I N F S Y S R E S E A R C H R E P O R T 2 0 0 2 INSTITUT FU¨R INFORMATIONSSYSTEME n a J ABTEILUNG WISSENSBASIERTE SYSTEME 6 1 ] O L s. COMPUTING PREFERRED ANSWER SETS BY c [ META-INTERPRETATION IN ANSWER SET 1 v PROGRAMMING 3 1 0 1 0 2 0 ThomasEiter WolfgangFaber NicolaLeone / s Gerald Pfeifer c : v i X r a INFSYS RESEARCH REPORT 1843-02-01 JANUARY 2002 Institutfu¨rInformationssysteme Abtg.WissensbasierteSysteme TechnischeUniversita¨tWien Favoritenstraße9-11 A-1040Wien,Austria Tel: +43-1-58801-18405 Fax: +43-1-58801-18493 sek@kr.tuwien.ac.at www.kr.tuwien.ac.at INFSYS RESEARCH REPORT INFSYS RESEARCH REPORT 1843-02-01, JANUARY 2002 COMPUTING PREFERRED ANSWER SETS BY META-INTERPRETATION IN ANSWER SET PROGRAMMING ThomasEiter1, WolfgangFaber1,NicolaLeone2, Gerald Pfeifer3 Abstract. Mostrecently,AnswerSetProgramming(ASP)isattractinginterestasanewparadigm forproblemsolving.Animportantaspectwhichneedstobesupportedisthehandlingofpreferences between rules, for which several approaches have been presented. In this paper, we consider the problemofimplementingpreferencehandlingapproachesbymeansofmeta-interpretersinAnswer Set Programming. In particular, we consider the preferredanswer set approachesby Brewka and Eiter,byDelgrande,SchaubandTompits,andbyWang,ZhouandLin. We presentsuitablemeta- interpreters for these semantics using DLV, which is an efficient engine for ASP. Moreover, we alsopresentameta-interpreterfortheweaklypreferredanswersetapproachbyBrewkaandEiter, which usesthe weak constraintfeatureof DLV as a toolfor expressingand solvingan underlying optimization problem. We also consider advanced meta-interpreters, which make use of graph- based characterizationsand often allow for more efficientcomputations. Our approachshows the suitability of ASP in generaland of DLV in particular for fast prototyping. This can be fruitfully exploitedforexperimentingwithnewlanguagesandknowledge-representationformalisms. 1Institutfu¨rInformationssysteme,AbteilungWissensbasierteSysteme,TechnischeUniversita¨tWien, Favoritenstraße9-11,A-1040Vienna,Austria. E-mail:{eiter,faber}@kr.tuwien.ac.at. 2DepartmentofMathematics,UniversityofCalabria,87030Rende(CS),Italy.E-mail:leone@unical.it. 3Institutfu¨rInformationssysteme,AbteilungDatenbankenundAI,TechnischeUniversita¨tWien, Favoritenstraße9-11,A-1040Vienna,Austria. E-mail:pfeifer@dbai.tuwien.ac.at. Acknowledgements: TheauthorswouldliketothankourcolleaguesandtheparticipantsoftheAAAI2001 SpringSymposiumonAnswerSetProgrammingfortheircommentsonthiswork. Thisworkwassupported bytheAustrianScienceFund(FWF)undergrantsP13871-INF,P14781-INF,andZ29-INF. Preliminary results of this paper appeared in Alessandro Provetti and Tran Cao Son, editors, Proceedings AAAI 2001 Spring Symposium on Answer Set Programming: Towards Efficient and Scalable Knowledge RepresentationandReasoning,StanfordCA,March2001,AAAIPress,ISBN1-57735-129-0. Copyright(cid:13)c 2008bytheauthors 2 INFSYS RR 1843-02-01 1 Introduction Handling preference information plays an important role in applications of knowledge representation and reasoning. Inthe context of logic programs and related formalisms, numerous approaches for adding preference information have been proposed, including [1, 4, 5, 8, 11, 12, 18, 22, 24, 25, 27, 29] to mention someofthem. Theseapproacheshavebeendesignedforpurposessuchascapturingspecificityornormative preference; seee.g.[7,11,25]forreviewsandcomparisons. Thefollowingexampleisaclassical situation fortheuseofpreference information. Example1(bird&penguin) Considerthefollowinglogicprogram: (1) peng. (2) bird. (3) ¬flies:-notflies, peng. (4) flies:-not¬flies, bird. This program has two answers sets: A = {peng, bird, ¬flies} and A = {peng, bird, flies}. 1 2 Assumethatrule(i)hashigherpriority than(j)iffi < j (i.e.,rule(1)hasthehighest priority andrule(4) the lowest). Then, A is no longer intuitive, as flies is concluded from (4), while (3) has higher priority 2 than(4),andthus¬fliesshouldbeconcluded. However,evenifthisexampleisverysimple,thevariouspreferencesemanticsarriveatdifferentresults. Furthermore, semantics whichcoincide onthisexamplemaywellyielddifferent resultsonotherexamples. Sinceevaluatingasemanticsonanumberofbenchmarkexamples,eachofwhichpossiblyinvolvingseveral rules, quickly becomes a tedious task, one would like to have a (quick) implementation of a semantics at hand, such that experimentation with it can be done using computer support. Exploring a (large) number of examples, which helps in assessing the behavior of a semantics, may thus be performed in significantly shortertime,andlesserrorprone,thanbymanualevaluation. In this paper, we address this issue and explore the implementation of preference semantics for logic programs by the use of a powerful technique based on Answer Set Programming (ASP), which can be seen asasort ofmeta-programming inASP.Inthis technique, agiven logic program P withpreferences is encoded by a suitable set of facts F(P), which are added to a “meta-program” P , such that the intended I answer setsofP aredetermined bytheanswer setsofthelogicprogram P ∪F(P). Thesalient feature is I that this P is universal, i.e., it is the same for all input programs P. We recall that meta-interpretation is I well-established inProlog-style logicprogramming, andisnotcompletely newinASP;asimilartechnique has been applied previously in [18] for defining the semantics of logic programming with defeasible rules (cf.Section6). We focus in this paper on three similar, yet different semantics for prioritized logic programs, namely the preference semantics by Brewka and Eiter [7], Wang, Zhou and Lin [27], and Delgrande, Schaub and Tompits [12], which we refer to as B-preferred, W-preferred, and D-preferred answer set semantics, respectively. We present ASP meta-programs P , P , and P such that the answer sets of P ∪F(P), IB IW ID IB P ∪F(P),andP ∪F(P)correspond(moduloasimpleprojectionfunction)preciselytotheB-,W-,and IW ID D-preferred answer sets of P. This way, by running F(P) together with the corresponding meta-program on the DLV system [15, 16], we compute the preferred answer sets of P in a simple and elegant way. For B-preferred answer sets we also provide an alternate meta-program P , which implements a graph-based Ig algorithmthatdeterministically checkspreferredness ofananswersetandismoreefficientingeneral. Note that, by suitable adaptions of the meta-programs, other ASP engines such as Smodels [23] can be used as well. INFSYS RR 1843-02-01 3 B-preferred answer setsemantics refinesprevious approaches foradding preferences todefault rules in [4, 5]. It is defined for answer sets of extended logic programs [17] and is generalized to Reiter’s default logic in[6]. Animportant aspect ofthisapproach isthatthedefinition ofpreferred answer setswasguided by two general principles which, as argued, a preference semantics should satisfy. As shown in [7], B- preferred answer sets satisfy these principles, while almost all other semantics do not. W- and D-preferred answersetsemanticsincreasingly strengthen B-preferredanswersetsemantics [26]. Since,ingeneral, programs havingananswersetmaylackapreferred answerset,alsoarelaxed notion of weakly preferred answer sets was defined in [7]. For implementing that semantics, we provide a meta- program P ,whichtakesadvantage oftheweakconstraints feature[9]ofDLV. I weak Theworkreportedhereisimportantinseveralrespects: • We put forward the use of ASP for experimenting new semantics by means of a meta-interpretation technique. The declarativity of logic programs (LPs) provides a new, elegant way of writing meta- interpreters, whichisverydifferentfromProlog-stylemeta-interpretation. Thankstothehighexpres- sivenessof(disjunctive) LPsandDLV’sweakconstraints, meta-interpreters canbewritteninasimple anddeclarative fashion. • The description of the meta-programs for implementing the various preference semantics also has a didactic value: it is a good example for the way how meta-interpreters can be built using ASP. In particular, wealso develop a core meta-program for plain extended logic programs under answer set semantics, whichmaybeusedasabuilding blockintheconstruction ofothermeta-programs. • Furthermore, the meta-interpreters provided are relevant per se, since they provide an actual implementation ofpreferred andweaklypreferred answersetsandallowforeasyexperimentation ofthese semantics in practice. Toourknowledge, this isthe firstimplementation of weakly preferred answer sets. An implementation of preferred answer sets (also on top of DLV) has been reported in [13], by mapping programs into the framework of compiled preferences [10]. Our implementation, as will be seen, is an immediate translation of the definition of preferred answer sets into DLV code. Weak constraints make the encoding of weakly preferred answer sets extremely simple and elegant, while thattaskwouldhavebeenmuchmorecumbersome otherwise. Insummary, theexperience reported inthispaper confirmsthepowerofASP.Itsuggests theuseofthe DLVsystem asahigh-level abstract machinetobeemployed alsoasapowerfultoolforexperimenting with newsemanticsandnovelKRlanguages. It isworthwhile noting that the meta-interpretation approach presented here does not aim at efficiency; rather,thisapproachfosterssimpleandveryfastprototyping, whichisusefule.g.intheprocessofdesigning andexperimenting newlanguages. Thestructure oftheremainder ofthispaper isasfollows: Inthenext section, weprovide preliminaries of extended logic programming and answer set semantics. We then develop in Section 3 a basic meta- interpreter program for extended logic programs under answer set semantics. After that, we consider in Section4meta-interpreter programsforB-preferred,W-preferred,andD-preferredansweranswersets. The subsequent Section5isdevoted totherefinement ofpreferred answersetstoweaklypreferred answersets. AdiscussionofrelatedworkisprovidedinSection6. ThefinalSection7summarizesourresultsanddraws someconclusions. 4 INFSYS RR 1843-02-01 2 Preliminaries: Logic Programs Syntax. LogicPrograms(LPs)useafunction-free first-orderlanguage. Asforterms,stringsstartingwith uppercase (resp., lowercase) letters denote variables (resp., constants). A(positive resp. negative) classical literal liseitheranatom aoranegatedatom ¬a,respectively; itscomplementary literal, denoted ¬l,is¬a anda,respectively. A(positive resp.,negative) negation asfailure(NAF)literalℓisoftheformlornot l, wherelisaclassical literal. Unlessstatedotherwise, byliteralwemeanaclassicalliteral. Arulerisaformula a v ··· v a :-b ,···,b , not b ,···, not b . (1) 1 n 1 k k+1 m where all a and b are classical literals and n ≥ 0, m ≥ k ≥ 0. The part to the left of “:-” is the i j head, the part to the right is the body of r; we omit “:-” if m = 0. We let H(r) = {a ,..., a } be the 1 n set of head literals and B(r)= B+(r)∪B−(r) the set of body literals, where B+(r) = {b ,..., b } and 1 k B−(r) = {b , ..., b } are the sets of positive and negative body literals, respectively. An integrity k+1 m constraint isarulewheren =0. A datalog program (LP) P is a finite set of rules. We call P positive, if P is not -free (i.e. ∀r ∈ P : B−(r)= ∅);andnormal,ifP is v-free(i.e.∀r ∈ P : |H(r)| ≤ 1). Aweakconstraint risanexpression oftheform :∼ b ,···,b ,not b ,···, not b .[w :l] 1 k k+1 m where every b isaliteral and l ≥ 1isthe priority level and w ≥ 1the weight among thelevel. Bothl and i w are integers and set to 1 if omitted. The sets B(r), B+(r), and B−(r) are defined by viewing r as an integrity constraint. WC(P)denotesthesetofweakconstraints inP. Asusual,aterm(atom,rule,...) isground, ifnovariables appearinit. Semantics. AnswersetsforLPswithweakconstraintsaredefinedbyextendingconsistentanswersetsfor LPsasintroduced in[17, 20]. Weproceed inthree steps: wefirstdefineanswer sets (1) ofground positive programs, then (2) of arbitrary ground programs, and (3) finally (optimal) answer sets of ground programs with weak constraints. As usual, the (optimal) answer sets of a non-ground program P are those of its ground instantiation Ground(P),definedbelow. For any program P, let U be its Herbrand universe and B be the set of all classical ground literals P P from predicate symbols inP overtheconstants ofU ;ifnoconstant appears inP,anarbitrary constant is P added toU . Foranyclause r,letGround(r)denote thesetofitsground instances. Then,Ground(P) = P Ground(r). Note that P is ground iff P = Ground(P). An interpretation is any set I ⊆ B of r∈P P ground literals. Itisconsistent, ifI ∩{¬l | l ∈ I} = ∅. S Inwhatfollows,letP beagroundprogram. (1) A consistent1 interpretation I ⊆ B is called closed under a positive program P, if B(r) ⊆ I P impliesH(r)∩I 6= ∅foreveryr ∈ P. AsetX ⊆ B isananswersetforP ifitisaminimalset(wrt.set P inclusion) closedunderP. (2) Let PI be the Gelfond-Lifschitz reduct of a program P w.r.t. I ⊆ B , i.e., the program obtained P fromP bydeleting • allrulesr ∈ P suchthatB−(r)∩I 6= ∅,and 1Weonlyconsiderconsistentanswersets,whilein[20]alsothe(inconsistent)setB maybeananswerset. P INFSYS RR 1843-02-01 5 • allnegativebodyliteralsfromtheremaining rules. Then, I ⊆ B is an answer set of P iff I is an answer set of PI. By AS(P) we denote the set of all P answersetsofP. Example2 Theprogram avb. bvc. dv¬d:-a,c. hasthreeanswersets: {a,c,d}, {a,c,¬d}, and{b}. (3) Given a program P with weak constraints, we are interested in the answer sets of the part without weakconstraintswhichminimizethesumofweightsoftheviolatedconstraintsinthehighestprioritylevel, andamongthemthosewhichminimizethesumofweightsoftheviolatedconstraintsinthenextlowerlevel, etc. Thisisexpressedbyanobjectivefunction forP andananswersetA: f (1) =1 P f (n)=f (n−1)·|WC(P)|·wP +1, n> 1 P P max HP = lmPax(f (i)· w ) A i=1 P N∈NA,P N i P P where wP and lP denote the maximum weight and maximum level of a weak constraint in P, respec- max max A,P tively; N denotes the set of weak constraints in level i which are violated by A, and w denotes the i N weight of the weak constraint N. Note that |WC(P)|·wP +1 is greater than the sum of all weights in max theprogram, andtherefore guaranteed tobegreater thananysumofweights ofasinglelevel. Ifweightsin level i are multiplied by f (i), it is sufficient to calculate the sum of these updated weights, such that the P updatedweightofaviolated constraint ofagreaterlevelisalwaysgreaterthananysumofupdatedweights ofviolated constraints oflowerlevels. Then, A is an (optimal) answer set of P, if A ∈ AS(P \WC(P)) and HP is minimal over AS(P \ A WC(P)). LetOAS(P)denotethesetofalloptimalanswersets. Example3 Letusenhancetheprogram fromExample2bythefollowing threeweakconstraints: :∼a,c.[2:1] :∼¬d.[1:1] :∼b.[3:1] Theresulting programP hasthesingleoptimalanswersetA = {a,c,d}withweight2inlevel1. 3 3 3 Meta-Interpreting Answer Set Programs Inthissectionweshowhowanormalpropositional answersetprogram canbeencoded forandinterpreted byagenericmeta-interpreter basedonthefollowingidea: Weprovidearepresentation F(P)ofanarbitrarynormalpropositionalprogramP 2asasetoffactsand combine these facts withageneric answer setprogram P such thatAS(P) = {π(A) | A ∈ AS(F(P)∪ Ia P )},whereπ isasimpleprojection function. Ia 2We assume that integrity constraints :- C.are writtenasequivalent rules bad:-C,not bad.where bad isapredicate not occurringotherwiseinP. 6 INFSYS RR 1843-02-01 3.1 Representing AnAnswer SetProgram Firstwetranslate thepropositional answersetprogram P intoasetoffactsF(P)asfollows: 1. Foreachrule c:-a ,...,a ,notb ,...,notb . 1 m 1 n oftheprogram P,F(P)contains thefollowingfacts: rule(r).head(c,r).pbl(a ,r). ... pbl(a ,r). 1 m nbl(b ,r). ... nbl(b ,r). 1 n wherer isauniqueruleidentifier. 2. For each pair of complementary literals ℓ,¬ℓ occurring in the program P we explicitly add a fact compl(ℓ,¬ℓ). Example4 The program of the bird & penguin example is represented by the following facts representing therules: rule(r1). head(peng,r1). rule(r2). head(bird,r2). rule(r3). head(neg flies,r3). pbl(peng,r3). nbl(flies,r3). rule(r4). head(flies,r4). pbl(bird,r4). nbl(neg flies,r4). andthefollowing factsrepresenting complementary literals: compl(flies,neg flies). 3.2 BasicMeta-interpreter program Several meta-interpreters that we will encounter in the following sections consist of two parts: a meta- interpreter program P forrepresenting ananswer set, andanother one forchecking preferredness. Inthis Ia section wewillprovide thefirstpartwhichiscommontomanymeta-interpreters showninthispaper. Representing an answer set. We define a predicate in AS(.) which is true for the literals in an answer setofP. Aliteral isinananswer setifitoccurs inthehead ofarule whosepositive bodyisdefinitely true andwhosenegativebodyisnotfalse. in AS(X):-head(X,R), pos body true(R), notneg body false(R). Thepositivepartofabodyistrue,ifallofitsliteralsareintheanswerset. Unfortunately wecannotencode such a universal quantification in one rule. We can identify a simple case: If there are no positive body literals, thebodyistriviallytrue. pos body exists(R):-pbl(X,R). pos body true(R):-rule(R),notpos body exists(R). INFSYS RR 1843-02-01 7 However, ifpositive body literals exist, wewillproceed iteratively. Tothis end, weuse DLV’sbuilt-in total orderonconstants fordefiningasuccessor relationonthepositivebodyliteralsofeachrule,andtoidentify thefirstandlastliteral, respectively, ofapositiverulebodyinthistotalorder. Technically, itissufficientto defineauxiliary relations asfollows. pbl inbetween(X,Y,R):-pbl(X,R),pbl(Y,R),pbl(Z,R),X < Z,Z < Y. pbl notlast(X,R):-pbl(X,R),pbl(Y,R),X < Y. pbl notfirst(X,R):-pbl(X,R),pbl(Y,R),Y < X. This information can be used to define the notion of the positive body being true up to (w.r.t. the built-in order)somepositivebodyliteral. Ifthepositivebodyistrueuptothelastliteral,thewholepositivebodyis true. pos body true upto(R,X):-pbl(X,R),notpbl notfirst(X,R),in AS(X). pos body true upto(R,X):-pos body true upto(R,Y),pbl(X,R),in AS(X), Y < X,notpbl inbetween(Y,X,R). pos body true(R):-pos body true upto(R,X),notpbl notlast(X,R). Thenegativepartofabodyisfalse,ifoneofitsliterals isintheanswerset. neg body false(Y):-nbl(X,Y),in AS(X). Each answer set needs to be consistent; we thus add an integrity constraint which rejects answer sets con- tainingcomplementary literals. :-compl(X,Y),in AS(X),in AS(Y). Therulesdescribedabove(referredtoasP inthesequel)areallweneedforrepresentinganswersets. Each Ia answersetofP ∪F(P)representsananswersetofP. Letπ bedefinedbyπ(A) = {ℓ |in AS(ℓ) ∈ A}. Ia Thenwecanstatethefollowing: Theorem1 Let P be a normal propositional program. Then, (i) if A ∈ AS(P ∪ F(P)) then π(A) ∈ Ia AS(P),and(ii)foreachA ∈ AS(P),thereexistsasingleA′ ∈ AS(P ∪F(P))suchthatπ(A′)= A. Ia Proof (i) π(A) must be a consistent set of literals from P, since for each ℓ s.t. in AS(ℓ) ∈ A, head(ℓ,r) must hold for some rule r, which, by construction of F(P), only holds for ℓ ∈ B , and since the con- P straint :-compl(ℓ,¬ℓ),in AS(ℓ),in AS(¬ℓ).mustbebesatisfied byA(againbyconstruction ofF(P)) {in AS(ℓ),in AS(¬ℓ)} 6⊆ Aandhence{ℓ,¬ℓ} 6⊆ π(A)forallℓ ∈B . P Thus,toshowthatπ(A) ∈ AS(P),itsufficestoshowthat(α)π(A)isclosedunderPπ(A),andthat(β) π(A) ⊆ T∞ musthold, whereT isthestandard T operator forP = Pπ(A). Let,forconvenience, Pπ(A) Pπ(A) P denoteQ = Ground(P ∪F(P)). Ia As for (α), we show that if r ∈ Pπ(A) such that B(r) ⊆ π(A), then there is a rule h ∈ Q such that r H(h ) = {in AS(h)} where H(r) = {h} and h is applied in A, i.e., A |= B(h ) and in AS(h) ∈ A. r r r Let r stem from a rule r′ ∈ P. Then, let h be the (unique) rule in Q such that H(h ) = {in AS(h)} r r (note that H(r) = H(r′) = {h}) and head(h,r′) ∈ B(h ). Since B(r) = B+(r′) ⊆ π(A), we have for r each ℓ ∈ B(r) that in AS(ℓ) ∈ A and, by construction of F(P), that pbl(ℓ,r′) ∈ A. Since < induces a linearordering ofB+(r′),itfollowsbyaninductive argument alongitthatpos body true upto(r′,ℓ) ∈ A holds for each ℓ ∈ B+(r′) and that pos body true(r′) ∈ A. Furthermore, since B−(r′) ∩ π(A) = ∅, 8 INFSYS RR 1843-02-01 it holds that A 6|= B(n ) for each rule n ∈ Q such that H(n ) = neg body false(r′). Therefore, r r r neg body false(r′) ∈/ A. Since head(h,r′) ∈ A by construction of F(P), it follows that A |= B(h ). r Thus,h isappliedinA,andhenceH(h )= in AS(h) ∈ A. Thisproves(α). r r As for (β), it suffices to show that if in AS(ℓ) ∈ Ti \Ti−1, i.e., in AS(ℓ) is added to A in the i-th QA QA stepoftheleastfixpointiterationforT ,i≥ 1,thenℓ ∈ T∞ holds. Additionofin AS(ℓ)impliesthat QA Pπ(A) ℓ = H(r) for some r ∈ P such that neg body false(r) ∈/ A and pos body true(r) ∈ A. This implies in AS(ℓ′)∈ Ti−1foreachℓ′ ∈B+(r)andin AS(ℓ′) ∈/ Aforeachℓ′ ∈ B−(r). Byaninductiveargument, QA weobtainB+(r)⊆ T∞ ;therefore, ℓ ∈T∞ holds. Thisproves(β). Pπ(A) Pπ(A) (ii)ForanyA ∈ AS(P),letA′ bedefinedasfollows( λ(ℓ) for each ℓ ∈ B−(r). Denote for any program P by P the set of rules in ir Ground(P ∪F(P)) whoserepresentation literals (i.e., literals overrule,head, bpl,nbl)aresatisfied by Ia F(P). Thenwehave: Proposition 1 Let P be a stratified normal propositional program. Then, P is locally stratified (i.e., ir stratified ifviewedasapropositional program). Thiscanbeseenbyconstructing fromastratification mappingλforP asuitable stratification mapping λ′ forP . LocallystratifiedprogramsareefficientlyhandledbyDLV.Since,byintelligentgroundingstrate- ir gies, P is efficiently computed in DLV, the overall evaluation of P ∪F(P) is performed efficiently by ir Ia DLVforstratifiedprogramsP.

Computing Preferred Answer Sets by Meta-Interpretation in Answer Set Programming

Similar Computing Preferred Answer Sets by Meta-Interpretation in Answer Set Programming