A Tableaux-based calculus for Abduction in Expressive Description Logics: Preliminary Results
Tommaso Di Noia (1), Eugenio Di Sciascio (1), Francesco M. Donini (2) (1) SisInfLab, Politecnico di Bari, Bari, Italy (2) Universit`a della Tuscia , Viterbo, Italy
{t.dinoia,disciascio}@poliba.it,[email protected]
1 Introduction
Abduction [12] is a well-known form of common-sense reasoning that has been widely exploited in Artificial Intelligence applications, including formalization of Diagnostic Reasoning [13] and Scene Interpretation [14]. Abduction has been also proposed as a logical tool for formalizing hypothetical reasoning in E-Commerce [7], Negotiation [15] and Web-service Discovery [10], among others. The computation of Propositional Abduction (PA) has been extensively studied, for nearly all criteria that can be adopted for choosing the preferred hypotheses [6]. Also Abductive Logic Programming is well established [9], and its computation resorts on a modification of SLD-resolution. Yet, when the representation language chosen for the application domain is a Description Logic (DL), the picture is far from being complete. Since concepts in a DL are in fact monadic predicates, the definition of Abduction should be extended from a proposi- tional to a first-order setting. A possible definition has been proposed by Di Noia et al.
[7]: given conceptsC(prior facts),D(target), and a TBoxT (background knowledge), a Concept Abduction Problem is finding another conceptH such that both(i)the con- junction ofC andH—denoted in DL byC uH—is satisfiable inT, and(ii)in all models ofT,CuHis subsumed byD(denoted byT |=CuH vD). A tableaux- based calculus for computing Concept Abduction in the DLALN was devised in [5].
However, althoughALNis computationally simple, it is a rather inexpressive DL. We propose here a form of Concept Abduction which is computed extending the frame- work of Concept Matching [2]. In a nutshell, we identify places in the description of C where a hypothesis can be added, and name such places with (all distinct) concept variablesH0, H1, H2, . . .We denoteChthe resulting concept. Then a solution to such an Abduction Problem is a substitutionσof concept variables with concepts, such that T |=σ(Ch)vD. Note that in Concept Matching the concept with variables isD, and a solution is a substitutionσsuch thatT |=Cvσ(D). The remaining of this paper is structured as follows. In the next section, we lay out some definitions and other prelim- inary notions. Then we propose a calculus based on tableaux for finding substitutions, and in Section 4 we propose some strategies for finding “good” substitutions. Finally, we compare our proposal with existing literature.
2 Structural Abduction in SH
Here, we admit only unfoldable TBoxes and we denote byv∗the transitive closure of role inclusion over the RBoxR. For the sake of conciseness we assume the RBox as part
of the TBox:R ⊆ T. We assume that concepts are always in Negation Normal Form (NNF), where negation in front of a concept is always ”pushed” inside connectives and quantifiers, recursively, till negation is in front of concept names only. In the following, we augment concept expressions by means ofconcept variables. We callconcept term [3] every concept expression formed by using the syntax above, plus the ability of using a concept variableh ∈ {h0, h1, h2, . . .}, in place of a concept name. To extend the semantics ofSHto concept terms, we letσbe an assignmentσ: h7→C ∈ SHthat interprets concept variables as concepts inSH, and we define the semantics of has (σ(h))I. We recall previous definition of Concept Abduction inALN[7].
Definition 1 (Concept Abduction).LetC,D, be two concepts inALN, andT be a set of axioms inALN, where bothCandDare satisfiable inT. AConcept Abduction Problem(CAP), denoted ashC, D,T,LiC, is finding a conceptH ∈ ALN such that T 6|=CuH ≡ ⊥, andT |=CuHvD.
While adequate for Description Logics asALN, Def. 1 shows its limits when dealing with languages where qualified existential quantification is allowed. This is the case of ALE and, more generally, ofALCandSH. We explain the need to generalize Def. 1 with the aid of a simple example.
Example 1. Let W1, W2 be two Web services,W1 having two payment methods: a Credit-card one and a non-https-based one (W1=∃howPay.CCu∃howPay.¬https), andW2for which it is only known that it has some payment method (∃howPay.>).
Suppose a user has to choose which service best satisfies her demandD=∃howPay.(CCu https). Only for sake of simplicity, let the TBox be empty. Obviously, bothW16vD andW2 6v D—i.e., neither service completely fulfills D—so the problem arises as to whether there is some logic-based method to compare them. Some researchers [7, 10] propose to automate the choice by solving two (kind of) abduction problems, and compare the results. Namely, they propose to compare some hypotheses H1 andH2 that, when added toW1 andW2 respectively, would make each one of them be sub- sumed by D. Then, an offer with a more generic H should be preferred over one with a more specificH. Following this criterion, an offer W for which H = >is the ”best” choice, since in this case alreadyW vD. Following Def. 1, we would have to find two concepts H1, H2 such that WiuHi v D (i = 1,2) and in this case it can be verified that both problems admit one subsumption-maximal solution, namely H1=H2=∃howPay.(CCuhttps) =D. Observe thatH1,2is trivially hypothesiz- ing the whole target conclusion, disregarding the fact that the conjunct∃howPay.CC inW1already “covers” part ofD. That is, Def. 1 cannot be used to prefer an offer that already partly covers the demand (W1) over another (W2) that does not1. 4 The explanation computed in Ex.1 according to Def. 1 involves the whole existential restriction even though only a part of it would be necessary—in this casehttps. This is because Concept Abduction (and all definitions of PA) only adds hypotheses as out- ermost conjunctions. Instead, solutions should take into account the structure of a for- mula; this amounts to hypothesize pieces of a formula to be added inside thequantifiers ofCand not just added as outermost conjunctions.
1We remark that also the approach in [4] would compute(CC∧https)(withthe modality associated withhowPay) for both abduction problems. Hence, also their proposal cannot be used to highlight themissinginformation inside quantifications.
Definition 2 (Abducible Concept – Hypotheses List). Let C and D be two SH- concepts in NNF, and letT be a set of axioms inSH. We defineabducible concept Ch .
=h0uRew(C), where the rewritingRew(C)defined recursively as Rew(A) =A; Rew(¬A) =¬A; Rew(C1uC2) =Rew(C1)uRew(C2)
Rew(∃R.C) =∃R.(hnewuRew(C)); Rew(∀R.C) = ∀R.(hnewuRew(C))where byhnewwe mean a concept variable not yet appearing in the rewriting2. We assume that concept variables are numbered progressively, and callhypotheses listofCh the listH=hh0, h1, h2, . . .i.
Observe that sinceCis in NNF, a case forRew(¬C), withCa generic concept, is not present in the definition ofRew(). Given an abducible conceptCh, its corresponding hypotheses listH=hh0, . . . , h`iand a listC =hC0, . . . , C`iofALEHR+ concepts, we denote withσ[H/C]the substitution{h0 7→C0, . . . , h`7→C`}and withσi[H/C], withi= 0, . . . , `, the substitution of the single variablehi.
Definition 3 (Structural Abduction).LetC, D∈ SH, be two concepts in NNF, andT be a set of axioms inSH, where bothCandDare satisfiable inT andT 6|=CuDv ⊥.
LetH = hh0, . . . , h`i be the hypotheses list of the abducible conceptCh andA˜ = hA0, . . . ,A`i(for Assumptions) be a list ofSHconcept sets. AStructural Abduction Problem(SAP) forSH, denoted ashC, D,T,Ai˜ S, is finding a list of conceptsH = hH0, . . . , H`iinALEHR+such that
Hi ∈ Ai for everyi= 0, . . . , ` (1)
T 6|=σ[H/H](Ch)v ⊥ (2)
T |=σ[H/H](Ch)vD (3)
We call a SAPGeneralwhenAi=SH, for everyi= 0, . . . , `.
Following [5] we usePS as a symbol for a SAP, and we denote withSOLSAP(PS) the set of all solutions to a SAPPS. Observe that we impose the hypotheses inHto be in the DLALEHR+, in which disjunction is not allowed, and negation is allowed only in front of concept names. This restriction is analogous to the one in PA—where the set of abduced hypotheses is always intended as a conjunction—and Abductive Logic Programming—where again the abduced atoms are (implicitly) taken conjunctively. As noted by Marquis [11] for First-order Abduction, if disjunction and full negation are allowed in solutions to an Abduction problem, the most general hypothesis is always
¬CtD, which in fact solves nothing. Hereafter, we always refer toGeneral SAPs.
Example 2. ConsiderW1andDas in Ex.1. According to Def. 2 and Def. 3 we have W1h=h0u ∃howPay.(CCuh1)u ∃howPay.(¬httpsuh2)
H=hh0, h1, h2i H=h>,https,>i
σ[H/H](Ch) => u ∃howPay.(CCuhttps)u ∃howPay.(¬httpsu >) Observe that the solution previously computed in Ex.1 is still a solution of the SAP, namely, it is the solutionH0=h∃howPay.CCuhttps,>,>i. 4
2A precise procedure to obtain this result would need a global counter, visible by all recursive calls. We skip this technical detail to simplify presentation.
Solutions can be compared according to two preference criteria, namely, component- wise subsumption, and subsumption in the abducible concept after substitution.
Definition 4 (Preference and Maximality).LetPSbe a SAP andH0=hC00, . . . , C`0i andH00=hC000, . . . , C`00ibe two solutions inSOLSAP(PS). We say that:
—H0isstructurally-preferred overH00, denoted byH0 vSH00, if fori= 0, . . . , `we haveT |=Ci00vCi0;
—H0issubsumption-preferred overH00, denoted asH0vH00, ifT |=σ[H/H00](Ch)v σ[H/H0](Ch);
A solution isStructural-maximalif no solution is structurally preferred to it, andSubsumption- maximalif no solution is subsumption-preferred to it.
However, it turns out that subsumption-preference includes structural preference, hence we can safely forget the latter.
Proposition 1. Given a SAPPS = hC, D,T,Ai˜ S and two solutionsH0 and H00 in SOLSAP(PS), ifH0 vSH00thenH0vH00.
Proof.Recall that (Def. 3) bothCandDare in NNF. For this form, substitutions are al- ways monotonic overv, that is, ifC1vC2then for everyi= 0, . . . , `,σi[hi/C1](Ch)v σi[hi/C2](Ch). This is because concept variables appear only in conjunctions, and positively—i.e., inside no negation. Iterating over the list of concept variables, the claim
follows. 2
Note that the requirement thatCis in NNF before rewriting inChis fundamental for the maximality criteria. ConsiderC=¬∃R.A; without NNF,Chwould beh0u ¬∃R.(Au h1), but due to the fact that h1 is added as a conjunction inside an odd number of negations, by substitutingh1with anything different from>we are in fact makingCh more generic, not more specific. So, in the criteria for subsumption-maximal solutions, we should now distinguish betweenh’s occurring inside an even number of negations andh’s occurring inside an odd number of negations (for which the criteria should be reverted: a more generic substitution yields a more specificCh). We preferred to use NNF, so that each concept variablehoccurs inside0negations inCh, and use uniform maximality criteria. Recalling Example2, we observe that none of the previous criteria allows us to preferHtoH0. This is becauseHandH0 are incomparable under struc- tural preference, whileσ[H/H](Ch) ≡ σ[H/H0](Ch). Hence a different preference criterion is needed.
Definition 5. Given an abducible concept Ch, we introduce the following preorder among variables inH:(i)h0 < hi withi 6= 0; (ii)hi < hp if hp is within a quan- tification inside the one ofhi; (iii)hi< hpifhpis within an existential quantification
∃R.(hpu· · ·),hiis within a universal quantification∀R.(hiu· · ·), and both quantifica- tions appear in the same quantification (i.e., they are siblings in the syntactic tree). Now letH0 =hC00, . . . , C`0iandH00 =hC000, . . . , C`00ibe two solutions inSOLSAP(PS).
We say thatH0≤ H00ifCi0vCi00for somei, and for allpsuch thathi< hp,Cp0 =Cp00. Example 3. Turning back to Example 2, we remember thatH={>,https,>}and H0 ={∃howPay.CCuhttps,>,>}. By Definition 5 we seeH ≤ H0. Indeed, we havehttpsv >(C1vC10) fori= 1and>=>(C2=C20) fori >1. 4
3 Calculus
We present a calculus and algorithms to compute a solution to a SAP as defined in Section 2 for the expressive DLSH. We assume the reader be familiar with tableau cal- culus [16]. Following [5], we will build a prefixed tableau using two labeling functions (to represent true/false prefixed tableaux)T() andF() instead of a single labelling one L().T() andF() map an individualnto a set of conceptsT(n) orF(n) or relatento another individualmvia a roleR. More formally, given two individualsnandmin a tableauτ, the formal semantics ofT() andF() is as follows. We say that an interpreta- tion(∆I,·I)satisfies two tableau labelsT(n)andF(n)if:
— for every conceptC∈T(n)and every conceptD∈F(n),nI∈CIandnI 6∈DI;
— for every roleR ∈T(n, m)and for every roleQ ∈F(n, m),(nI, mI)∈RIand (nI, mI)6∈QI;
— an interpretation satisfies a branchB ofτ if it satisfiesT(n),F(n),T(n, m)and F(n, m)for every individualn, and for every pair of individualsn, minB.
Given two individual namesnandm,mis called anR-successor ofnif, for someR0 withR0 v∗ R,mis a successor ofnand eitherR0 ∈ T(n, m)or ¬R0 ∈ F(n, m).
Ancestors are definerd as usual. If nis an ancestor ofm, we saymis blocked by n if either T(m) ⊆ T(n)or F(m) ⊆ F(n). In order to compute a solution to a SAP, we will build a prefixed tableauτ using the rules in Fig. 1. Since we use two labelling functions, for each classical tableau rule we have we both aT) version and its dualF) version. The only optimization technique we include here is lazy unfolding [1, 8]. We assume concepts are always simplified in NNF and we useCto represent the NNF of a conceptC. Given aSHconcept in NNF, whenever we have a roleQ∈F(n, m), it is of the form¬R. HenceQ∈F(n, m)means, in fact,(nI, mI)∈RItoo. Unfolding rules for TBox axioms are presented in Fig. 2. We already discussed the need of havingCin NNF. For what concernsD, it is just a matter of not doubling the tableaux rules (ifD were not in NNF, we would need, e.g., a rule foruand another rule for¬(...u....)).
In the following we use the definition of homogeneous and heterogeneous clash as pro- posed in [5]. We recall here the definition of homogeneous and heterogeneous clash as proposed in [5].
Definition 6 (Clash).A branchBcontains ahomogeneous clashif it contains one of the following:
1. either⊥ ∈ T(n) or > ∈ F(n), for some individual n; 2. eitherA,¬A ∈ T(n) (T-homogeneous) orA,¬A∈F(n)(F-homogeneous) for some individualnand some concept nameA;Bcontains aheterogeneous clashif it contains one of the following:
1.T(n)∩F(n)contains eitherAor¬Afor some individualnand some concept name A;
We say a branchBiscompleteiff for each individual namenoccurring inAno new rule application is possible both toT(n)andF(n). A complete branch isopenif it contains no clash, otherwise it isclosed. A complete tableau is open if it contains at least one open branch, otherwise it is closed. Soundness and completeness of the calculus follow from the version without prefixes [8].
u- rules :
T) ifCuD∈T(n), then add bothCandDtoT(n).
F) ifCtD∈F(n), then add bothCandDtoF(n).
t- rules :
T) ifCtD∈T(n), then add eitherCorDtoT(n).
F) ifCuD∈F(n), then add eitherCorDtoF(n).
∃- rules :
T) if∃R.C ∈ T(n),nis not blocked, andnhas noR-successormwith eitherC ∈ T(m)or
¬C∈F(m), then pick up a new individualm, addRtoT(n, m), and letT(m) :={C}.
F) if∀R.C ∈ F(n),nis not blocked, andnhas noR-successormwith eitherC ∈ T(m)or
¬C∈F(m), then pick up a new individualm, add¬RtoF(n, m), and letF(m) :={C}.
∀- rules :
T) if ∀R.C ∈ T(n), nis not blocked, and there exists an individualmsuch thatmis anR- successor ofn, then addCtoT(m).
F) if∃R.C ∈ F(n), nis not blocked, and there exists an individualmsuch thatmis anR- successor ofn, then addCtoF(m).
∀+- rules :
T) if∀R.C∈T(n),nis not blocked, withTrans(R)∈ Rand:
– Rv∗S;
– there exists an individualmsuch thatmis anS-successor ofn;
then add∀R.CtoT(m).
F) if∃R.C∈F(n),nis not blocked, withTrans(R)∈Rand:
– Rv∗S;
– there exists an individualmsuch thatmis anS-successor ofn;
then add∃R.CtoF(m).
Fig. 1.Expansion Rules
4 Algorithm and Substitution Strategies
We observe that given a TBoxT and two conceptCandD, it resultsT |=CvDiff the tableauτstarting fromC∈T(1), D∈F(1)is closed. Moreover, we can say that an heterogeneous clash inτrepresents in some way an “interaction” between the concepts CandD. An heterogeneous clash can be seen a “clue” that the relationT |=CvD might hold. Whenever a complete tableauτ has one or more open branchesBj, if we want the relation T |= C v D hold, then we have to force the closure of all open branchesBjinτ. We observe that if we had a tableauτ0starting withC∈T(1), andτ0 is closed thenT |=Cv ⊥,i.e.,Cis unsatisfiable w.r.t.T. Also notice that in this case, since we have onlyT() label, all branches will close with aT-homogeneous clash.
Proposition 2. LetC,Dbe two concepts inSH,T be a TBox inSHwithT |=Cv ⊥.
A complete tableaux τ starting withC ∈ T(1)andD ∈ F(1), is such that for each branchBthere is at least oneT-homogeneous clash.Proof.IfT |=C v ⊥, andτ is complete, as a direct consequence, it will surely close all its branches with at least one
T-homogeneous clash. 2
v- rules :
T) ifnis an individual such thatA∈T(n)in the branch, andAvC∈ T, then addCtoT(n).
F) ifnis an individual such that¬A∈F(n)in the branch, andAvC∈ T, then addCtoF(n).
≡- rules :
T) ifnis an individual such thatA∈T(n)in the branch, andA≡C∈ T, then addCtoT(n).
T) ifnis an individual such that¬A∈T(n)in the branch, andA≡C∈ T, then addCtoT(n).
F) ifnis an individual such that¬A∈F(n)in the branch, andA≡C∈ T, then addCtoF(n).
F) ifnis an individual such thatA∈F(n)in the branch, andA≡C∈ T, then add¬AtoF(n).
Fig. 2.Unfolding Rules
Note that in this case, in order to catch the inconsistency ofC, we needτto be complete.
Given two conceptsC,Dand a TBoxT inSH, we callH-Tableau a complete tableau built starting fromCh∈T(1)andD∈F(1).
Example 4. Suppose to have the following SAP:C =∃R.(A1u(A2t ¬A3)); D =
∃R.(A1uA2)u ∃R.(A3t ¬A2); T =∅The tableau computed following rules in Fig.
1 and Fig. 2 is depicted in Fig. 3.
T(1) ={h0u ∃R.(A1u(A2t ¬A3)uh1)}
F(1) ={∃R.(A1uA2)u ∃R.(A3t ¬A2)}
T(1) ={h0,∃R.(A1u(A2t ¬A3)uh1)}
T(1,2) ={R}
T(2) ={A1, A2t ¬A3, h1}
F(1) ={∃R.(A1uA2)}
F(2) =A1uA2
F(2) ={A1}
× heterogeneous clash A1∈F(2)andA1∈T(2)
F(2) ={A2}
T(2) ={A2}
× heterogeneous clash A2∈F(2)andA2∈T(2)
T(2) ={¬A3} open branchB0
F(1) ={∃R.(A3t ¬A2)}
F(2) ={A3,¬A2, A3t ¬A2}
T(2) ={A2} open branchB1
T(2) ={¬A3} open branchB2
Fig. 3.A H-Tableau computed applying expansion rules in Fig. 1 and Fig. 2.
Looking at Fig. 3 we observe that we could close the tableau by substituting the two concept variablesh0andh1with concepts generating a clash in the open branchesB0, B1 andB2. Once the tableau is closed we know thatT |=σ(Ch)vD. Actually this relation holds even when triviallyT |=σ(Ch)v ⊥. Nevertheless, if we want thatσbe a solution to a SAP, by condition (2) of Def. 3, we have to avoid that. By Proposition 2, we know that in order to have T 6|= σ(Ch) v ⊥, it suffices to have at least one branch closed by a heterogeneous clash. Hence, in order to computeσwe will look for
instantiation of variable in the hypotheses listHofChsuch that, given a complete open tableauτ, there is at least one open branchB ∈ τ such thatσ(B)closes without any homogeneous clash.
Proposition 3. LetT be a TBox,Chbe an abducible concept,H=hh0, . . . , hlibe its corresponding hypotheses list andτ be an open complete tableau starting fromCh ∈ T(1), D ∈F(1). For each open branchBj ∈τ, withj ∈[0, . . . , k], ifσj[H/Hj]is a substitution such thatσj[H/Hj](Bj)is closed then
σ[H/H] ={h07→σ00[H/H0]u. . .uσ0k[H/Hk], . . . , hl7→σ0l[H/H0]u. . .uσlk[H/Hk]}
is such thatT |=σ[H/H](Ch)vD. We callσj[H/Hj]abranch substitution.Proof.
Ifσiˆj[H/Hˆj]closesBˆjthen, byu-rule T) in Fig. 1,Bˆjis closed also byσiˆj[H/Hˆj]uC, beingCa generic concept. IfC=uj6=ˆjσij[H/Hj]the proposition holds. 2 For the sake of clarity, when no confusion arises, from now on we will writeσto denote σ[H/H],σito denoteσi[H/H],σj to denoteσj[H/Hj]andσji to denoteσji[H/Hj].
Algorithm 1 shows how to compute a solution to SAPs using prefixed tableaux.
Some Desiderata forσ[H/H]: In Algorithm 1 we propose a greedy procedure to
Algorithm 1: An Algorithm to compute a solution to a SAP Algorithm:abduce
Input:SHconceptsC,D, acyclic TBoxT Output: substitutionσ
begin
1
σ:={H07→ >, . . . , Hn7→ >};
2
compute a H-Tableau;
3
ifτis not closedthen
4
repeat
5
Θ:=∅;
6
foreachopen branchBj∈τdo
7
find a branch substitutionσjsuch thatBjis closed with a heterogeneous
8
clash;
Θ:=Θ∪ {σj};
9
end
10
foreachσj∈Θdo
11
σi:=σiuσij;
12
end
13
until T 6|=σ(Ch)v ⊥;
14
end
15
returnσ;
16
end
17
compute a solution to a General SAPhC, D,T,Ai˜S. There a practical problem still
remains: the computation of the branch substitution σj in line 8. There are different strategies to computeσjdepending on the characteristics we desire to be shown byσ.
In this section we identify and discuss some properties ofσthat can be easily computed (implemented) given a H-Tableau, lead by information minimal criteria. In other words, we would like to have a practical (from an implementation point of view) procedure able to compute a solution to a SAP that shows some information minimal properties. The first practical issue we face is: given a H-Tableau, in case we want to solve a General SAP, how could we select concepts inSHin order to close open branches? We could perform a syntactic search of such concepts looking at the H-Tableau itself:
(C0)For each open branchBjin a H-Tableau we select all the sets labeled withF(n).
If there is a concept variablehi ∈ T(n), we could just pick up a conceptC ∈ F(n) and perform the substitutionσij ={hi7→C}. It is easy to see thatσji(Bj)closes with at least one heterogeneous clash.(C1)For each branch substitution, in order to close a branch it suffices to have at least one variable whose value is different from>3. Note that, depending on the substitution we choose, we have different solutions for the same SAP. Suppose we select the substitutionσ1 = {h0 7→ >, h1 7→ A3}for B1in Fig. 3. It is easy to see thatσ1closes bothB1with a heterogeneous clash andB2with a homogeneous clash. Hence, in this case we do not need to compute a substitutionσ2 to closeB2. Since we are looking for information minimal hypotheses, in case we are willing to compute, for each variable,conjunction minimalsubstitutions:
(C2) we should avoid variable substitution in conjunctive form. In fact, if a conjunc- tive substitution hi 7→ D1uD2 closes a branch Bthen by u-rule T) in Fig. 1 ei- ther hi 7→ D1 or hi 7→ D2 closesB;(C3) when computing a branch substitution for an open branch, we have to take into account also the substitutions of the other open branches. With respect to Example 4 we see that there are three different sub- stitutions for the same variable h0. By Proposition 3 we know that the conjunction
∀R.A2u ∀R.(A3t ¬A2)u ∃R.(A3t ¬A2)is a substitution forh0 closingB0,B1 andB2. Hence, if a variable substitution closes more than one open branch, i.e., the same variable substitution appears in more than one branch substitution, it should be preferred over the others. This is the case, for instance, ofh07→ ∀R.(A3t ¬A2)forσ1 andσ2;
Going back to Condition (C0) and Condition (C1), given a H-Tableau, in case there is more than one individualnsuch thathi∈T(n), how to choose the individual? What is the best candidate? Moreover, once we choose and individualn, which concept inF(n) should we pick-up? It depends on the solution we want to compute.
(C4) With respect to Example 4, if we consider eitherσ0in row 3 orσ0in row 4 and σ1in row 11 we obtain4:
σ[H/H00] ={h07→ >, h17→A2uA3} (4) σ[H/H0] ={h07→ ∃R.(A1uA2), h17→A3} (5) Note that bothH0 andH00are computed satisfying all the above conjunction minimal conditions. Nevertheless, we see that solutions (4) is more “fine-grained” than solu-
3Branch substitutions with all the variables inHbut one equal to>tend to generate structural maximal solutions.
4In this case it suffices to haveh17→A3inσ1to close bothB1andB2.
tion (5). Also notice thatσ[H/H0](Ch) v σ[H/H00](Ch). In other words, solutions (4) is subsumption-maximal w.r.t. solution (5). With respect to Definition 5 it results H0 ≤ H00. Hence, when closing an open branchBto compute a subsumption maximal solution we tend to select variablehi ∈ T(n)such that there is nohp ∈ T(m)with hi < hp. In Algorithm 2 we propose a procedure to compute a branch substitutionσj taking into account (C0), (C1), (C2), (C3) and (C4). Moreover, from Proposition 2 we know that if there is at least one branchBjin a H-Tableau such thatBjdoes not contain anyT-homogeneous clash, thenT 6|=σ(Ch) v ⊥. In order to catch this situation as soon as possible, while computingσjwe should avoid, if possible, to closeBjgenerat- ing also aT-homogeneous clash.5
— In line 1 of Algorithm 2 we take into account Condition (C3). Indeed, givenBjwe try to reuse a substitutions already computed for previous branchesBj−k.
— Line 4 formalizes conditions (C0) and (C4). We select an individualnsuch that both T(n)containshiand there is no otherh-variablehpsuch that it is “bigger” thanhi.
— Since we selectednsuch thatF(n)6=∅, we can choose a concept inF(n)to close Bj. Furthermore, while closingBj we try to avoid heterogeneous clashes in order to discard inconsistent substitutions (see Proposition 2).
It is noteworthy that the algorithms we presented in this section compute a sub-optimal
Algorithm 2: Computation of a branch substitution for information minimal so- lutions.
ifhi∈T(n)and there exists a substitutionσj−ki , withk= 1, . . . , jsuch that
1
σij:=σj−ki closesBjthen σji :=σij−k;
2
else
3
selectan individualnsuch thatT(n)6=∅,F(n)6=∅and there exists no other
4
individualmsuch that bothT(m)6=∅,F(m)6=∅andhi< hpwithhi∈T(n)and hp∈T(m);
choosea conceptE∈F(n)such thatEdoes no contain a conjunction and,if
5
possible,Bj∪ {E∈T(n)}does not close with aT-homogeneous clash;
σji :=E;
6
end
7
foreachhk∈ Hwithk6=i do
8
σjk:=>;
9
end
10
solution due to their greedy nature. In fact, we have no guarantee that Algorithm 2 finds a substitution such that σ[H/H](τ) have at least one branch closed by no T- homogeneous clash. However, the algorithms we illustrated in this section are very useful to understand the issues related to the computation of a solution to a SAP.
Dealing with RBoxes:It is easy to see that Algorithm 2 works nicely whenR=∅. Let us take a look at what happens when we have to deal with a non-empty RBox.
5We minimizeT-homogeneous clashes in order to avoid trivial solutions.
Example 5. Suppose we have to solve the following SAP.C=∀R.(A1u∃S.A2);D=
∀T.∃S.A3; T =∅; R={Trans(R), SvR, T vR}Part of the tableau we compute following rules in Fig. 1 is represented in Fig. 4.
T(1) ={h0u ∀R.(h1uA1u ∃S.(h2uA2))}
F(1) ={∀T.∃S.A3}
T(1) ={h0,∀R.(h1uA1u ∃S.(h2uA2))}
F(1,2) ={¬T}
F(2) ={∃S.A3}
T(2) ={h1, A1,∃S.(h2uA2),∀R.(h1uA1u ∃S.(h2uA2))}
T(2,3) ={S}
T(3) =h2, A2, h1, A1,∃S.(h2uA2),∀R.(h1uA1u ∃S.(h2uA2)) F(3) =A3
. . . . . .
Fig. 4.Part of the tableau computed for Example 5
Looking at Example 5 we see that, due toR, bothh1andh2appears inT(3). Hence, following Algorithm 2 we can close the branch with one of the following two substi- tutions:σ[H/H0] = {h0 7→ >, h1 7→ >, h2 7→ A3};σ[H/H00] = {h0 7→ >, h1 7→
A3, h27→ >}
Due toTrans(R), we haveσ[H/H00](Ch)vσ[H/H0](Ch). Similarly to what we do in (C4), if we are looking for subsumption-maximal solutions we will preferH0 over H003. On the other hand we see thatH0≤ H00. Hence, (C4) holds also in presence of a non-empty RBox.
5 Remarks and Conclusion
We claim that our proposal is a proper extension of PA. In fact, given a theory T, some observationsO (both propositional formulas) and a set of possible assumptions A={a1, . . . , an}(atoms), PA is the problem of finding a subsetS ⊆ Asuch thatT∧S is consistent, andT ∧S|=O. Then, PA can be translated into SAP by(i)lettingC=>
(henceCh=> uH0≡H0),(ii)expressingT andOasALCconcepts—just replace
∧,∨withu,t—and(iii)solving the SAPT |=H0vO, whereT ={> vT}. Since in propositional Logic the Deductive Theorem holds, one could also letC=T,T =∅, and solve the SAP|=T uH0 v O. In both cases only substitutions involving atoms inAshould be considered. Hence, SAP properly extends PA. Given thatALCis a syn- tactic variant of the Modal LogicKn, and that (the Modal Logic version of) transitive roles is present in the Modal LogicS4, the work most similar to ours is [4]. Compared to our work, Cialdea & Pirri did not have our known factsC. Moreover, they looked for formulasαsuch thatT ∪ {α} |=O, that in our setting could be rephrased as CA of Section 2, or alternatively, as a SAP having onH0as variable to be substituted. Fi- nally, they just looked for subsumption-maximal abductions onH0, while our approach allows us to find several abduction solutions depending on the preference criterion for the variousHi’s. On the other hand, every solution we find can be turned into a solution in their framework: in fact, given a solutionσ[H/H], sinceT |=σ[H/H](Ch)vD, one can always letα=σ[H/H](Ch)as a solution.
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