (5) follo ws (6) ∀ y 0 , y 1 , y 2 : h y 0 , y 1 i ∈ IEXT( I ( ex:p )) ∧ h y 2 , y 1 i ∈ IEXT( I ( ex:p )) ⇒ y 0 = y 2 . F rom (2), (6) and the semantic condition for in verse-functional prop erties (OWL2/T ab.13, “ ← ”) follows (7) I ( ex:p ) ∈ ICEXT( I ( owl:InverseFunctionalProperty )) . Finally , from (7) and the RDFS semantic condition for ICEXT (“ ← ”) follows (8) h I ( ex:p ) , I ( owl:InverseFunctionalProperty ) i ∈ IEXT( I ( rdf:type )) . 028 Inferred Prop ert y Characteristics I I I (Pro of ) Let I b e an OWL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e a z such that (1 a ) h I ( ex:InversesOfFunctionalProperties ) , z i ∈ IEXT( I ( owl:equivalentClass )) , (1 b ) h z , I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) , (1 c ) h z , I ( owl:inverseOf ) i ∈ IEXT( I ( owl:onProperty )) , (1 d ) h z , I ( owl:FunctionalProperty ) i ∈ IEXT( I ( owl:someValuesFrom )) . F rom (1a) and the seman tic condition for class equiv alence (OWL2/T ab5.9, “ → ”) follows (2) ∀ x : x ∈ ICEXT( I ( ex:InversesOfFunctionalProperties )) ⇔ x ∈ ICEXT( z ) . 62 Mic hael Schneider and Geoﬀ Sutcliﬀe F rom (1b) – (1d) and the semantic condition for existential property restrictions (O WL2/T ab5.6) follo ws (3) ∀ x : x ∈ ICEXT( z ) ⇔ ∃ y : [ h x, y i ∈ IEXT( I ( owl:inverseOf )) ∧ y ∈ ICEXT( I ( owl:FunctionalProperty )) ] . F rom (2) and (3) follows (4) ∀ x : x ∈ ICEXT( I ( ex:InversesOfFunctionalProperties )) ⇔ ∃ y : [ h x, y i ∈ IEXT( I ( owl:inverseOf )) ∧ y ∈ ICEXT( I ( owl:FunctionalProperty )) ] . Let p b e an arbitrary individual suc h that (5) p ∈ ICEXT( I ( ex:InversesOfFunctionalProperties )) . W e receiv e (5) and (4): (6) ∃ y : h p, y i ∈ IEXT( I ( owl:inverseOf )) ∧ y ∈ ICEXT( I ( owl:FunctionalProperty )) . According to (6) there is a q for p such that (7) h p, q i ∈ IEXT( I ( owl:inverseOf )) ∧ q ∈ ICEXT( I ( owl:FunctionalProperty )) . F rom (7) and the seman tic condition for inv erse prop erties (OWL2/T ab5.12, “ → ”) follows (8) ∀ x, y : h x, y i ∈ IEXT( p ) ⇔ h y , x i ∈ IEXT( q ) . F rom (7) and the semantic condition for functional prop erties (OWL2/T ab5.13, “ → ”) follows: (9) ∀ x, y 1 , y 2 : h x, y 1 i ∈ IEXT( q ) ∧ h x, y 2 i ∈ IEXT( q ) ⇒ y 1 = y 2 . F rom (8) and (9) follows (10) ∀ y 1 , y 2 , x : h y 1 , x i ∈ IEXT( p ) ∧ h y 2 , x i ∈ IEXT( p ) ⇒ y 1 = y 2 . F rom (7) and the prop ert y extension of owl:inverseOf (OWL2/T ab.5.3) follo ws (11) p ∈ IP . F rom (11), (10) and the semantic condition for in verse-functional prop erties (O WL2/T ab5.13, “ ← ”) follows (12) p ∈ ICEXT( I ( owl:InverseFunctionalProperty )) . Reasoning in OWL 2 F ull using First-Order A TP 63 Since (12) follo ws from (5) and since p has b een arbitrarily c hosen, w e receiv e (13) ∀ x : x ∈ ICEXT( I ( ex:InversesOfFunctionalProperties )) ⇒ x ∈ ICEXT( I ( owl:InverseFunctionalProperty )) . F rom (1a) and the prop ert y extension of owl:equivalentClass (OWL2/T ab5.3) follo ws (14) I ( ex:InversesOfFunctionalProperties ) ∈ IC . F rom O WL2/T ab5.3 follo ws for owl:InverseFunctionalProperty (15) I ( owl:InverseFunctionalProperty ) ∈ IC . Finally , from (14), (15), (13) and the OWL 2 semantic condition for class sub- sumption (OWL2/T ab5.8, “ ← ”) follows (16) h I ( ex:InversesOfFunctionalProperties ) , I ( owl:InverseFunctionalProperty i ∈ IEXT( I ( rdfs:subClassOf )) . 029 Ex F also Quo dlibet (Pro of ) Let I b e an OWL 2 RDF-Based interpre- tation and B b e a blank no de mapping for the blank no des in the premise graph suc h that I + B satisﬁes the premise graph. Let x , y , l 1 , l 2 b e individuals such that (1 a 1) h I ( ex:A ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 b 1) h I ( ex:B ) , I ( owl:Class ) i ∈ IEXT( I ( rdf:type )) , (1 c 1) h I ( ex:w ) , x i ∈ IEXT( I ( rdf:type )) , (1 d 1) h x, l 1 i ∈ IEXT( I ( owl:intersectionOf )) , (1 e 1) h l 1 , I ( ex:A ) i ∈ IEXT( I ( rdf:first )) , (1 e 2) h l 1 , l 2 i ∈ IEXT( I ( rdf:rest )) , (1 e 3) h l 2 , y i ∈ IEXT( I ( rdf:first )) , (1 e 4) h l 2 , I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) , (1 f 1) h y , I ( ex:A ) i ∈ IEXT( I ( owl:complementOf )) . F rom (1d1), (1e1) – (1e4), and the semantic condition for class intersection (O WL2/T ab5.4, “ → ”, binary) follo ws (2) ∀ z : z ∈ ICEXT( x ) ⇔ z ∈ ICEXT( I ( ex:A )) ∧ z ∈ ICEXT( y ) . F rom (1f1) and the semantic condition for class complement (O WL2/T ab5.4, “ → ”) follows (3) ∀ z : z ∈ ICEXT( y ) ⇔ z / ∈ ICEXT( I ( ex:A )) . F rom (2) and (3) follows (4) ∀ z : z ∈ ICEXT( x ) ⇔ z ∈ ICEXT( I ( ex:A )) ∧ z / ∈ ICEXT( I ( ex:A )) . F rom (1c1) and the RDFS seman tic condition of ICEXT (“ → ”) follo ws (5) I ( ex:w ) ∈ ICEXT( x ) . 64 Mic hael Schneider and Geoﬀ Sutcliﬀe F rom (5) and (4) follows a con tradiction, i.e. the set of premises is contradictory . F rom a con tradiction follo ws arbitrary ( “ex falso se quitur quo d lib et” ), hence we receiv e: (6) h I ( ex:w ) , I ( ex:B ) i ∈ IEXT( I ( rdf:type )) . 030 Bad Class (Pro of ) Let I b e an O WL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank nodes in the premise graph suc h that I + B satisﬁes the premise graph. Let there be an x , such that (1 a ) h I ( ex:c ) , x i ∈ IEXT( I ( owl:complementOf )) , (1 b ) h x, I ( owl:Restriction ) i ∈ IEXT( I ( rdf:type )) , (1 c ) h x, I ( rdf:type ) i ∈ IEXT( I ( owl:onProperty )) , (1 d ) h x, I ( "true"8sd:boolean ) i ∈ IEXT( I ( owl:hasSelf )) . F rom (1a) and the semantic condition for class complemen t (OWL2/T ab5.4, “ → ”) follows (2) ∀ y : y ∈ ICEXT( I ( ex:c )) ⇔ y / ∈ ICEXT( x ) . F rom (1b), (1c), (1d) and the semantic condition for self-restrictions (OWL2/T ab5.6) follo ws (3) ∀ z : z ∈ ICEXT( x ) ⇔ h z , z i ∈ IEXT( I ( rdf:type )) . No w the following equiv alence holds: h I ( ex:c ) , I ( ex:c ) i ∈ IEXT( I ( rdf:type )) ⇔ I ( ex:c ) ∈ ICEXT( x ) : by (3) ⇔ I ( ex:c ) / ∈ ICEXT( I ( ex:c )) : b y (2) ⇔ h I ( ex:c ) , I ( ex:c ) i / ∈ ICEXT( I ( rdf:type )) : by ICEXT deﬁnition Hence, we receiv e a contradiction solely from the original settings (1a), (1b), (1c) and (1d). That is, the original setting is an inconsisten t on tology . 031 Large Universe (Pro of ) Let I b e an OWL 2 RDF-Based interpretation and B b e a blank no de mapping for the blank no des in the premise graph such that I + B satisﬁes the premise graph. Let there b e x and l , such that the follo wing holds: (1 a ) h I ( owl:Thing ) , x i ∈ IEXT( I ( owl:equivalentClass )) , (1 b ) h x, l i ∈ IEXT( I ( owl:oneOf )) , (1 c ) h l, I ( ex:w ) i ∈ IEXT( I ( rdf:first )) , (1 d ) h l, I ( rdf:nil ) i ∈ IEXT( I ( rdf:rest )) . F rom (1a) and the seman tic condition for class equiv alence (OWL2/T ab5.9, “ → ”) follows (2) ∀ z : z ∈ ICEXT( I ( owl:Thing )) ⇔ z ∈ ICEXT( x ) . Reasoning in OWL 2 F ull using First-Order A TP 65 F rom (1b), (1c), (1d) and semantic condition for en umeration classes (OWL2/T ab5.5, “ → ”) follows (3) ∀ z : z ∈ ICEXT( x ) ⇔ z = I ( ex:w ) . Since I is a simple interpretation and from the class extension of owl:Thing (O WL2/T ab5.2, “ ← ”) we receiv e (4 a ) I ( owl:Thing ) ∈ ICEXT( I ( owl:Thing )) , (4 b ) I ( owl:Nothing ) ∈ ICEXT( I ( owl:Thing )) . Applying (2) and (3) to (4a) and (4b), resp ectiv ely , leads to (5 a ) I ( owl:Thing ) = ex:w , (5 b ) I ( owl:Nothing ) = ex:w , and therefore (6) I ( owl:Thing ) = I ( owl:Nothing ) . F rom (6) and (4b) follows (7) I ( owl:Nothing ) ∈ ICEXT( I ( owl:Nothing )) . Ho wev er, from the class extension of owl:Nothing (O WL2/T ab5.2) follows (8) ∀ z : z / ∈ ICEXT( I ( owl:Nothing )) . By (7) and (8) w e get a contradiction. Hence the original setting (1a), (1b), (1c) and (1d) is an inconsisten t on tology . 032 Datatype Relationships (Pro of ) Let I b e an O WL 2 RDF-Based in- terpretation that satisﬁes the empt y graph. As a consequence of OWL2/Def4.2, I must b e sp eciﬁed with resp ect to some O WL 2 RDF-Based datatype map D . According to OWL2/Def4.1, D m ust include the datat yp es denoted b y the URIs xsd:string , xsd:integer , and xsd:decimal . The denotations are given by name-datat yp e pairs “( u, d )” pro- vided by the datatype map, and the v alue spaces are given as “VS( d )”. Accord- ing to the “gener al semantic c onditions for datatyp es” in the sp eciﬁcation of D- en tailment, the datatypes are identiﬁed by “ I ( xsd:string )”, “ I ( xsd:integer )”, and “ I ( xsd:decimal )”, resp ectiv ely . Secondly , the datatypes I ( u ), for u one of “ xsd:string ”, “ xsd:integer ”, and “ xsd:decimal ”, are instances of the set ICEXT( I ( rdfs:Datatype )). O WL2/T ab5.2 implies ICEXT( I ( rdfs:Datatype )) = IDC. F rom OWL2/T ab5.1 follo ws that IDC is a sub set of IC. F rom O WL2/T ab5.2 follo ws that ICEXT( I ( owl:Class )) = IC. Hence, we get: (1 a ) I ( xsd:string ) ∈ IC ; (1 b ) I ( xsd:integer ) ∈ IC ; (1 c ) I ( xsd:decimal ) ∈ IC . F urther, according to the “gener al semantic c onditions for datatyp es” in the sp eciﬁcation of D-entailmen t the datatypes hav e the follo wing v alue spaces: 66 Mic hael Schneider and Geoﬀ Sutcliﬀe ICEXT( I ( xsd:string )), ICEXT( I ( xsd:integer )), and ICEXT( I ( xsd:decimal )). According to O WL2/Def4.1 (referring to the OWL 2 Structural Sp eciﬁcation), the v alue spaces of the three datatypes are deﬁned according to the XSD Datat yp e sp eciﬁcation. This has the follo wing consequences. Firstly , the v alue spaces of xsd:decimal and xsd:string are disjoin t sets: (2 a ) ∀ x : ¬ [ x ∈ ICEXT( I ( xsd:decimal )) ∧ x ∈ ICEXT( I ( xsd:string )) ] . Secondly , the v alue space of xsd:integer is a subset of the v alue space of xsd:decimal : (2 b ) ∀ x : x ∈ ICEXT( I ( xsd:integer )) ⇒ x ∈ ICEXT( I ( xsd:decimal )) . Using (1c), (1a), (2a), and the “ ← ” direction of the semantic condition for class disjoin tness (O WL2/T ab5.9), w e get: (3 a ) h I ( xsd:decimal ) , I ( xsd:string ) i ∈ IEXT( I ( owl:disjointWith )) . Using (1b), (1c), (2b), and the “ ← ” direction of the OWL 2 seman tic condition of class subsumption (OWL2/T ab5.8), w e get: (3 b ) h I ( xsd:integer ) , I ( xsd:decimal ) i ∈ IEXT( I ( rdfs:subClassOf )) . The combination of (3a) and (3b) was the conjecture. Reasoning in OWL 2 F ull using First-Order A TP 67 C T ranslation into TPTP In Section 3 it was explained how RDF graphs and the semantics of OWL 2 F ull (see Section 2.1) are translated into FOL, and Section 4.1 men tioned that the TPTP language [14] is used as a concrete FOL serialization syntax. In this app endix, the translation in to TPTP are demonstrated b y means of a concrete example. The translation is demonstrated using the test case 020 Logical Complications from the test suite of char acteristic OWL 2 F ul l c onclusions , which has been deﬁned in App endix B. The example translation will b e complete in the sense that the resulting TPTP enco ding can b e used with F OL A TPs that understand the TPTP language 7 in order to obtain the reasoning result of the test case. The TPTP translations for the example test case and for all other c haracteristic conclusions test cases are included in the electronic version of the test suite; see App endix B for p oin ters. In addition, the supplementary material for this pap er (see the download link at the beginning of Section 4) con tains a translation of a large fragmen t of the O WL 2 F ull seman tics into TPTP (see Section 4.1 for a characterization of the fragment) and provides an executable softw are to ol for the conv ersion of arbitrary RDF graphs in to TPTP . C.1 RDF Graphs and T est Case Data In this section it is shown how RDF graphs and test case data are con verted in to the TPTP language. Example translations are giv en for the premise and conclusion graphs of the en tailment test case 020 L o gic al Complic ations from the “characteristic OWL 2 F ull conclusions” test suite. According to Section B.1, the pr emise gr aph of the example test case is given in T urtle syn tax 8 as: @prefix ex: . @prefix rdf: . @prefix rdfs: . @prefix owl: . ex:c owl:unionOf ( ex:c1 ex:c2 ex:c3 ) . ex:d owl:disjointWith ex:c1 . ex:d rdfs:subClassOf [ owl:intersectionOf ( ex:c [ owl:complementOf ex:c2 ] ) ] . 7 The reasoners av ailable online as part of the SystemOnTPTP service can be used for this purp ose: http://www.tptp.org/cgi- bin/SystemOnTPTP/ . 8 T urtle RDF syntax: http://www.w3.org/TeamSubmission/turtle/ 68 Mic hael Schneider and Geoﬀ Sutcliﬀe This encoding uses some of the “syntactic sugar” that T urtle oﬀers for con- cisely representing certain language constructs, such as RDF collections. F or the purp ose of translating the RDF graph into TPTP , it is advisable to restate the ab o ve representation into an equiv alent form that consists of only RDF triples: @prefix ex: . @prefix rdf: . @prefix rdfs: . @prefix owl: . ex:c owl:unionOf _:lu1 . _:lu1 rdf:first ex:c1 . _:lu1 rdf:rest _:lu2 . _:lu2 rdf:first ex:c2 . _:lu2 rdf:rest _:lu3 . _:lu3 rdf:first ex:c3 . _:lu3 rdf:rest rdf:nil . ex:d owl:disjointWith ex:c1 . ex:d rdfs:subClassOf _:xs . _:xs owl:intersectionOf _:li1 . _:li1 rdf:first ex:c . _:li1 rdf:rest _:li2 . _:li2 rdf:first _:xc . _:li2 rdf:rest rdf:nil . _:xc owl:complementOf ex:c2 . Premise graphs of entailmen t test cases are translated into TPTP axiom form ulae. F ollowing the explanation in Section 3 on ho w to translate RDF graphs in to F OL, the translation in to TPTP is as follows: fof(testcase_premise, axiom, ( ? [B_xs, B_xc, B_lu1, B_lu2, B_lu3, B_li1, B_li2] : ( iext(uri_owl_unionOf, uri_ex_c, B_lu1) & iext(uri_rdf_first, B_lu1, uri_ex_c1) & iext(uri_rdf_rest, B_lu1, B_lu2) & iext(uri_rdf_first, B_lu2, uri_ex_c2) & iext(uri_rdf_rest, B_lu2, B_lu3) & iext(uri_rdf_first, B_lu3, uri_ex_c3) & iext(uri_rdf_rest, B_lu3, uri_rdf_nil) & iext(uri_owl_disjointWith, uri_ex_d, uri_ex_c1) & iext(uri_rdfs_subClassOf, uri_ex_d, B_xs) & iext(uri_owl_intersectionOf, B_xs, B_li1) & iext(uri_rdf_first, B_li1, uri_ex_c) & iext(uri_rdf_rest, B_li1, B_li2) & iext(uri_rdf_first, B_li2, B_xc) & iext(uri_rdf_rest, B_li2, uri_rdf_nil) & iext(uri_owl_complementOf, B_xc, uri_ex_c2) ))) . Reasoning in OWL 2 F ull using First-Order A TP 69 The T urtle representation of the c onclusion gr aph of the test case is given as: @prefix ex: . @prefix rdf: . @prefix rdfs: . @prefix owl: . ex:d rdfs:subClassOf ex:c3 . Conclusion graphs of entailmen t test cases are translated into TPTP c onje c- tur e formulae, which is done as follo ws: fof(testcase_conclusion, conjecture, ( iext(uri_rdfs_subClassOf, uri_ex_d, uri_ex_c3) )) . C.2 Seman tic Conditions of the OWL 2 RDF-Based Semantics In this section it is sho wn ho w the semantic conditions of the O WL 2 RDF-Based Seman tics are translated into the TPTP language. An example translation is giv en for a small subset of s eman tic conditions that are suﬃcient to en tail the conclusion graph of the en tailment te st case 020 L o gic al Complic ations from its premise graph. The selection of the small suﬃcient subset of semantic conditions w as made based on the correctness proof for the test case, as giv en in Section B.2. The following semantic conditions are used to prov e correctness: – extension of prop ert y owl:disjointWith (Section 5.3 of O WL 2 RDF-Based Seman tics); – class complemen t (Section 5.4 of OWL 2 RDF-Based Semantics); – binary class intersection (Section 5.4 of OWL 2 RDF-Based Semantics); – ternary class union (Section 5.4 of OWL 2 RDF-Based Semantics); – class subsumption, OWL v ersion (Section 5.8 of OWL 2 RDF-Based Seman- tics); – class disjoin tness (Section 5.9 of OWL 2 RDF-Based Semantics). Seman tic conditions are translated in to TPTP axiom formulae, since, tech- nically , they act as further premises in addition to the axiom that represen ts the premise graph of a test case. F ollowing the explanation in Section 3 on how to translate semantic conditions into F OL, the translation into TPTP is as follows: % extension of property owl:disjointWith % (Section 5.3 of OWL 2 RDF-Based Semantics) fof(owl_prop_disjointwith_ext, axiom, ( ! [X, Y] : ( iext(uri_owl_disjointWith, X, Y) => ( ic(X) & ic(Y) )))) . 70 Mic hael Schneider and Geoﬀ Sutcliﬀe % class complement % (Section 5.4 of OWL 2 RDF-Based Semantics) fof(owl_bool_complementof_class, axiom, ( ! [Z, C] : ( iext(uri_owl_complementOf, Z, C) => ( ic(Z) & ic(C) & ( ! [X] : ( icext(Z, X) <=> ~ icext(C, X) )))))) . % binary class intersection % (Section 5.4 of OWL 2 RDF-Based Semantics) fof(owl_bool_intersectionof_class_002, axiom, ( ! [Z, S1, C1, S2, C2] : ( ( iext(uri_rdf_first, S1, C1) & iext(uri_rdf_rest, S1, S2) & iext(uri_rdf_first, S2, C2) & iext(uri_rdf_rest, S2, uri_rdf_nil) ) => ( iext(uri_owl_intersectionOf, Z, S1) <=> ( ic(Z) & ic(C1) & ic(C2) & ( ! [X] : ( icext(Z, X) <=> ( icext(C1, X) & icext(C2, X) )))))))) . % ternary class union % (Section 5.4 of OWL 2 RDF-Based Semantics) fof(owl_bool_unionof_class_003, axiom, ( ! [Z, S1, C1, S2, C2] : ( ( iext(uri_rdf_first, S1, C1) & iext(uri_rdf_rest, S1, S2) & iext(uri_rdf_first, S2, C2) & iext(uri_rdf_rest, S2, S3) & iext(uri_rdf_first, S3, C3) & iext(uri_rdf_rest, S3, uri_rdf_nil) ) => ( Reasoning in OWL 2 F ull using First-Order A TP 71 iext(uri_owl_unionOf, Z, S1) <=> ( ic(Z) & ic(C1) & ic(C2) & ic(C3) & ( ! [X] : ( icext(Z, X) <=> ( icext(C1, X) | icext(C2, X) | icext(C3, X) )))))))) . % class subsumption, OWL version % (Section 5.8 of OWL 2 RDF-Based Semantics) fof(owl_rdfsext_subclassof, axiom, ( ! [C1, C2] : ( iext(uri_rdfs_subClassOf, C1, C2) <=> ( ic(C1) & ic(C2) & ( ! [X] : ( icext(C1, X) => icext(C2, X) )))))) . % class disjointness % (Section 5.9 of OWL 2 RDF-Based Semantics) fof(owl_eqdis_disjointwith, axiom, ( ! [C1, C2] : ( iext(uri_owl_disjointWith, C1, C2) <=> ( ic(C1) & ic(C2) & ( ! [X] : ( ~ ( icext(C1, X) & icext(C2, X) ))))))) .

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