Elimination of Complex RIAs without Automata
František Simanˇcík
Department of Computer Science, University of Oxford, UK
Abstract. We present an algorithm that eliminates complex role inclusion ax- ioms (RIAs) from aSROIQontology preserving all logical consequences not involving non-simple roles. Unlike other existing methods, our algorithm does not explicitly construct finite automata recognizing the languages generated by the RIAs. Instead, it is formulated as a recursive expansion of universal restric- tions, similar to well-known encodings of transitivity axioms.
1 Introduction
Complex role inclusion axioms (RIAs)R1◦. . .◦Rn v R are an important feature by which the web ontology language OWL 2 [2], based on the description logic (DL) SROIQ[5], extends the earlier standard OWL DL. Unrestricted use of complex RIAs causes undecidability of the basic reasoning tasks already in case of fairly inexpres- sive DLs and modal logics such asALC[1]; decidability is recovered by requiring that RIAs, when viewed as context-free grammar rulesR → R1. . . Rn, generate regular languages [3, 4, 7]. However, checking if a given set of RIAs has this property is already difficult; this problem is related to checking regularity ofpurecontext-free grammars [10] (which do not distinguish terminal and non-terminal symbols) whose decidability appears to have been long open [7]. To avoid this difficulty, a stronger syntacticregular- itycondition that requires RIAs to be acyclic (apart from a few selected cases such as transitivity axioms) is imposed inSROIQ. This condition is easy to check and allows for an effective construction of the underlying finite automata [5].
The standard automata construction forSROIQ[5] is an inductive procedure that performs non-trivial manipulations such as taking the mirrored copy and the disjoint union of previously constructed automata. To implement the procedure directly, de- velopers of OWL 2 reasoners have to either look for suitable third-party libraries that support such operations, or resort to writing their own automata library. In this paper, we present an algorithm that eliminates complex RIAs from aSROIQontology with- out explicitly using finite automata. Instead, our algorithm is formulated as a simple recursive expansion of universal restrictions, similar to well-known encodings of tran- sitivity axioms, using acyclicity of RIAs directly to ensure termination. For this reason, we believe that our method might be easier to implement in practice. Furthermore, we illustrate that, by introducing new rules for handling universal restrictions, the tableau algorithm forSROIQ[5] can be modified to apply our elimination on the fly. In that case, no pre-construction due to complex RIAs is needed at all.
Our recursive expansion of universal restrictions is inspired by and directly simu- lates the recursion in the standard automata construction. Therefore, for many purposes, the result of our elimination algorithm can be regarded as equivalent to the standard automata-based encoding [6].
2 Preliminaries
2.1 The DLSROIQ
For a gentle introduction to DLs we refer the readers to the DL primer [8]. In this section we merely recall the definition (syntax only) ofSROIQ[5], together with the notions of a regular RBox and of polarity of concept occurrence. We follow the approach of Shearer [11] in assuming that the set of simple roles is given in the signature and calling a RIAwvRcomplex iffRis non-simple (even ifwis of length1).
A signatureΣ = hΣS, ΣR, ΣC, ΣIiconsists of mutually disjoint sets ofatomic rolesΣR,atomic conceptsΣC, andindividualsΣI, together with a distinguished sub- set ΣS ⊆ ΣR of simple atomic roles. The set of roles (over Σ) is R := ΣR ∪ {R−|R∈ΣR}; the set ofsimple rolesisS :=ΣS ∪ {S−|S∈ΣS}. Arole chain is an expression of the formR1·. . .·Rnwithn≥1and eachRi ∈ R. The function inv(·)is defined on roles byinv(R) := R− andinv(R−) := RwhereR ∈ ΣR, and extended to role chains byinv(R1·. . .·Rn) :=inv(Rn)·. . .·inv(R1).
The setCofSROIQconcepts(overΣ) is defined recursively as follows:
C:=ΣC| {ΣI} |(CuC)|(CtC)| ¬C| ∃R.C| ∀R.C|>nS.C|6nS.C| ∃S.Self. Arole inclusion axiom(RIA) is either asimple RIAof the formS1 v S2where S1, S2 ∈ S, or a complex RIA of the form w v R where w is a role chain and R ∈ R\S. Arole assertionis an axiom of the formRef(R)(reflexivity),Irr(S)(ir- reflexivity),Uni(R)(universality), orDis(S1, S2)(role disjointness), whereR∈Rand S(i)∈S. Transitivity and symmetry must be expressed asR·RvRandinv(R)vR respectively. AnRBoxis a finite set of RIAs and role assertions.
Aregular order≺is an irreflexive transitive binary relation on the set of rolesR satisfyingR1≺R2iffinv(R1)≺inv(R2). An RBoxRis≺-regularif each RIA inR is of one of the following forms:
(R1) R1·. . .·RnvRwithRi≺Rfor all1≤i≤n;
(R2) R·R1·. . .·RnvRwithRi≺Rfor all1≤i≤n;
(R3) R1·. . .·Rn·RvRwithRi≺Rfor all1≤i≤n;
(R4) R·RvR;
(R5) inv(R)vR.
An RBoxRisregularif it is≺-regular for some regular order≺. For a regular RBox R, let≺R be the intersection of all regular orders≺such thatRis≺-regular; thedepth ofRis the maximalnfor which there exists a sequenceR1≺R. . .≺RRn. It is easy to show that ifRis regular, then it is ≺R-regular.
ATBoxis a finite set ofgeneral concepts inclusions(GCIs) of the formC v D whereC, D ∈C. To keep the presentation simple, we do not allow ABox assertions;
these can be expressed as GCIs using nominals. ASROIQontology (overΣ) is a pair O=hR,T iwhereRis a regular RBox andT a TBox.
Polarities of occurrencesofSROIQconcepts in concepts and GCIs are defined in- ductively as follows:Coccurs positively inC. IfCoccurs positively (resp. negatively) inC0, thenCoccurs positively (resp. negatively) inC0uD,DuC0,C0tD,DtC0,
∃R.C0,∀R.C0,>n R.C0, andD v C0, andC occurs negatively (resp. positively) in
¬C0,6n R.C0, andC0vD.
2.2 Conservative Encodings
We use the framework of conservative extensions [9] to prove correctness of our encod- ing of RIAs. LetObe an ontology over a signatureΣ=hΣS, ΣR, ΣC, ΣIi, and letQ be an ontology over a (not necessarily strictly) larger signature. ThenQis aconserva- tive encodingofO(alsoQis conservative overO) if
(i) for every modelI ofOthere exists a model J ofQsuch thatI andJ have the same domain and coincide on the interpretation ofΣR,ΣC, andΣI, and
(ii) for every modelJ ofQthere exists a modelI ofOsuch thatI andJ have the same domain and coincide on the interpretation ofΣR,ΣC, andΣI.
Since this definition is sensitive toΣ, to avoid ambiguity, we will assume that each on- tology carries its signature with it, so that each ontology is over only one signature. The signature may, however, contain symbols not occurring in the ontology. Note that, un- like the standard notion of a conservative extension, the above notion of a conservative encoding does not require thatQcontains all axioms fromO.
We define asimple-conservative encodinganalogously except that the models I andJ are only required to coincide onΣS,ΣC, andΣI. As observed by Lutz et al.
[9], this model-theoretic notion of conservativity implies that the two ontologies en- tail the same consequences overΣS,ΣC, andΣI. We will prove that our encoding of complex RIAs produces a simple-conservative encoding of the input ontology. Thus, in particular, the two ontologies have the same classification (ignoring the extra atomic concepts introduced in the encoding). Furthermore, by introducing new concept defini- tionsA ≡CandB ≡ Din the original ontology, one can check subsumptions even between conceptsCandDwhich contain non-simple roles.
2.3 Languages Generated by RIAs
Each RIAwvRcan be expressed equivalently asinv(w)vinv(R); to avoid having to keep this in mind, letRc:=R∪{inv(w)vinv(R)|wvR∈ R}be thecompletionof the RBoxR. Note thatRandRcare equivalent andRis≺-regular iffRcis≺-regular.
The languagesLR(R) are defined inductively by (i)R ∈ LR(R) for each role R, and (ii) ifR1·. . .·Rn v R ∈ Rcandwi ∈ LR(Ri)for all1 ≤ i ≤ n, then w1·. . .·wn ∈LR(R). Intuitively,LR(R)is the language generated from the roleR by the grammar rules{R→w|wvR∈ Rc}. Horrocks et al. [5] showed that ifRis regular, then eachLR(R)is a regular language, the finite automata recognizingLR(R) can be effectively constructed by induction over≺R, and the size of these automata is at most exponential in the depth ofR.
An interpretation function·Iis extended to the languagesLR(R)as follows:
LR(R)I ={hx, yi |there existsw∈LR(R)such thathx, yi ∈wI}. (1) One can easily prove by induction on the definition ofLR(R)thatw∈LR(R)implies R |=wvR. The following proposition is a direct consequence of this fact.
Proposition 1. IfI |=R, thenLR(R)I=RI.
3 Motivation
In this section we motivate and present (first approximation of) our RIA-elimination algorithm. For simplicity, we do not yet consider symmetry axioms (R5) in this section.
ManySROIQconstructors are restricted to simple roles and therefore do not in- teract with complex RIAs. The key step in eliminating complex RIAs is to capture the propagation of universal restrictions over non-simple roles. Consider the following property:
ifx∈(∀R.C)Iandhx, yi ∈LR(R)I, theny∈CI. (2) Every modelIof the RBoxRsatisfies (2) simply becauseLR(R)I =RI by Propo- sition 1, in which case (2) coincides with semantics of universal restrictions. The main idea behind all methods for dealing with complex RIAs (e.g., [4–6]) is to axiomatise (using a finite number of GCIs) the property (2) for all∀R.Coccurring in the ontology, and use this axiomatisation to simulate the presence of complex RIAs fromR.
In the simplest case, when all RIAs inRare of the form (R1), this can be achieved by a simple recursive expansion of all universal restrictions occurring in the ontology.
To expand∀R.C, for each RIAR1·. . .·RnvR∈ Rcintroduce the axiom
∀R.Cv ∀R1.∀R2. . .∀Rn.C, (3) and recursively expand all the nested universal restrictions on the right-hand side of (3).
If all RIAs are of the form (R1), thenRi≺RRfor each roleRiin (3), so the depth of the recursion is bounded by the depth ofRand the expansion terminates.
In case there are other forms of RIAs inR, e.g., transitivity axioms, the recursion would never terminate. On the other hand, transitivity axioms can be eliminated using several well-known encodings. For example, to capture (2) for the concept∀R.Cwith respect to the transitivity axiomR·RvR, introduce two new atomic conceptsIand F not in the signature of the the ontology and assert
∀R.CvI, FvC, Iv ∀R.F, F vI. (4) This encoding is inspired by the fact that the RIA R·R v R generates the regular languageR+which is recognised by the following finite automaton:
i
start f
R
The encoding simulates the run of this automaton from all instances of∀R.Cin a model of (4). ConceptsIandF respectively correspond to the initial stateiand the final state f, the first axiom initialises the automaton at all x ∈ (∀R.C)I, the second axiom ensures thaty∈CIfor allyin the final state, and the last two axioms encode the tran- sitions of the automaton. This method generalises easily to all cases when the language LR(R)is given by a finite automaton [6].
We propose a RIA-elimination algorithm that does not assume that the automaton is already fully constructed. Our method recursively expands all universal restrictions similarly to (3), but uses a two-state automaton at each step to handle the cyclic forms of RIAs similarly to (4). More specifically, to expand the universal restriction∀R.C, introduce two new atomic conceptsIandF, assert
(E0) ∀R.C vI,F vC, andIv ∀R.F,
(E1) Iv ∀R1. . . .∀Rn.Ffor each RIAR1·. . .·RnvR∈ Rcof form (R1);
(E2) F v ∀R1r. . . .∀Rrn.Ffor each RIAR·Rr1·. . .·RrnvR∈ Rcof form (R2), (E3) Iv ∀R1l. . . .∀Rnl.Ifor each RIARl1·. . .·Rln·RvR∈ Rcof form (R3), (E4) F vIifR·RvR∈ Rc,
and recursively expand all the universal restrictions introduced in (E1)–(E3), but not the∀R.F introduced in (E0). Regularity ofRensures that the depth of the recursion is bounded by the depth ofR. This encoding is inspired by the following automaton (the -edge is present iffR·R vR ∈ Rc) used in the construction by Horrocks et al. [5]
that recognises the language generated fromRby the RIAs referred to in (E1)–(E4) : i
start f
R
Rl1. . . Rln Rr1. . . Rrn
R1. . . Rn
Example 1. We will now demonstrate the RIA-elimination algorithm on an example.
LetΣS = {P, R},ΣR = {P, R, S, T},ΣC = {A, B, C, D},ΣI =∅, and letO = hR,T ibe the following ontology over the signatureΣ=hΣS, ΣR, ΣC, ΣIi:
R={T·S vT, T·T vT, R·S vS, P vR}, (5) T ={AvDt ∀T.¬C, Bv ∃T.∃P.∃S.C}. (6) Note that Ris regular andP ≺R R≺R S≺R T. The RIAP v R is simple; the remaining RIAs inRare complex. To expand the universal restriction∀T.¬Coccurring inT, introduce new atomic conceptsI1andF1, and assert the following axioms:
∀T.¬CvI1, F1v ¬C, I1v ∀T.F1 by (E0) (7) F1v ∀S.F1 by (E2) forT·SvT (8)
F1vI1 by (E4) forT·T vT (9)
Then, to recursively expand the new universal restriction∀S.F1occurring in (8), intro- duce new atomic conceptsI2andF2, and assert the following axioms:
∀S.F1vI2, F2vF1, I2v ∀S.F2 by (E0) (10) I2v ∀R.I2 by (E3) forR·SvS (11) Finally, since we intend to keep all simple RIAs in the RBox and the roleRis sim- ple, the new universal restriction∀R.I2introduced in (11) does not need to be further expanded. LetU be the TBox consisting of the new axioms (7)–(11). The results of the next section will establish that the ontologyQ = h{P vR},T ∪ U i is simple- conservative overO, so, in particular, the two ontologies entail the same consequences overΣS,ΣC, andΣI. For example, bothOandQentailP v RandAuB vD.
Note that this cannot be strengthened to all consequences overΣsince, for example,O entailsT·P·SvT andBv ∃T.C, butQdoes not entail either of these axioms.
4 The RIA-Elimination Algorithm
In this section we formally present our RIA-elimination algorithm and prove that it produces a simple-conservative encoding of the input ontology. Some of the proofs are rather technical, in those cases we present only brief sketches here. More detailed proofs can be found in the appendix.
There are several important points about the algorithm that were omitted in the previous section in favour of simplicity; this is amended in this section. Firstly, ifR is a symmetric role, i.e.,inv(R) v R ∈ Rc, then the concept∀R.C is additionally expanded in the same way as∀inv(R).Cwould be. Secondly, since we do not assume that the ontology is in negation normal form, we have to treat negative occurrences of existential restrictions similarly to positive occurrences of universal restrictions. Finally, the expansion rule (E0) is inefficient because it introduces the axiom∀R.C v I in which∀R.C occurs negatively. Negative occurrences of universal restrictions are not Horn, i.e., they lead to non-determinism in reasoning. To avoid this problem, instead of asserting∀R.C vI, the algorithmreplacesall positive occurrences of∀R.Cin the original ontology byI. This way we obtain a Horn-preserving encoding.
To keep track of the progress of the algorithm, we label those concepts∀R.Cand
∃R.C that still need to be expanded withRas defined below.
Definition 1 (R-labelled concepts).Given an RBoxR, we introduce new concept con- structors∀RR.Cand∃RR.CcalledR-labelled universalsandR-labelled existentials respectively. Their semantics is defined as follows:
(∀RR.C)I={x| ∀y:hx, yi ∈LR(R)I →y∈CI}, (∃RR.C)I={x| ∃y:hx, yi ∈LR(R)I∧y∈CI}.
R-labelled conceptsareSROIQconcepts thatmayadditionally containR-labelled universals and existentials. To distinguish them from normal SROIQconcepts, we sometimes call the latter unlabelled. Similarly, we speak of R-labelled (resp. unla- belled) ontologies.
Note that the semantics ofR-labelled concepts is irrelevant for the execution of the algorithm. It does, however, greatly simplify the proofs, since with this semantics we can prove that each intermediate expansion step of the algorithm already produces a simple-conservative encoding of the input ontology.
Given an input ontologyO=hR,T i, the initial step of the algorithm is to remove all complex RIAs fromO(keeping all simple RIAs and all role assertions) and label all positive occurrences of universal restrictions and all negative occurrences of existential restrictions inOwithRto indicate that they need to be expanded. This is defined more formally in the following two definitions.
Definition 2 (labelling).LetRbe an RBox. Forxan unlabelled concept or a TBox, let σR(x)be the result of labelling each positive occurrence of each universal restriction and each negative occurrence of each existential restriction in xwithR. Dually, let
¯
σR(x)be the result of labelling each negative occurrence of each universal restriction and each positive occurrence of each existential restriction inxwithR.
Definition 3 (initialisation). Let O = hR,T i be an unlabelledSROIQ ontology over a signatureΣ. LetRs:=R \ {wvR∈ R |wvRis a complex RIA}. Theini- tialisationofOis theR-labelled ontologyhRs, σR(T)iover the same signatureΣ.
The next theorem proves that initialisation produces a simple-conservative encod- ing ofO. This captures the intuition that positive universal restrictions and negative existential restrictions are the onlySROIQfeatures that interact with complex RIAs.
Theorem 1. The initialisation ofOis simple-conservative overO.
Proof (sketch).We must show that (i) for each modelI ofO there is a modelJ of Q=hRs, σR(T)ithat agrees withIonΣS,ΣC, andΣI, and (ii) vice versa.
For (i), we show that each modelI ofOis already a model ofQ. Trivially,I |= Rimplies I |= Rs sinceRs ⊆ R. By Proposition 1, we haveLR(R)I = RI, so (∀R.C)I = (∀RR.C)Iand(∃R.C)I = (∃RR.C)Ifor all concepts∀R.Cand∃R.C.
This means thatR-labelling does not affect the interpretation of concepts inI, soI |= T impliesI |=σR(T). ThereforeI |=Q.
For (ii), each modelJ ofQcan be transformed to a modelI ofOby extending the interpretation of roles toRI =LR(R)J. FromJ |=Rsone proves thatIandJ agree on simple roles. Checking thatI |=Ris routine. The key step to proveI |=T is to show by structural induction for each unlabelled conceptDthatσR(D)J ⊆DI ⊆
¯
σR(D)J; sinceσR(T) ={¯σR(C)vσR(D)|CvD∈ T }, we can then inferI |=
T fromJ |=σR(T). HenceI |=O. ut
After initialisation, the algorithm repeatedly expands allR-labelled universals and existentials until it arrives at an unlabelled ontology. Before we define these expansions, in the next definition we first introduce an auxiliary functionexpand(Iv ∀RR.F)that encodes the axiomIv ∀RR.Conly using universals∀RS.DwithS≺RR. The func- tion implements the transition function of the two-state automaton from the previous section, and additionally deals with symmetric rolesRby expandingI v ∀RR.C in the same way asIv ∀Rinv(R).C. The correctness of this encoding is expressed in the following Proposition 2. Its proof can be found in the appendix.
Definition 4 (expand).LetRbe a regular RBox,Ra role, andIandF atomic con- cepts. We defineexpand0(Iv ∀RR.F)to be the set consisting of the following GCIs:
1. Iv ∀R.F,
2. Iv ∀RR1. . .∀RRn.Ffor eachR1·. . .·RnvR∈ Rc, 3. Fv ∀RR1. . .∀RRn.Ffor eachR·R1·. . .·RnvR∈ Rc, 4. Iv ∀RR1. . .∀RRn.Ifor eachR1·. . .·Rn·RvR∈ Rc, 5. FvIifR·RvR∈ Rc,
where eachRiis distinct fromR. Finally, we defineexpand(Iv ∀RR.F)to be (expand0(Iv ∀RR.F)∪expand0(Iv ∀Rinv(R).F) ifinv(R)vR∈ Rc,
expand0(Iv ∀RR.F) otherwise.
Proposition 2. IfI |=expand(Iv ∀RR.F), thenI |=Iv ∀RR.F.
We are now ready to define the expansions ofR-labelled concepts. Similarly to structural transformation,∀R-expansion uses atomic conceptsIandF as new names for the concepts∀RR.CandCrespectively, replaces all positive occurrences of∀RR.C byI, addsF vC, and, instead of assertingIv ∀RR.F, it usesexpand(Iv ∀RR.F) to encode the same property. ∃R-expansion works similarly but it additionally uses the equivalence of ∃R.I v F withI v ∀inv(R).F; it uses atomic concepts I and F as new names for the conceptsC and∃RR.C respectively, addsC v I, replaces all negative occurrences of∃RR.CbyF, and, instead of asserting∃RR.IvF, it uses expand(Iv ∀Rinv(R).F)to encode the same property. More formal definitions follow.
Definition 5 (substitution).For conceptsColdandCnew, andxa concept or a TBox, letρ+[Cnew/ Cold](x)resp.ρ−[Cnew/ Cold](x)be the result of simultaneously replacing each positive resp. negative occurrence ofColdinxbyCnew.
Definition 6 (∀R- and∃R-expansions).LetRbe a regular RBox and letO=hR0,T i be anR-labelled ontology over a signatureΣ=hΣS, ΣR, ΣC, ΣIi.
∀R-expansion:
Let∀RR.C be anR-labelled universal occurring inO. LetIandF be two different atomic concepts not inΣC. The∀RR.C-expansionofOis the ontology
hR0, ρ+[I /∀RR.C](T)∪ {F vC} ∪expand(Iv ∀RR.F)i.
∃R-expansion:
Let∃RR.Cbe anR-labelled existential occurring inO. LetIandF be two different atomic concepts not inΣC. The∃RR.C-expansionofOis the ontology
hR0, ρ−[F /∃RR.C](T)∪ {CvI} ∪expand(Iv ∀Rinv(R).F)i.
In both cases, the resulting ontology is over the signaturehΣS, ΣR, ΣC∪ {I, F}, ΣIi.
Note that the initialisation of an ontology contains only positive occurrences of R-labelled universals and only negative occurrences ofR-labelled existentials. Fur- thermore, both expansions introduce only positive occurrences ofR-labelled univer- sals. Therefore, when used in the context of the RIA-elimination algorithm, the above expansions will actually eliminate alloccurrences of ∀RR.C resp.∃RR.C from the ontology. The reason for making the substitutions polarity-sensitive was to make the following theorem hold for arbitrary ontologies.
Theorem 2. LetRbe a regular RBox, letObe anR-labelled ontology, and letQbe a
∀RR.C- or∃RR.C-expansion ofO. ThenQis conservative overO.
Proof (sketch).Let O = hR0,T1i be an R-labelled ontology over Σ and let Q = hR0,T2ibe a∀RR.C-expansion ofO. We need to show that (i) for each modelIofO there exists a modelJ ofQthat agrees withIonΣ, and (ii) vice versa.
For (i), each modelIofOcan be extended to a model ofQby interpreting the new conceptsII := (∀RR.C)IandFI := (∃Rinv(R).∀RR.C)I. The substitution merely replaces some occurrences of∀RR.C byI, and(∀RR.C)I =II, soI |=T1implies I |=ρ+[I /∀RR.C](T1). To conclude thatIis a model ofQ, it remains to check that
I satisfies each axiom in{FvC} ∪expand(I v ∀RR.F). This is done using the definition ofIIandFI; for example,I |=F vCholds because∃Rinv(R).∀RR.Cv Cis a tautology. The remaining axioms can be checked similarly.
For (ii), we show that each modelJ ofQis already a model ofO. To proveJ |=T1, the key step is to prove by structural induction for all conceptsDoverΣthat
ρ+[I /∀RR.C](D)J ⊆DJ ⊆ρ−[I /∀RR.C](D)J; (12) thenJ |= ρ+[I /∀RR.C](T1) ⊆ T2 impliesJ |= T1. The only non-trivial case in the induction isD =∀RR.C, where (12) reduces toIJ ⊆(∀RR.C)J; to show this, we apply Proposition 2 toJ |= ({FvC} ∪expand(I v ∀RR.F)) ⊆ T2 to infer J |=Iv ∀RR.C, which is equivalent to the requiredIJ ⊆(∀RR.C)J.
The proof of the case whenQis an∃RR.C-expansion ofO, is similar: Each model I ofOcan be extended to a model ofQby interpretingII := (∀Rinv(R).∃RR.C)I andFI := (∃RR.C)I. The substitution merely replaces some occurrences of∃RR.C byF, and(∃RR.C)I =FI, soI |=T1impliesI |=ρ−[F /∃RR.C](T1). To conclude thatI is a model ofQ, one can check that the definition ofII andFIsatisfies each axiom in{CvI} ∪expand(Iv ∀Rinv(R).F).
To show that each modelJ ofQis a model ofO, prove by structural induction for all conceptsDoverΣthatρ−[F /∃RR.C](D)J ⊆DJ ⊆ρ+[F /∃RR.C](D)J. This, in the only non-trivial caseD=∃RR.C, reduces to(∃RR.C)J ⊆FJ; to show this, apply Proposition 2 toJ |= ({CvI} ∪expand(Iv ∀Rinv(R).F))⊆ T2to infer J |=Cv ∀Rinv(R).F, which is equivalent to the required(∃RR.C)J ⊆FJ. ut The following theorem is our main result. It ensures that the RIA-elimination al- gorithm produces a simple-conservative encoding of the input ontology, and that the number of expansions is at most exponential in the depth ofR. Therefore, since each expansion is linear in the size ofR, the algorithm can be implemented to run in time exponential in the depth ofR, which is optimal since complex RIAs are known to incur an exponential increase in the complexity of reasoning [6].
Theorem 3. LetO=hR,T ibe an unlabelledSROIQontology, lethRs,T0ibe the initialisation ofO, and let(Ti)ni=1be any sequence of TBoxes such thathRs,Ti+1iis obtained fromhRs,Tiiby a∀R- or∃R-expansion. ThenhRs,Tniis simple-conservative overO. Moreover,nis bounded bykT k ·(2· kRk)dwheredis the depth ofR.
Proof. The proof uses the observation that ifO1is simple-conservative overO, andO2
is conservative overO1, thenO2is also simple-conservative overO, which follows di- rectly from the respective definitions. With this, it is easy to prove by induction onithat eachhRs,Tiiis simple-conservative overO: the base case is established in Theorem 1, and the induction step follows from Theorem 2 and the above observation. Therefore, in particular,hRs,Tniis simple-conservative overO.
To obtain the bound onn, letr= 2· kRk. As remarked earlier, each∀RR.C- resp.
∃RR.C-expansion eliminates all occurrences of∀RR.Cresp.∃RR.Cfrom the ontol- ogy, and introduces at mostrnew concepts∀RS.D(the factor2 is due to symmetric roles) all satisfyingS≺RR. Therefore, each∀R.Cand∃R.Coccurring inO(of which there are at mostkT k) can altogether generate at most1 +r+r2+. . .+rd−1< rd R-labelled universals and existentials, which yields the required bound ofkT k ·rd. ut
I-intro: if ∀R.C∈ L(x), thenL(x)+=I[∀R.C], and
L(x)+=I[∀inv(R).C]ifinv(R)vR∈ Rc.
F-intro: if I[∀R.C]∈ L(x)and (R∈ L(x, y)orinv(R)∈ L(y, x)), thenL(y)+=F[∀R.C].
F-elim: if F[∀R.C]∈ L(y), thenL(y)+=C.
I-exp: if I[∀R.C]∈ L(x),
thenL(x)+=∀R1. . .∀Rn.F[∀R.C]for eachR1·. . .·RnvR∈ Rc, and L(x)+=∀R1. . .∀Rn.I[∀R.C]for eachR·R1·. . .·RnvR∈ Rc. F-exp: if F[∀R.C]∈ L(y),
thenL(y)+=∀R1. . .∀Rn.F[∀R.C]for eachR1·. . .·Rn·RvR∈ Rc, and L(y)+=I[∀R.C]ifR·RvR∈ Rc.
Fig. 1.Tableau rules for expansion of complex RIAs
Finally, as already observed in Example 1, the algorithm can be optimised by re- placing all concepts ∀RS.C and∃RS.C with a simple roleS directly by∀S.C and
∃S.C respectively, omitting their expansion. Interestingly, this makes the algorithm work without further modifications even in the presence of arbitrary cyclic simple RIAs;
it is enough that complex RIAs are acyclic to ensure termination.
5 Elimination of Complex RIAs in the Tableau Algorithm
In this section we briefly sketch how the tableau algorithm forSROIQ[5] can be mod- ified to perform our encoding of complex RIAs on the fly. We assume that the readers are already familiar with the tableau algorithm. We use the standard notationL(x)and L(x, y)for labels of nodes and edges in the completion graph. We assume that with each concept∀R.C we can uniquely associate new conceptsI[∀R.C] andF[∀R.C];
these will be used in the expansion of∀R.C. Since the tableau algorithm operates with concepts in negation normal form, it can never encounter a negative occurrence of an existential restriction, therefore expansion is only applicable to universal restrictions.
To obtain the modified tableau algorithm, replace all rules relating to universal re- strictions (rules∀1,∀2,∀3in [5]) by the rules in Fig. 1, whereL(x)+=Cis a shorthand forL(x) :=L(x)∪ {C}and eachRiis implicitly distinct fromR. These rules can be readily compared to the expansion rules (E0)–(E4) from Section 3: rulesI-intro,F- intro, andF-elim implement the axioms∀R.C vI,I v ∀R.F, andF vCof (E0), ruleI-exp implements the expansions (E1) and (E3), and ruleF-exp implements the expansions (E2) and (E4). The rules can be extended with blocking conditions as usual.
Note that the first three rules together subsume the standard∀-rule, but additionally introduceI[∀R.C]inL(x)andF[∀R.C]inL(y), even in case there are no complex RIAs in the ontology at all. To eliminate this overhead, similarly to the optimisation above, one can restrict ruleI-intro to non-simple rolesR, and apply the standard∀-rule to universal restrictions with simple roles.
6 Conclusions
We presented an algorithm that encodes complex RIAs inSROIQwithout construct- ing finite automata. The algorithm can also be applied in weaker DLs: apart from GCIs involving atomic concepts and concepts already occurring in the ontology, the algo- rithm introduces only GCIs of the formIv ∀R.F whereIandF are atomic concepts, andRa possibly inverse role. Inverse roles are not strictly required either: if desired, eachIv ∀R−.Fwith an inverse roleR−can be replaced by the equivalent∃R.I vF. Our algorithm shares many theoretical properties of the traditional approaches based on automata, e.g., it is Horn-preserving and runs in time exponential in the depth of the RBox. On the other hand, a notable difference between the two approaches is that, in the automata construction, one can apply standard techniques for minimising the number of automata states and thus potentially reduce the number of new concepts introduced in the encoding. While it might be difficult to provide similarly robust optimisation for our algorithm, several simple optimisations, such as the one presented here that restricts expansion of universal restrictions to non-simple roles, might already help in many realistic cases. Experimental evaluation of the algorithm is left for future work.
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