Bridging the Gap between Tableau and Consequence-Based Reasoning

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Bridging the Gap between Tableau and Consequence-Based Reasoning

Yevgeny Kazakov and Pavel Klinov The University of Ulm, Germany {yevgeny.kazakov, pavel.klinov}@uni-ulm.de

Abstract. We present a non-deterministic consequence-based procedure for the description logicALCHI. Just like the similar style (deterministic) procedures forELand Horn-SHIQ, our procedure explicitly derives subsumptions between concepts, but due to non-deterministic rules, not all of these subsumptions are consequences of the ontology. Instead, the consequences are only those subsumptions that can be derived regardless of the choices made in the application of the rules. This is similar to tableau-based procedures, for which an ontology is inconsistent if every expansion of the tableau eventually results in a clash. We report on a preliminary experimental evaluation of the procedure using a version of SNOMED CT with disjunctions, which demonstrates some promising potential.

1 Introduction and Motivation

Consequence-based and tableau-based methods are two well-known approaches to reasoning in Description Logics (DLs). The former work by deriving logical consequences of axioms in the ontology using inference rules while the latter by building countermod- els for conjectures. Historically, consequence-based methods have been applied to Horn DLs, most prominently, theELfamily [1, 10] and Horn-SHIQ[6], for which they are more efficient than tableau, both theoretically and empirically. Tableau-based methods, in turn, continued to dominate for non-Horn logics and very expressive fragments of the DL family, e.g.,SROIQ. This was mostly because, first, reasoning in expressive logics requires various forms of case analysis, for which designing complete goal-directed inference rule systems has not been easy (but tableau can handle cases via backtracking), and second, the tableau algorithms have been deemed easier to extend to new constructors, because they follow more closely their semantics.

The most straightforward attempt to combine the best from both worlds is to implement both a consequence-based and a tableau-based procedure and use one or another depending on the ontology. Indeed, several tableau reasoners, including Pellet and Konclude, switch to a consequence-based algorithm forELinputs. A more advanced approach, is to automatically split the ontology into parts and use a consequence-based algorithm for the deterministic part (which often covers nearly all of the ontology) and tableau for the rest. Such reasoners as MORe [11] and Chainsaw [14] use ontology modularity techniques for such splitting and then combine the results.

Another line of research is to extend consequence-based calculi to deal with non- determinism. One notable approach has been developed for ConDOR, a reasoner for

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fullALCH[13]. The ConDOR’s procedure deals with disjunctions using deterministic inference rules akin to ordered resolution [3]. The calculus is complete, retains the optimal worst-case complexity, and still enables the “one pass” classification. It shows im- pressive performance speed-up over the existing (hyper)tableau-based reasoners [13].

In this paper we present another consequence-based procedure for a non-Horn logic, this time forALCHI, but instead of resolution we incorporate some of the tableau features into our inference rule system. Specifically, we present non-deterministic rules with alternative conclusions to deal with non-determinism (instead of deriving disjunctions) and prove soundness, completeness, and termination of the corresponding saturation procedure. We show that our procedure is another step towards combining the best from the consequence-based and tableau-based worlds: It retains the optimal worst-case complexity, goal directedness, and the “granularity” property of consequence-based reasoning (i.e. computation of subsumers for different concepts can be done with lim- ited interaction which enables concurrent [9] and incremental reasoning [7]). At the same time, the non-deterministic rules enable tableau-style backtracking to be used during consequence-based reasoning which has advantages over the resolution-style procedure, for example, can re-use memory instead of storing all derived disjunctions.

Finally, we present results of a preliminary evaluation using Horn ontologies and the same disjunctive extension of the anatomical part of SNOMED CT which was used to evaluate ConDOR (SCT-SEP) [13]. The results demonstrate that the procedure im- proves over ConDORs on SCT-SEP, is comparable with ELK [10] on largeEL⁺ontologies, and several times faster than the fastest available tableau reasoner Konclude [5].

2 Preliminaries

The syntax ofALCHI is defined using a vocabulary consisting of countably infinite sets ofatomic rolesandatomic concepts. We use the lettersR, S for roles,C, Dfor concepts, andA, Bfor atomic concepts. Complexconceptsare defined by the grammar

C(i)::=> | ⊥ |A|C1uC2|C1tC2| ¬C| ∃R.C| ∀R.C,

whereRis either an atomic role or its inverse (writtenR⁻). Semantics is defined in the standard way; formal details can be found in the technical report [8].

AnALCHIontologyis a finite set ofconcept inclusionandrole inclusionaxioms of the formCvDandRvS, respectively. Theontology classification taskrequires to compute all entailed subsumptions between atomic concepts occurring inO.

AnALCHIontologyOisHornif for every axiomCvD, the conceptsCandD satisfy, respectively, the following grammar definitions:

C_(i)::=> | ⊥ |A|C1uC2|C1tC2| ∃R.C, D_(i)::=> | ⊥ |A|D₁uD₂| ∃R.D| ∀R.D| ¬C.

That is, negations and universal restrictions should not occur on the left-hand side of concept inclusions, and disjunctions should not occur on the right-hand side.

A consequence-based procedure for Horn-ALCHIcan be presented using the rules in Fig. 1. These rules are similar to the rules for Horn-SHIQ[6]; the main difference

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R0

HvC:H=KuC R⁺_>

Hv >:> ∈sub(O) Rv

HvC

HvD:CvD∈ O R⁻u

HvC1uC2

HvC1 HvC2

R⁻¬

HvC Hv ¬C Hv ⊥ R^←_⊥ Hv ∃R.K Kv ⊥

H v ⊥

R⁺u

HvC1 H vC2

H vC1uC2

:C1uC2∈sub(O)

R⁺t

HvC1|HvC2

HvC1tC2

:C1tC2∈sub(O)

R^←_∀ Hv ∃R.K Kv ∀S.C

HvC :Rv^∗OS⁻ R^←_∃ Hv ∃R.K KvC

Hv ∃S.C : ∃S.C∈sub(O) Rv^∗OS R^→_∀ Hv ∃R.K Hv ∀S.C

Hv ∃R.(KuC) :Rv^∗OS R^→_∃ Hv ∃R.K HvC

Hv ∃R.(Ku ∃S.C): ∃S.C∈sub(O) Rv^∗OS⁻ Fig. 1.The consequence-based rules for reasoning in Horn-ALCHI

is that they can operate directly on complex concepts without requiring the ontology to be normalized. Here we denote by sub(O)the set of concepts occurring in the ontology O(possibly as subconcepts). The rules derivesubsumptionsof the formH vU with

U :: =⊥ |C| ∃R.K, (1) H, K:: =d

i∈IC_i, (2)

whereC_i ∈ sub(O) (i ∈ I) andC ∈ sub(O). The premises of each rule (if any) are given above the horizontal line, the conclusion below, and theside conditionsthat determine when each rule is applicable (in terms ofO) after the colon.¹Note that the concept inclusion axioms inOare used not as premises of the rules but as side condition ofR_v. The side conditions of the rulesR^←_∀ ,R^←_∃ ,R^→_∀ , andR^→_∃ use the precomputed role hierarchyv^∗_O, which is the smallest reflexive transitive relation on roles such that R v S ∈ OimpliesR v^∗_O SandR⁻ v^∗_O S⁻. It can be shown (and follows from the more general result of this paper) that the inference system in Fig. 1 issoundand completefor deriving all entailed subsumptions with concepts occurring in the ontology, namely that for everyC∈sub(O)and everyHof the form (2), we haveO |=H vC iff eitherH v ⊥orH vCis derivable by the rules in Fig. 1 forO.

3 Non-Deterministic Consequence-Based Reasoning in ALCHI

Extending consequence-based procedures beyond Horn DLs is difficult because dealing with non-deterministic constructors, such as disjunction, requires reasoning by case. In tableau-style procedures which construct (representations of) models, dealing with non- deterministic constructors is straightforward: if a domain element is an instance ofBtC then it should be an instance ofBorC. In consequence-based procedures that derive entailed subsumptions directly, this kind of principle is not valid: ifAvBtCis entailed,

1RuleR⁺t stands for two rules with single premisesHvC1andHvC2respectively.

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then it is not true that eitherAvBorAvCis entailed. The consequence-based procedure for (non-Horn)ALCH[13] solves this problem by retaining disjunctions and recombining them using resolution-style inference rules. For example, if ontology O containsB vDandCvD, then fromAvBtCusingBvD∈ O, one can derive AvDtC, from which similarly “resolving” onCvD∈ Oone obtainsAvDtD, which is factorized toAvDby merging duplicate disjuncts.

One of the well-known disadvantages of resolution-based procedures compared to search-based (DPLL-style) procedures, is that resolution can produce many long disjunctions which can occupy a considerable amount of space and make inferences slow because merging disjunctions is not a trivial operation. In this paper we, thus, reconsider the former idea of splitting disjunctions using a non-deterministic rule:

R⁻_t H vC1tC2

H vC1 | H vC2

(3) Just like in tableau-based procedures, this rule creates two branches in which subsumptions can be independently derived. As discussed above, it is not true that all subsumptions derived in one of the branches will be entailed by the ontology. However, it can be the case for the subsumptions that are derivedon every branch, i.e., regardless of the non-deterministic choices made. For example, if we split subsumptionAvBtC onA v B andA v C, then in both branches we can deriveA v D using axioms B v D ∈ O andC v D ∈ Orespectively. This idea of considering common conclusions in all non-deterministic branches is somewhat reminiscent of the Stålmarck’s procedure for propositional logic (see, e.g., [12]). Unfortunately, adding just the rule R⁻_t to the system in Fig. 1 is not sufficient for obtaining a sound inference system:

Example 1. Consider the ontologyOcontaining the following axioms:

Av ∃R.A, AvBtC, ∃S.CvB, RvS.

Then the subsumers ofAcan computed using two branches after the splitting rule:

AvA byR0(),

Av ∃R.A byR_v(AvA):Av ∃R.A∈ O, AvBtC byR_v(AvA):AvBtC∈ O, AvB AvC byR⁻_t(AvBtC),

Av ∃S.C byR^←_∃ (Av ∃R.A,AvC):Rv^∗_O S, AvB byR_v(Av ∃S.C):∃S.CvB∈ O.

As can be seen,AvBcan be derived in both of these branches, butO 6|=AvB.

To understand the problem in Example 1, we first need to understand how to inter- pret rules like (3). Given an interpretationI = (∆Î,·Î), we say that an elementa∈∆Î satisfiesa subsumptionCvDinI(orCvDis satisfiedinIbya) ifa∈CÎimplies a∈DÎ. So, ruleR⁻_t can be understood as follows: for eacha∈∆Î ifasatisfies the

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premiseH vC₁tC₂ then it satisfies one of the conclusionsH vC₁or H v C₂. Clearly, this is true for this rule (and everyI), however, this property fails for rulesR^←_⊥, R^←_∀ , andR^←_∃ in Fig. 1, even ifI |=O. For a counter-example, takeI = (∆^I,·^I)with

∆Î=AÎ={a, b},BÎ ={b},CÎ={a}, andSÎ =RÎ={ha, bi,hb, ai}. Clearly,I is a model ofOin Example 1, andasatisfies bothAv ∃R.AandAvC, but,adoes not satisfy the conclusionAv ∃S.Cobtained from these premises byR^←_∃ . Hence, the inference byR^←_∃ in Example 1 is unsound.

To “fix” ruleR^←_∃ , we use a similar observation as in ConDOR [13]. Specifically, if asatisfiesH v ∃R.K inIbut does not satisfyH v ∃S.C, andI |=R vS, thena should satisfyH v ∃R.(Ku ¬C). Thus, we can generalize ruleR^←_∃ as follows:

R^←_∃ H v ∃R.K KvC

H v ∃S.C | H v ∃R.(Ku ¬C): ∃S.C∈sub(O)

Rv^∗_O S (4) Example 2. Consider the ontologyOfrom Example 1. Using the updated rule (4), we can now obtain the following inferences forA:

AvA byR0(),

Av ∃R.A byR_v(AvA):Av ∃R.A∈ O, AvBtC byR_v(AvA):AvBtC∈ O, AvB AvC byR⁻_t(AvBtC,Av ¬B),

Av ∃S.C AvB

Av ∃R.(Au ¬C) byR^←_∃ (Av ∃R.A,AvC).

Note that in the rightmost branch, we derive the subsumptionA v ∃R.(Au ¬C)to which ruleR^←_∃ can potentially be applied. In order to check if this rule is applicable, we need to derive subsumptions forAu ¬C:

Au ¬CvA byR0(),

Au ¬Cv ∃R.A byR_v(Au ¬CvA):Av ∃R.A∈ O, Au ¬CvBtC byR_v(Au ¬CvA):AvBtC∈ O, Au ¬CvB Au ¬CvC byR⁻_t(Au ¬CvBtC),

Au ¬Cv ⊥ byR_⊥(Au ¬CvC).

If we now consider the rightmost branch in the derivation forAand the leftmost branch forAu ¬Cwe obtain a set of subsumptions for which all inferences are applied but neitherAvBnorAv ⊥is derived. Hence, we can conclude thatO 6|=AvB.

Figure 2 presents a generalization and extension of the rules in Fig. 1 to (full) ALCHI using the idea similar to rule (4). The new rules now derive subsumptions

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R0

HvC:H =KuC R⊥ HvC

Hv ⊥:H =Ku ¬C R⁺_>

Hv >:> ∈sub(O) Rv

H vC

HvD:CvD∈ O R⁻u

H vC1uC2

HvC1 H vC2

R⁻t

H vC1tC2

HvC1|H vC2

R⁻¬

HvC Hv ¬C Hv ⊥ R^←_⊥ Hv ∃R.K Kv ⊥

Hv ⊥

R⁺u

H vC1 HvC2

HvC1uC2

:C1uC2∈sub(O)

R⁺t

H vC1|HvC2

HvC1tC2

:C1tC2∈sub(O) R⁺¬

H v ¬C|HvC:¬C∈sub(O) R⁺_∀

H v ∀R.C|Hv ∃R.¬C:∀R.C∈sub(O) R^←_∀ H v ∃R.K Kv ∀S.C

H vC|H v ∃R.(Ku ¬∀S.C): Rv^∗_OS⁻ R^←_∃ H v ∃R.K KvC

H v ∃S.C|H v ∃R.(Ku ¬C): ∃S.C∈sub(O) Rv^∗OS R^→_∀ H v ∃R.K H v ∀S.C

H v ∃R.(KuC) : Rv^∗_OS R^→_∃ Hv ∃R.K H vC

H v ∃R.(Ku ∃S.C): ∃S.C∈sub(O) Rv^∗OS⁻

Fig. 2.The non-deterministic consequence-based rules for reasoning inALCHI

of the formH vU, whereU is of the form (1), and H, K:=l

i∈I

C_iu l

j∈J

¬D_j, (5)

whereCi ∈ sub(O)(i ∈ I), andDj ∈ sub(O)(j ∈ J). Note that the ruleR^←_⊥ has remained the same, and is still formally unsound under our new interpretation. This rule will be dealt with by our procedure in a special way, which does not affect soundness.

3.1 The Rule Application Strategy

Just like for other consequence-based procedures [1, 6, 13], every set closed under the rules in Fig. 2 contains all “relevant” subsumptions entailed by the ontology, that is, our rules arecomplete. Specifically, we say that a setM of subsumptions isclosedunder a ruleRif, whenever all premises ofRbelong toM, all its conclusions (respectively one of the conclusions ifRis non-deterministic) belong toM.

Theorem 1 (Completeness).LetObe anALCHIontology,Ha conjunction of form (5),Da concept occurring inOsuch thatO |=H vD, andM a set of subsumptions closed under all rules in Fig. 2. Then eitherHvD∈M orH v ⊥ ∈M.

The proof of Theorem 1 (cf. [8]) works along the same lines as forEL[1] and Horn-SHIQ[6], i.e., by constructing a canonical model ofOfromM.

Remark 1. Just like forEL, if the set ofgoalsubsumptionsH vDfor which entailment should be checked is known in advance (e.g., for classification), it is possible to

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sharpen Theorem 1 by relaxing the requirement on the closure ofM [10]. Specifically, it is necessary to produceH vU by rules only if there is a goal subsumptionH vD for someD, or ifK v ∃R.H has been derived for someKearlier. Furthermore, the rulesR⁺_>,R⁺_u,R⁺_u,R⁺_¬,R⁺_∀, andR^←_∃ introducing new concepts in sub(O)(see the side conditions), need to be applied only if the new concept is a subconcept of the left-hand side of some concept inclusion inOor of someDin the right-hand side of goal subsumptions. Both optimizations can be particularly useful to mitigate “blind guessing”

in rulesR⁺_¬ andR⁺_∀, which do not have any premises, and thus apply to everyH.

It is possible to prove the converse of Theorem 1, namely that ifO 6|= H v D, then there exists a set of subsumptionsM closed under the rules in Fig. 2 that contains neitherH vDnorHv ⊥. This result, however, is not very useful in practice because in order to check whetherO |=HvDusing this result, one would need to enumerate allM closed under the rules in Fig. 2. Just like tableau-based procedures do not need to enumerate all completion graphs, we do not need to enumerate all such closed setsM. In particular, subsumptions derived for different conjunctionsHof the form (5) can be in most cases considered independently.

The followingconcept satisfiability procedure(short CSAT) checks satisfiability of conjunctionsH of the form (5) using the rules in Fig. 2:

1. We apply the inference rules in Fig. 2 (introducing conjunctionsHas necessary according to Remark 1) and record every derived subsumptionsH vUin a separate local branchfor eachH, just like in Example 2.

2. If a subsumptionH vU was derived using a non-deterministic rule, we also re- member the alternative conclusion of this rule in abranching point.

3. If we derive alocal clashH v ⊥, we remove all conclusions on the branch for Hstarting from the last non-deterministically derived conclusion, and produce the alternative conclusion, which we now consider asdeterministically derived.

4. If local clash H v ⊥is derived and there are no non-deterministically derived conclusions on the branch forHanymore, we markHasinconsistent.

5. RuleR^←_⊥ is applied only if for the right premise of the ruleKv ⊥(local clash in the branch forK), the conjunctionKis marked as inconsistent.

6. The procedure terminates when no more rules can be applied and for each derived local clashH v ⊥(on the branch forH),His marked as inconsistent.

Note that even though we apply ruleR^←_⊥ in a restricted way, because of the last condition, the resulting setM of subsumptions will be closed under all rules in Fig. 2.

Also note that if some conclusionH v U was derived using a premiseKvV with K6=H(using rulesR^←_⊥,R^←_∀ orR^←_∃ ), we do not removeKvV when we backtrack H v U as we remove only conclusion on the local branch for H. Likewise, if we remove the premiseKvV, we do not need to remove the conclusionH vU derived from this premise. In other words, premises from other local branches can never result in inconsistency ofH, and so, deriving and backtracking of conclusions can be performed for each local branch independently.

By Theorem 1, eachH that is not marked as inconsistent by our procedure must be satisfiable sinceH v ⊥∈/ M for the resulting closureM. We will also prove the converse, namely that only unsatisfiableHcan be marked as inconsistent by CSAT.

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The rules in Fig. 2 can be used not only for checking satisfiability ofH, but also for computing its entailed subsumers. The followingbranch exploration procedure(short BEXP) continues after CSAT, and computes the set S(H)of concepts D such that H vDappear on every branch ofH that does not contain a local clash:

1. For eachH not marked as inconsistent, create a setS(H)ofcandidate subsumers consisting of thoseD∈sub(O)such thatH vDis on the branch forH.

2. Remove all conclusions on the branch forHstarting from the last non-deterministic conclusion, and produce the alternative conclusion.

3. Apply the rules like in CSAT (this may extend other branches), backtracking if necessary, but do not markHas inconsistent when no backtracking inHis possible.

4. After all rules are reapplied and there is no local clash inH, remove fromS(H)all Dfor whichH vDno longer occurs on the branch.

5. Repeat from Step 2 until no non-deterministic conclusions are left on the branch.

It follows from Theorem 1, thatS(H)contains all conceptsD ∈sub(O)such that O |= H v D sinceD is removed fromS(H)only when we findM closed under the rules in Fig. 2 such thatH vD /∈M. In the next section we prove the converse, namely that ifO 6|=H vDthen there is a branch ofHthat does not containH vD.

3.2 Soundness, Termination, and Complexity

To prove soundness of the procedures CSAT and BEXP described in the previous section, we need to formally define our non-deterministic derivation strategy.

Definition 1 (Tableau, Local Tableau).Atableau(for O) is a tripleT = (H,≺, t), whereHis a set of conjunctions of the form(5),≺is a partial order onH, andtis a function that assigns to everyH ∈Ha treet(H)called thelocal tableau forH whose nodes are labeled by subsumptionsHvU (with thisH),Uof the form(1), such that:

1. For each nodenoft(H), there is a rule in Fig. 2 forOsuch the label ofnis a conclusion of this rule and all premises of the formH vU(with thisH) are labels of some ancestors ofn(the nodes on the path from the root oft(H)tonwithoutn);

if the inference is byR^←_⊥, the second premiseKv ⊥must be such thatK≺H. 2. If the label is obtained by a non-deterministic rule (R⁻_t,R⁺_¬,R⁺_∀,R^←_∀ ,R^←_∃ ), the

node should have a sibling node labeled by the alternative conclusion of the rule.

3. All labels of the nodes on the same branch int(H)must be different.

4. IfK≺Hthen all branches oft(K)must contain the local clashKv ⊥.

Intuitively, each local tableaut(H) for H describes the set of branches that are encountered forH by procedure BEXP described in Sect. 3.1. For example, the two trees in Example 2 are local tableaux respectively forH =AandH =Au ¬C. The partial order≺is the order in which the conjunctionsH are marked as inconsistent.

According to the last condition, we setK≺HifKis marked as inconsistent andHis not. Thus, ruleR^←_⊥ derivesH v ⊥fromKv ⊥by our procedure only ifK≺H. Theorem 2 (Soundness).LetT = (H,≺, t)be a tableau forO. Then for eachH ∈H and each conceptD (not necessarily occurring inO) such thatO 6|= H vD, there exists a branch int(H)that contains neitherH v ⊥norH vD.

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Proof. The proof is by induction on≺. Assume that the claim holds for eachK≺H, and prove it forH. Since O 6|= H v D, there is a modelI = (∆Î,·Î)of Oand a ∈∆Îsuch thatadoes not satisfyH vD inI, i.e.,a∈ HÎ\DÎ. We prove that there is a branch int(H)such that every labelHvUon this branch is satisfied inIby a, that isa∈UÎ. This implies that neitherH v ⊥norH vDoccurs on this branch.

It is sufficient to show that for every noden int(H), if a satisfies the labels of all ancestors ofnthenasatisfies the label ofnor of its sibling node (if there is one).

According to Definition 1, these labels are obtained by some rule in Fig. 2 in which all premises of the formH v U must be the labels of some ancestors ofn. That is, all premises of the rule with the left-hand sideHare satisfied inIbya. We will show that in this case, the conclusion of the rule (respectively, one of the non-deterministic conclusions if the rule is non-deterministic) is also satisfied inIbya:

– RulesR0,R_⊥,R⁺_>,R_v,R⁻_u,R⁻_u,R⁻_¬,R⁺_u,R⁺_t,R⁺_¬,R^→_∀ ,R^→_∃ : These are deterministic rules that contain only premises with the left-hand sideH. It is easy to see that for all these rules the conclusion is a logical consequence of the premises and O. SinceI |=O,asatisfies the conclusion because it satisfies all premises.

– RulesR⁻_t,R⁺_¬,R⁺_∀,R^←_∀ ,R^←_∃ : These are non-deterministic rules. For each rule we show how to choose one of the conclusions that is satisfied ina.

ForR⁻_t, sincea ∈ (C1tC2)^I, we have eithera ∈ C1I or a ∈ C2I. Hencea satisfies eitherH vC₁orH vC₂. The proofs forR⁺_¬ andR⁺_∀ are similar.

ForR^←_∀ , we havea ∈ (∃R.K)Î. That is, there existsb ∈ KÎsuch that ha, bi ∈ RÎ. SinceR v^∗_O S⁻, we havehb, ai ∈ SÎ. Ifa ∈ CÎ, thenasatisfies the left conclusionH vC. Otherwise,a /∈ CÎ and sob /∈ (∀S.C)Î sincehb, ai ∈ SÎ. Henceb ∈ (Ku ¬∀S.C)Î anda ∈ ∃R.(Ku ¬∀S.C). So, asatisfies the right conclusionH v ∃R.(Ku ¬∀S.C). The proof forR^←_∃ is similar.

– RuleR^←_⊥: This is a deterministic rule, but for its right premiseKv ⊥we may have K6=H. By Definition 1, however, we must haveK≺H, and so, every branch of t(K)containsK v ⊥. SinceK ≺H, by induction hypothesis,O |=K v ⊥(if O 6|=Kv ⊥there would be a branch int(K)withoutKv ⊥). So, this case is not possible sinceamust satisfy the premiseH v ∃R.K, buta /∈(∃R.K)^I=∅.

Remark 2. Note that the second premises of rulesR^←_∀ andR^←_∃ were not used in the proof of Theorem 2. That is, these premises do not have any impact on soundness.

The purpose of these premises, is to restrict the non-deterministic “guessing” made by these rules to the cases that are sufficient for completeness. Note also that if the second premises of these rules are entailed by the ontology, e.g., they are derived deterministically, then the right conclusion is equivalent toH v ⊥, thus it cannot be satisfied bya that does not satisfyH vD. Hence the right conclusion is not necessary in this case, and so, these rules could be applied deterministically as in Fig. 1. This strategy makes sure that our procedure is deterministic when applied to Horn-ALCHIontologies.

To combine Theorems 1 and 2 and obtain soundness and completeness for our procedure, we need to define when the tableau is fully expanded under inferences.

Definition 2 (Fully Expanded Tableau).LetT = (H,≺, t)be a tableau. A branch of t(H)isclosedifH v ⊥occurs on this branch. Otherwise, the branch is calledopen. A

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branch isfully expandedif there exists a setM of subsumptions closed under the rules in Fig. 2 such that all subsumptions of the formH vU inM (with thisH) occur on this branch.Tis fully expanded if every open branch of everyt(H)is fully expanded.

Note that each time all rules are reapplied without producing a local clash at Step 4 of procedure BEXP in Sect. 3.1, we obtain a setM closed under the rules (possibly different from the previous one). So, we “fully expand” every branch in our procedure.

However, BEXP is far from enumerating all possible setsMclosed under the inference rules. E.g., we do not explore all possible combinations of branches in local tableaux.

Corollary 1 (Correctness).LetT = (H,≺, t)be a fully expanded tableau forO. Then for everyH∈Hand everyDoccurring inO, we haveO |=H vDiff the subsumption H vDis derived in every open branch oft(H).

It is possible to prove that BEXP runs in exponential time in the size ofO. Infor- mally, the running time is bounded by the sum of sizes for each local tableau. There are at most exponentially many local tableaux (one per each conjunctH), and it can further be shown that the size of each local tableau is also exponential. See the technical report [8] for more formal and detailed arguments.

4 Discussion and Preliminary Evaluation

The branch exploration procedure BEXP described in Sect. 3.1 can be readily used for classification, i.e., computing the entailed subsumption between atomic concepts inO.

We need to iterate over all singleton conjunctsH of atomic concepts, and for each one compute the setS(H)of atomic concepts, subsumptions with which occur on every branch. Another way to compute atomic subsumers forH, is to reduce subsumption entailment to concept satisfiability testing, as done in tableau-based procedures:O |= H vDiffO |=Hu ¬D v ⊥.We can use our first procedure CSAT for this purpose.

The next example demonstrates the key differences between these two approaches.

Example 3 (Long Fork).Consider the ontologyOcontaining the following axioms:

AvBtC, BvD, CvD, DvD1u · · · uDn, (n >0).

The two branches in the tree forH =Acontain the following sets of atomic subsumers forA:S₁(A) ={A, B, D, D1, . . . , D_n}, andS₂(A) ={A, C, D, D1, . . . , D_n}. Hence the procedure BEXP can find the set of common subsumersS(A) =S₁(A)∩S₂(A) = {A, D, D₁, . . . , D_n}inO(n)steps. If we use CSAT instead and test satisfiability for each candidateA vD_i,(1≤i≤n), then we potentially needO(n²)steps since for eachH_i=Au ¬D_iwe can produceO(n)conclusionsAu ¬D_ivD_j,1≤j≤n.

This difference between the two strategies can be observed not only on toy examples but also on real ontologies (cf. [8] for evaluation results).

Tableau reasoners usually implement further optimizations exploiting a partially constructed taxonomy that can reduce the number of subsumption tests [2, 4]. If in Example 3 the reasoner computes the subsumers forDfirst (usingO(n)steps because they are derived deterministically), then after the first (positive) subsumption test for AvD, all subsumers forAare completely determined. This optimization can be also applied for BEXP to reduce the set of candidate subsumersS(H)to be verified.

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Table 1.Comparison of the performance (in seconds) for classification of ontologies between the new implementation and other reasoners. ELKNstands for ELK with N parallel worker threads.

Total time includes CSAT, BEXP, and taxonomy construction (transitive reduction).

Ontology CSAT BEXP Total ConDOR Konclude ELK1 ELK4

SNOMED CT 22.7 – 29.8 38.2 45.2 25.7 13.1

GALEN-EL 2.3 – 2.6 3.6 2.7 2.1 1.0

SCT-SEP 16.9 15.9 37.5 59.4 246.6 N/A N/A

4.1 Preliminary Experimental Evaluation

We have implemented our non-deterministic procedure, currently for a subset ofALCH without universals and negations (but with disjunctions). The implementation supports transitive roles via the well-known encoding [6]. The main goal is to make a preliminary comparison of the new non-deterministic procedure with the deterministic resolution- style procedure of ConDOR [13], which supports this fragment. We have also made a comparison with Konclude (v.0.5.0), which was the fastest tableau-based reasoner for our test ontologies, and with ELK (v.0.4.1) onEL⁺ ontologies. The latter is done to evaluate the potential overheads (our procedure works deterministically on Horn ontologies, so, in theory, it should behave like theEL⁺procedure onEL⁺ontologies).

We used the July 2013 release of SNOMED CT,² an EL⁺-restricted version of GALEN, and the version of SNOMED CT with disjunctions mentioned in the introduction (SCT-SEP). Performance is tested on classification and the results are averaged over 5 runs (excluding 2 warm-ups). We used a PC with Intel Core i5-2520M 2.50GHz CPU, running Java 1.6 with 4GB of RAM available to JVM.

Performance results of classification are presented in Table 1. The procedure works in two steps as described in Sect. 3.1: satisfiability testing for all atomic concepts using CSAT, then exploring branches to compute subsumers using BEXP. First, the results show that our implementation is slightly faster than ConDOR and considerably faster than Konclude on SCT-SEP. Second, it is comparable to ELK on deterministicEL⁺ ontologies. The numbers for ELK with 1 and 4 worker threads demonstrate that the difference is mostly due to concurrency: our implementation does not perform compu- tations in parallel yet. BEXP is not necessary for this case as everything is deterministic.

4.2 Summary and Future Research Directions

In this paper, we presented a non-deterministic consequence-based reasoning procedure forALCHI, proved its correctness, and presented some promising preliminary evaluation. The procedure retains many nice properties of previously known consequence- based methods, such as optimal worst-case complexity, but also employs some tableau- like features, such as backtracking, without need for blocking to achieve termination.

Our future plan is to extend the implementation to fully supportALCHI, integrate some optimizations used in ELK (including concurrency), and investigate extensions of the procedure with other constructors, such as number restrictions. We conjecture that it is easier to support new features using non-deterministic rules rather than resolution.

2http://www.ihtsdo.org/snomed-ct/

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