Towards Closed World Reasoning in Dynamic Open Worlds (Extended Version)

Under c onsider ation for public ation in The ory and Pr actic e of Lo gic Pr o gr amming 1 T owar ds Close d World R e asoning in Dynamic Op en Worlds (Extende d V ersion) MAR TIN SLOT A ∗ and JO ˜ AO LEITE CENTRIA & Dep artamento de Inform´ atic a Universidade Nova de Lisb o a 2829-516 Cap aric a, Portugal Note : This is an extended version con taining all pr o ofs of the ar ticle published in T he ory and Pr actic e of L o gic Pr o gr amming , 10 (4-6): 547 – 564, July . c  2010 Cambridge Universit y Press . Changes on July 23, 201 0 : some minor substitutions and additions to be in line with the TPLP version; full re ference to the journal version added. Abstract The need for integration of ontolog ies with nonmonotonic rules has b een gaining imp or- tance in a num b er of areas, such as the Semantic W eb. A num b er of researc hers add ressed this problem by prop osing a uniﬁed semantics for hybrid know le dge b ases composed of b oth an ontology ( expressed in a fragment of ﬁ rst- order logic) and non mon otonic rules. These semantics hav e matured ov er the years, bu t only provide solutions for the static case when kn o wledge do es n ot need to ev olve. In this pap er we take a ﬁrst step tow ards addressing the dynamics of hybrid kno wledge bases. W e focus on kno wledge up dates and, considering the state o f t he art of belief update, ontol ogy up date and rule up date, w e show that current solutions are only partial and diﬃcult to com bine. Then w e extend the existing w ork on ABo x up dates with rules, provide a semantics for such evolving hybrid know ledge b ases and stud y its basic prop erties. T o th e b est of our kn o wledge, this is the ﬁ rst time that an up date op erator is p roposed for hybrid k n o wledge bases. KEYWORDS : belief c han ge, b elief up date, hybrid know ledge bases, on tologies, rules, d e- scription logics, answer set programming, seman tic w eb 1 In tro duction In this pa per w e address upda tes o f h ybrid knowledge bases compo sed of a Descrip- tion Logic o ntology and Logic P rogra mming rules . W e prop ose an op erator to b e used when a hybrid theory is up dated b y new obser v atio ns of a changing world, examine its prop erties, and discuss op en problems p ointing to future r esearch. ∗ Supported by F CT Sc holarship SFRH/BD/38214/2 007. P articipation on conference supp orted b y FLoC 2010 Studen t T rav el Support and b y APPIA Study Sc holarshi p. 2 M. Slota and J. L eite The Seman tic W eb was initiated almost a decade ago with an ambitious plan regar ding the sharing of metadata and knowledge in the W eb, enhanced with rea- soning services for adv anc e d new applications ( Berners- L ee et al. 20 0 1 ). Since then, the considerable amo un t of resea rch devoted to this endeavour originated imp or- tant fo undational results and a deep er understanding o f the issues inv olved, while ident ifying imp ortant conclusions regarding future developmen ts, namely that: 1. On tolo gies are necessary and useful for knowledge represe ntation in the Se- mantic W eb. The fo rmalisms develop e d, e.g. OWL, ar e powerful enoug h to capture existing mo delling langua ges used in softw are engineering , and ex- tend their capabilities. Ontologies are usually based on decidable, as w ell as tractable, fragments of Classica l Logic, such a s the Descr iptio n L o gics (DL) ( Baader et al. 20 03 ). They a dopt the op en world assumption (O W A) i.e. they view a knowledge base, by assumption , to b e p otent ially inco mplete, hence a prop osition p is false only if the knowledge bas e is inc o nsistent with p . This suits well the o p en nature of such systems where complete knowledge a bo ut the environmen t cannot b e assumed. 2. Rules are fundamental to overcome the limitations found in OWL. They enjoy formal, declarative a nd well-understoo d semantics, the stable mo del seman- tics ( Gelfond and Lifschitz 1988 ) and its tractable approximation, the three- v a lued wel l-founde d semant ics ( Gelder et al. 199 1 ) b eing the most prominent and widely accepted. These semantics a dopt the close d world assumption (CW A) i.e. the knowledge base is assume d to contain complete infor mation. Consequently , a pr op osition p is cons idered false whenever it is not entailed to b e true. This t yp e o f negatio n is usua lly dubb e d default ne gation or we ak ne gation , to distinguish it from the classic al ne gation used in Class ical Logic. Rules can naturally express assumptions, policies, preferences, norms and laws, and pr ovide co nstructs which are more natura l for soft ware develope r s (as used in Relational Databa s es and Log ic Pr ogra mming ). 3. The op en and dynamic c haracter of the Semantic W eb requires new knowledge based systems to b e equipp ed with mechanisms to evolve. Indeed, the growing av ailability of information requires the supp ort of dynamic data and applica tion integration, a uto mation and interoper ation of business pro- cesses and pro blem-solving in v arious domains, to enforc e co r rectness of decisio ns, and to allow tracea bilit y of the knowledge used and of the decisions taken. In thes e scenarios , ontologies provide the logical foundation of intelligen t access a nd infor - mation integration, while rules are used to represent busine s s po lic ies, re g ulations and declar a tive guidelines ab out infor mation, and ma pping s b etw een diﬀerent in- formation sources. Over the last deca de, there hav e b een man y prop osals for integrating DL ba sed monotonic on tolo gies with no nmonotonic rules (see ( Hitzler and Parsia 2 009 ) for a survey). Recently , in ( Motik and Ros a ti 2 007 ), Hybrid MKNF Knowledge Bases were introduced, allowing pr edicates to be deﬁned concurr en tly in b oth an ontol- ogy and a set o f rules, while enjoying several imp orta n t prop erties. There is even a tractable v ariant based on the well-founded semantics that allows for a top-down T owar ds Close d World R e asoning in Dynamic Op en Worlds 3 querying pro cedure ( Alferes et al. 200 9 ), making the approach amena ble to pra cti- cal applications that need to deal with lar ge ontologies. But this o nly addresses pa rt of the pr oblem. The highly dynamic character of the Semantic W eb calls for the developmen t of w ays to deal with updates of these h ybrid knowledge bases composed of b oth rules and on tolo gies, and the incons istencies that may arise. The dyna mics of hybrid knowledge ba s es, to the be st of our knowledge, has never be en addressed b efore. How ever, the problems a sso ciated with knowledge evolution have b e en extensively studied, over the years, by res e a rchers in diﬀeren t research communities, na mely in the context of Classical Logic, and in the context of Log ic Pr ogra mming . They prov ed to b e extr emely diﬃcult to solve, a nd existing s o lutions, even within each communit y , are s till sub ject of active debate as they do not seem adequate in a ll kinds of situations in which their a pplica tion is des irable. In the context of Classica l Log ic, the se minal work by Alchourr´ on, G¨ ardenfor s and Mak inson (AGM) ( Alc hour r´ on et al. 1 985 ) pro po sed a set o f desira ble prop- erties o f b elief change o pe r ators, now called AGM p ostulates . Subsequently , in ( Katsuno and Mendelzon 1991 ), up date and r evision ha ve b een distinguished as t wo very rela ted but ultimately diﬀerent b elief change o per ations. While re vi- sion deals with inc o rp orating new informatio n a bo ut a static world, upda te takes place whe n changes o ccur ring in a dynamic world are recorde d. The authors of ( Katsuno and Mendelzon 1991 ) formulated a separate set of p os tula tes for up dates. One of the sp eciﬁc up date o per ators that satisﬁes these p ostulates is Winslett’s minimal c hang e update op era to r ( Winslett 19 90 ). Though w e belie v e that revisio n op erators for hybrid knowledge bas es p ose an interesting and imp ortant r esearch topic, in this pa p er we fo cus on up date op era tors and do not tackle revisio n any further. F ur ther research showed that, in mos t cases, b elief up date op erators cannot b e directly applied to Descriptio n Lo gic o n tolo gies. The existing work cons ider s o nly ABox up dates, allowing only for static a c y clic TBo xes which are “expanded” befo re the update takes place ( Liu et al. 2006 ), or s ta tic gener al TBoxes in the for m o f int egr it y constr a int s ( Giacomo et al. 200 7 ). The main re asons for these restrictio ns were ex pr essibility and co mputabilit y of the up dated ontology . But w e b elieve there is a mor e fundamental problem with using belie f update op erators to up date TBoxes bec ause it frequently yields counterin tuitive res ults, a s illustrated here: Example 1.1 ( Counterintuitive TBox up date ) Suppo se we wan t to up date the descr iptio n logic TBox T = { B ⊑ A } and we wan t to up date it with the new infor mation U = { C ⊑ B } . In other w or ds, we introduce a new sub concept C of co ncept B . Using Winslett’s up date o per ator we obtain the up dated knowledge base { C ⊑ B , B ⊓ ¬ C ⊑ A } . Thus, the sub concept a xiom from T is severely weakened. Using other op era tors (see ( Herzig and Riﬁ 1999 ) fo r a survey) it ma y even get completely forgo tten. Suc h a for getful b e haviour canno t be explained b y the sole fact that w e are recording a change tha t o ccurr e d in the mo delled environmen t – new sub concepts may arise without disturbing other relations the target concept may have. 4 M. Slota and J. L eite Thu s, appropr ia te ways o f up dating ontologies in gener al, and TBoxes in partic- ular, still need to b e explored and p ose a n imp or tant op en proble m on its own. In our current pap er we follow the men tioned ontology update literature and fo cus on ABox updates , leaving the TBox sta tic thro ughout the upda te pro cess. Updates w ere also in vestigated in the con text of Log ic Progr ams. Earlier ap- proaches based on literal inertia ( Marek and T ruszczynski 199 8 ) prov ed not suﬃ- ciently expressive for dea ling with rule up dates, leading to the development o f rule upda te semantics ba sed on diﬀerent intuitions, pr inciples a nd constructions, when compared to their cla ssical counterparts. F or example, the in tro duction o f the c ausal r eje ction principle ( Leite and Pereira 199 7 ) lead to several appro aches to r ule up- dates ( Alferes et al. 200 0 ; Leite 20 0 3 ; E iter et al. 2002 ; Alferes et al. 200 5 ), all of them with a strong syntactic ﬂav our whic h makes them v er y har d to combine with belie f update op erator s that are semantic in their nature. Other existing approaches to upda tes of Log ic Pr ograms ( Sak ama and Ino ue 2 003 ; Zhang and F o o 2005 ; Delgrande et al. 20 0 8 ) hav e diﬀeren t problems, such as, for example, not be ing immune to tautologica l up- dates. It has b een shown in ( Eiter et al. 2002 ) that the ab ov e men tioned rationality po stulates, set fo r th in the co n text of Classical Logic , are inappro priate for dealing with upda tes of Logic Prog rams. In order to develop a n a ppropriate upda te ope r ator for hybrid knowledge ba ses, one has to so mehow com bine these apparently irr econcilable a pproaches to upda tes, a problem that is far aw ay from having an appro priate so lution. In this pa per , we take an imp ortant ﬁrst step in address ing the up dates of hybrid knowledge bases. F ollowing the state of the art in ontology up dates ( Liu et al. 2006 ; Giacomo et al. 2007 ), w e c ho ose a cons trained scena rio – which is, nev er theles s, rich enough to encompass many practical applications of h ybr id theories – in which only the ABox is allow ed to ev olve, while the TBox is k ept static. W e a dd rule s uppor t to this scena rio by a ugment ing the tra ditional immedia te co nsequence oper ator used in logic prog r amming with the classica l up date op era tor. The resulting framework is signiﬁcantly more expr essive than tho se o f ( Liu et al. 2006 ; Giacomo et al. 2007 ) and a llows for a seamless tw o- way interaction b etw een Logic Pr o gramming rules and Descriptio n Logic a xioms. The co nsequences of rules are also sub ject to up date through the ABox upda tes, making it p ossible to use rules to represent default preferences or b ehaviour and later dire c tly imp ose exceptions to tho se rules. The resulting upda te semantics enjoys several desirable prop erties, namely it: • generalises the stable mo del semantics ( Gelfond and Lifschitz 19 88 ). • generalises, under reas onable as sumptions, the MK NF s emantics for hybrid knowledge bases ( Motik and Rosa ti 2 007 ). • generalises, under rea sonable a ssumptions, the minimal change upda te op er- ator ( Winslett 1990 ). • adheres to the principle of primacy of new informatio n ( Dalal 1988 ), so every mo del resulting from the up date by a n ABox A is a model of A . • is syn tax - independent w.r .t. the TBox and ABox, i.e. yields the sa me r esult with equiv alent TBoxes and when up dating by eq uiv alent ABoxes. T o the b est of our knowledge, this is the ﬁr st pr o po sal of an up date se ma n tics T owar ds Close d World R e asoning in Dynamic Op en Worlds 5 for hybrid knowledge bas es in a sing le framework. This semantics not only provides an appropriate solution to the constr ained scenario we chose, but it unv eils a set of impo rtant issues, op ening the do or for interesting future research endeav ours. The rema inder of this pap er is structured as follows: In Sect. 2 we intro duce the notions needed throug ho ut the rest of the pa per , and discuss some of the choices we made. Section 3 contains the deﬁnition of our op erator while in Sect. 4 w e exa mine its prop erties. In Sect. 5 w e conclude and sketch some directions for future work. 2 Preliminaries In this sectio n we present the necessa ry preliminaries that w e need to deﬁne the hybrid up date op erator , and discuss s ome of the choices we made. As the basis for the formal part o f o ur in vestigation, we cho ose the sa me notation a nd no tions as those used for Hybrid MKNF Knowledge Bas es ( Motik and Rosati 200 7 ). This makes it p o ssible to treat ﬁrst-order formulae and nonmono tonic rules in a uniﬁed manner and also compare our semantics to the one of Hybr id MK NF more easily . 2.1 MKNF The logic of minimal knowledge and negation as failure (MKNF) is an exten- sion o f ﬁrs t- order log ic with tw o mo dal op erator s: K and not . In the follow- ing, w e follow the presentation o f syntax and s emant ics of this logic as given in ( Motik and Rosati 2007 ). W e use a function-free ﬁrst-order syntax extended by the men tioned mo dal o per ators in a natural way . Similarly as in ( Motik and Rosati 200 7 ), we consider only Herbra nd interpretations in our semantics. W e begin with the deﬁnition o f syntax o f MKNF for m ulas . Fir st we need to int ro duce the la ng uage o f MK NF: Deﬁnition 2.1 ( MKNF L anguage ) An MKNF language contains 1. lo gic al c onne ctives ¬ and ∧ ; 2. the quantiﬁer ∃ ; 3. mo dal op er ators K a nd no t ; 4. punctuation symb ols “(”, “)” and “ , ”; 5. a c ount ably inﬁnite set of variables V = { x, X, y , Y , . . . } ; 6. a s et of c onst ant symb ols C = { c , d, . . . } a nd 7. a set of pr e dic ate symb ols P = { P, Q, . . . } , each with a n asso ciated natura l nu mber that we called its arity . Each MKNF la nguage is deter mined by sp ecifying the set o f constant symbols C and the set of predicate s y m b ols P . Such a languag e is denoted by L MKNF ( C , P ). The languag e is alwa ys a ssumed to contain at least o ne pre dic a te symbol and at least one constant symbol. F r om now onw ards, we assume that the MK NF langua ge L = L MKNF ( C , P ) is given a nd use it implicitly in the text b elow. Almost all the deﬁned notions ar e with 6 M. Slota and J. L eite resp ect to this langua ge but we do no t stress this fact in the deﬁnitions. So instead of deﬁning an “ MK NF formula of L ”, w e simply deﬁne an “MKNF form ula”, lea ving out the words “of L ”. Similarly , instead of deﬁning an “MKNF structure ov er L ”, we simply de ﬁne an “ MKNF structur e”, leaving out the words “ov er L ”. Other deﬁnitions follow this pattern as well. F ur thermore, while in the deﬁnitions the notions are deﬁned with their full names (e.g. “MKNF language”, “MKNF for m ula” , . . . ), further in the text w e occasio nally drop the w ord “MK NF” . W e believe these s impliﬁca tions do not cause any confusion while signiﬁcantly improving the re a dability of the text. W e cont inue with the deﬁnitio n of a n MKNF formu la : Deﬁnition 2.2 ( MKNF F ormula ) A term is a v aria ble or a constant. A ﬁrst-or der atom is every expressio n of the form P ( t 1 , t 2 , . . . , t n ) where P is a predica te s ymbol of arity n and each t i is a term. The set of MKNF formulas is the sma llest set satisfying the following conditions: 1. Every ﬁrst-or der ato m is an MKNF formula. 2. If φ, ψ are MKNF formulas and x is a v ariable, then ¬ φ , ( φ ∧ ψ ), ( ∃ x : φ ), K φ a nd not φ are also MKNF formulas. Where it do esn’t cause confusion, the par ent hesis ar e r emov ed for the sake of read- ability . F urther more, ( φ ∨ ψ ), ( φ ⊃ ψ ), ( φ ⊂ ψ ), ( φ ≡ ψ ), true , false and ( ∀ x : φ ) are used as shortcuts for ¬ ( ¬ φ ∧ ¬ ψ ), ( ¬ φ ∨ ψ ), ( φ ∨ ¬ ψ ), ( φ ⊃ ψ ) ∧ ( φ ⊂ ψ ), ( p ∨ ¬ p ), ( p ∧ ¬ p ) a nd ¬ ( ∃ x : ¬ φ ), resp ectively , wher e p is a ﬁxed gro und ﬁrst-or der atom from the language. 1 An MK NF formula of the form K φ is called a mo dal K - atom , a nd a for m ula of the form no t φ is called a mo dal not -atom ; collectively , mo dal K - a nd no t - atoms are called mo dal atoms . An MKNF formula φ is a sentenc e if it has no free v a riable o ccurences; φ is op en if a ll its v aria ble o c c ur ences a r e free; φ is gr ound if it do es not cont ain v aria bles; φ is p ositive if it do es not con tain oc- currences of not ; φ is ﬁrst-or der or obje ctive if it do es no t contain mo dal oper a- tors. By φ [ t 1 /x 1 , t 2 /x 2 , . . . , t n /x n ] we de no te the formula o btained by s im ultane- ously replacing in φ all free o ccurences of the v ariable x i by the term t i for every i ∈ { 1 , 2 , . . . , n } . A se t of MK NF sentences is an MKNF the ory . An MKNF theo r y has prop erty X if all its mem b ers do (for instance, an MKNF theor y is ﬁ rs t-or der if all sentences inside it are ﬁrst-or de r ). Now we can deﬁne the sema ntics of MKNF formulas. W e use Herbrand interpre- tations, a ssuming that apart from the constants from C o ccurring in the formulas, the signa ture contains a coutably inﬁnite supply of co nstants not o ccurring in the 1 As stated in abov e, we assume that at least one predicate sym b ol and at least one constan t symbol exist in the language, f rom which at least one ground ﬁrs t- or der atom can be formed. T owar ds Close d World R e asoning in Dynamic Op en Worlds 7 formulas. The Her brand Universe of suc h a s ignature is denoted by ∆ and has the prop erty C ⊆ ∆. If not stated o therwise, we assume that o ne ﬁxed Herbrand Univ er se ∆ with these prop erties is used as the universe for all interpretations. Deﬁnition 2.3 ( First-Or der Interpr etation and Mo del ) A ﬁrst- or der interpr etation I is a relational structure that contains for every pr ed- icate sym b ol P ∈ P of arit y n a relation P I ⊆ ∆ n . The set of all ﬁrst- o rder int er pr etations is deno ted b y I . Each ﬁr st-order in terpre ta tion determines a unique truth assignment to all ﬁrst- order s en tences. The s atisﬁability of a ﬁr s t-order sentence φ in I is deﬁned induc- tively as follows: 1 ◦ If φ is a ground ﬁrst-or der a tom P ( c 1 , c 2 , . . . , c n ), then φ is true in I if and only if ( c 1 , c 2 , . . . , c n ) ∈ P I ; 2 ◦ If φ is a ﬁr s t-order formula o f the form ¬ ψ , then φ is tr ue in I if and only if ψ is not true in I ; 3 ◦ If φ is a ﬁrst-or der formula of the form φ 1 ∧ φ 2 , then φ is true in I if and only if φ 1 is true in I and φ 2 is true in I ; 4 ◦ If φ is a ﬁrst-orde r formula of the fo r m ( ∃ x : ψ ), then φ is true in I if and only if ψ [ c/x ] is true in I for some constant c ∈ ∆. The fact that φ is true in I is deno ted by I | = φ . A formula φ is false in I if and only if it is not true in I , denoted by I 6| = φ . F or a ﬁrst-order theory S w e say that S is true in I , denoted by I | = S , if I | = φ for each φ ∈ S . O therwise, S is fals e in I , denoted by I 6| = S . If I | = φ , then we say that I is a mo del of φ . Similarly , if I | = S , then I is a mo del of S . The set of all mo dels of φ is denoted by mo d ( φ ). The set of a ll mo dels of S is denoted b y mo d ( S ). The satisﬁability of MKNF form ulas is deﬁned with resp ect to MKNF structures . Deﬁnition 2.4 ( MKNF Structu r e ) An MKNF stru ct ur e is a triple h I , M , N i where I is a ﬁr st-order interpretation and M , N ar e sets of ﬁrs t-order in ter pr etations. 2 Every MKNF s tructure has three compo nen ts. The ﬁrst is a ﬁrst-or der interpre- tation used to interpret the ob jective parts of a formula. The second and thir d ar e sets of ﬁrst-o rder in terpr etations used to interpret the parts of a formula under the K and not moda lit y , resp ectively . Deﬁnition 2.5 ( MKNF Satisﬁability ) Let h I , M , N i b e an MKNF structure. The satisﬁability of an MK NF sentence φ in h I , M , N i is deﬁned inductively a s follows: 1 ◦ If φ is a gr ound ﬁrst-o rder atom P ( c 1 , c 2 , . . . , c n ), then φ is true in h I , M , N i if and only if ( c 1 , c 2 , . . . , c n ) ∈ P I ; 2 In diﬀerence to ( Motik and Rosati 2007 ), we allo w for empt y M , N in this deﬁnition as later on it will be useful to hav e satisﬁabilit y deﬁned even for this marginal case. How eve r , the empty set is still not considered an MKNF int erpretation as can b e seen further in Deﬁnition 2.6 8 M. Slota and J. L eite 2 ◦ If φ is a ﬁrst-order formula o f the form ¬ ψ , then φ is tr ue in h I , M , N i if a nd only if ψ is not true in h I , M , N i ; 3 ◦ If φ is a ﬁrst-or der formula of the for m φ 1 ∧ φ 2 , then φ is true in h I , M , N i if and only if φ 1 is true in h I , M , N i and φ 2 is true in h I , M , N i ; 4 ◦ If φ is a ﬁrs t-order fo r mu la of the form ( ∃ x : ψ ), then φ is true in h I , M , N i if and only if ψ [ c/x ] is true in h I , M , N i for some constant c ∈ ∆; 5 ◦ If φ is a formula of the form K ψ , then φ is true in h I , M , N i if a nd o nly if ψ is true in h J, M , N i for each J ∈ M ; 6 ◦ If φ is a formula o f the form not ψ , then φ is true in h I , M , N i if and only if ψ is not true in h J, M , N i for some J ∈ N . The fact that φ is true in h I , M , N i is denoted by h I , M , N i | = φ . A formula φ is false in h I , M , N i if and only if it is not true in h I , M , N i , denoted by h I , M , N i 6| = φ . Now we are rea dy to intro duce the notions of MK NF in terpr etation and model. Deﬁnition 2.6 ( MKNF Interpr etation and Mo del ) An MKNF interpr etation M is a no n- empt y set of ﬁrst-or der interpretations. By M = 2 I we denote the set of a ll MKNF interpretations tog ether with the empty set. Let φ b e an MKNF sentence, S an MKNF theory and M ∈ M . W e say φ is true in M , denoted b y M | = φ , if h I , M , M i | = φ for each I ∈ M . 3 Otherwise φ is false in M , denoted b y M 6| = φ . S is true in M , denoted b y M | = S , if M | = φ for each φ ∈ S . Otherwise, S is fals e in M , denoted by M 6| = S . If M ∈ M is non-empty 4 , then M is • an S 5 mo del of φ if M | = φ ; • an S 5 mo del of S if M | = S ; • an MKNF mo del of φ if M is an S5 mo del of φ and for every MKNF inter- pretation M ′ ) M there is some I ′ ∈ M ′ such that h I ′ , M ′ , M i 6| = φ ; • an MKNF mo del of S if M is an S5 mo del of S and for every MKNF in- terpretation M ′ ) M there is so me I ′ ∈ M ′ and some φ ∈ S suc h that h I ′ , M ′ , M i 6| = φ . If ther e exists the greatest S5 mo del of φ , then it is deno ted by mo d ( φ ). If φ has no S5 mo del, then mo d ( φ ) denotes the empty set. F or the rest of MKNF sentences, mo d ( · ) stays undeﬁned. If there exists the greatest S5 mo del of S , then it is denoted by mo d ( S ). If S ha s no S5 model, then mo d ( S ) denotes the empty set. F or the rest of MKNF theories, mo d ( · ) stays undeﬁned. 3 Notice that if M is empt y , this condition is v acuously satisﬁed for an y sen tence φ , so any sent ence is true in ∅ . 4 As seen ab o ve, ev ery for mu la i s true i n ∅ , so ∅ is not considered an MKNF inte rpr etation and for the same reason it is neve r given the status of a mo del. T owar ds Close d World R e asoning in Dynamic Op en Worlds 9 2.2 Description L o gics Description Log ics (DLs) ( Baader et al. 20 03 ) are (mostly) decida ble fragments of ﬁrst-order logic that are fr e q uen tly used for knowledge repr esentation in practical applications. In the following we assume that some Description Logic is used to describ e a n ontology . W e do no t cho ose any sp eciﬁc Descr iptio n Log ic , we only assume tha t the ontology expressed in it is comp osed of t wo disting uis hable parts: a TBox w ith c oncept and r ole deﬁnitions using the constructs of the under lying description logic, and an ABox with individual as sertions, i.e. asser tio ns of the form C ( a ) and R ( a, b ) where a, b are constants, C is a concept expressio n and R is a role expression of the un derly ing description lo gic. This distinction is imp or tant to us as we treat the tw o t yp es of knowledge in diﬀerent wa ys – the TBox is considered static while the ABox is a llow ed to evolv e. As was no ted in the introductio n, our main reason fo r this is that we b elieve e x isting upda te op er a tors to b e unsuitable for up da ting co ncept deﬁnitions contained in the TBox. W e also a ssume that the axioms of the underlying DL can b e transla ted in to ﬁrst-o r der logic and for the sake of simplicity w e assume that the TB ox and ABox already cont ain these translations instead of the syntactic co ns tructs of the under lying DL. 2.3 Hybrid MKNF Know le dge Bases W e make use o f the genera l MKNF fra mew o r k to give a semantics to hybrid knowl- edge bases comp osed of an o n tolo g y and a no rmal logic progr am. The fo llowing deﬁnition introduces the notion of a rule as we use it in the fo llowing: Deﬁnition 2.7 ( Rule ) A rule is any o pen MKNF for m ula of the form K p ⊂ K q 1 ∧ K q 2 ∧ · · · ∧ K q k ∧ not s 1 ∧ not s 2 ∧ · · · ∧ not s l (1) where k , l are non-neg ative integers a nd p, q i , s j are ﬁrst-or der atoms for any i ∈ { 1 , 2 , . . . , k } , j ∈ { 1 , 2 , . . . , l } . Given a rule r of the form ( 1 ), the following notation is also deﬁned: H ( r ) = K p , H ∗ ( r ) = p , B + ( r ) = { K q 1 , K q 2 , . . . , K q k } , B − ( r ) = { no t s 1 , no t s 2 , . . . , not s l } , B ( r ) = B + ( r ) ∪ B − ( r ) . H ( r ) is dubb ed the he ad of r , H ∗ ( r ) the ﬁ rst-or der he ad of r , B + ( r ) the p ositive b o dy of r , B − ( r ) the ne gative b o dy of r and B ( r ) the b o dy of r . A rule r is called deﬁnite if its negative b o dy is empty . A r ule r is ca lled a fact if its b o dy is empty . A pr o gr am is a set of rules. A deﬁnite pr o gr am is a s et of deﬁnite rules. As was shown in ( Lifschitz 19 9 1 ), the MKNF semantics genera lis es the stable mo del semantics for logic pro g rams. In particular, every lo gic progr amming rule of 10 M. Slota and J. L eite the form p ← q 1 , q 2 , . . . , q k , n ot s 1 , n ot s 2 , . . . , not s l . can be translated into the MKNF for m ula ( 1 ) and the sta ble mo dels o f sets of such rules (i.e. o f norma l logic prog rams) directly corre s po nd to MKNF mo dels of the set of translated rules. W e are now ready to deﬁne a hybrid knowledge base and its semantics. Deﬁnition 2.8 ( Hybrid know le dge b ase ) Let O b e an o n tolo gy and P a progra m. The pair K = hO , P i is then called a hybrid know le dge b ase . W e say K is deﬁnite if P is deﬁnite and we say K is P -gr ound if P is ground. The s emantics of hybrid k nowledge bases is given in terms of a translatio n π into a set of MKNF formulas which is deﬁned a s follows: Deﬁnition 2.9 F o r an ontology O , a rule r with the vector of free v ariables x , a progr am P a nd the hybrid knowledge base K = hO , P i , we deﬁne: π ( O ) = { K φ | φ ∈ O } , π ( r ) = ( ∀ x : r ) , π ( P ) = { π ( r ) | r ∈ P } , π ( K ) = π ( O ) ∪ π ( P ) . W e say an MKNF interpretation M is an S 5 mo del of K if M is an S5 mo del of π ( K ). W e say M is an MKNF mo del of K if M is an MK NF mo del of π ( K ). In this pap er, we are not co ncerned with decidability of rea soning, s o w e refra in from introducing a safety condition on our rules a s was done in ( Motik and Rosati 200 7 ). 2.4 Classic al Up dates As a basis for our up date op era tor, we adopt an up date semantics called the minimal change up date semantics (sometimes also called the p ossible mo dels ap- pr o ach (PMA)) as deﬁned in ( Winslett 1990 ) for upda ting ﬁrst-order theor ies. There are a num be r of reasons for this c hoice . First, it satisﬁes all of Ka tsuno and Mendelzon’s upda te po stulates ( Katsuno and Mendelzon 19 91 ). This means, for insta nc e , that unlik e some other up date semantics, such as the standar d seman- tics ( Winslett 1990 ), it is not sensitive to syntax of the o riginal theory or of the upda te. Seco nd, it is based on an intuitiv e ide a , tr eating ea c h classical model of the original theory as a po ssible world and modifying it as little as p ossible in order to bec ome consistent with the new informatio n. This ide a ha s its ro ots in reas oning ab out action ( Winslett 198 8 ) and upda tes of relational theories ( Winslett 19 90 ). Third, the op erator has already b een successfully used to deal with ABox up dates ( Liu et al. 2006 ; Giacomo et al. 2007 ). This semantics uses a notion of closeness of ﬁrst-or der interpretations w.r .t. a ﬁxed ﬁrst-or der in terpr etation I . T his notion is ba sed on the set o f g round ﬁrst- order atoms that are int er pr eted diﬀeren tly than in I . T owar ds Close d World R e asoning in Dynamic Op en Worlds 11 Deﬁnition 2.10 ( Int erpr etation distanc e ) Let P be a predica te symbol and I , J be ﬁrst-order interpretations. The di ﬀer enc e in the interpr etation of P b etwe en I and J , written di ﬀ ( P , I , J ), is a r elation containing the set of tuples ( P I \ P J ) ∪ ( P J \ P I ). Given ﬁrst-o rder interpretations I , J, J ′ , we say that J is at least a s close to I as J ′ , denoted by J ≤ I J ′ , if for every pre dic a te symbol P it holds that diﬀ ( P, I , J ) is a subset of diﬀ ( P , I , J ′ ). W e also say that J is closer to I than J ′ , denoted by J < I J ′ , if J ≤ I J ′ and J ′  I J . W e now give a deﬁnition of the minimal c hange up date semantics but in diﬀerence to ( Winslett 199 0 ), we use a sp eciﬁc vo cabulary which is closer to the setting of this pap er. In par ticular, we deﬁne the s e ma n tics o f up dating an initial theory S b y an ABox A in the context of the TBox T . The TBox is trea ted as static integrity constraints for the w ho le up date pro cess. The minimal change update semantics chooses those models of T ∪ A tha t are the closest w.r.t. the relation ≤ I to some mo del I of T ∪ S . F or ma lly: Deﬁnition 2.11 ( Winslett ’s minimal change up date semantics ) Let S b e a ﬁrst-orde r theory , T a TBox, A an ABox, I a ﬁrst-or der interpretation and M a set of ﬁrs t- o rder interpretations. W e deﬁne: incorporate T ( A , I ) = { J ∈ mo d ( T ∪ A ) | ( ∄ J ′ ∈ mo d ( T ∪ A ))( J ′ < I J ) } , incorporate T ( A , M ) = [ I ∈ M incorporate T ( A , I ) , mo d ( S ⊕ T A ) = i ncorporate T ( A , mo d ( T ∪ S )) . If mo d ( S ⊕ T A ) is nonempty , we call it the minimal change up date mo del of S ⊕ T A . The previous deﬁnition ca n b e na turally gener alised to a llow for s equences of ABoxes. Star ting from the models of the or iginal theory , for each ABox in the sequence w e transform the set of mo dels a ccording to the minimal change up date semantics deﬁned ab ove. The resulting set of mo dels then determines the up dated theory . F o rmally: Deﬁnition 2.12 ( Up date by a se quenc e of ABoxes ) Let S be a ﬁrst-order theory , T a TBox, A = ( A 1 , A 2 , . . . , A n ) a sequence of ABoxes and M a set of ﬁrs t- o rder interpretations. W e inductively deﬁne: incorporate T ( A , M ) = incorporate T (( A 2 , . . . , A n ) , in corporate T ( A 1 , M )) , mo d ( S ⊕ T A ) = incorporate T ( A , mo d ( T ∪ S )) . If mo d ( S ⊕ T A ) is nonempty , we call it the minimal change up date mo del of S ⊕ T A . 3 Hybrid Up date Op erator T ur ning to the formal part of our prop osal, our aim is to prop ose a semantics for a prog ram P up dated b y a sequence of ABoxes ( A 1 , A 2 , . . . , A n ) in the context of a TBox T . W e assume pro gram P to b e ﬁnite and gro und, a common a ssumption when dealing with reasoning under the stable mo del sema n tics. 12 M. Slota and J. L eite W e follow a path similar to how the stable mo dels o f no r mal log ic pr ograms were originally deﬁned ( Gelfond and Lifschitz 19 88 ), and start b y deﬁning how a deﬁnite progra m can b e upda ted by a sequence of ABoxes, a nd o nly afterwards dea l w ith progra ms containing default neg ation. As with the lea st mo del of a deﬁnite logic progr am, our resulting mo del is the least ﬁxed po int of a n immedia te consequence op era tor. Our o p er a tor is in a wa y similar to the usua l immediate consequence o per ator T P commonly used to draw consequences from a logic pro gram P . The crucial diﬀerence b etw een T P and our op erator is that in the latter, the consequences ar e subsequently up dated b y the sequence of ABoxes A using the clas s ical up date op er a tor. F or mally: Deﬁnition 3.1 ( Up dating imme diate c onse quenc e op er ator T P ⊕ T A ) Let P b e a ﬁnite ground deﬁnite program, T a TBox and A a sequence of ABoxes. W e deﬁne the o per ator T P ⊕ T A for any M ⊆ I as follows 5 : T P ⊕ T A ( M ) = mo d ( { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ⊕ T A ) An imp or tant prop erty of an immediate conseq uence op era tor is c ontinu ity b e- cause it guara ntees the ex istence of a lea s t ﬁxed p oint and also provides a way of computing this least ﬁxed po in t (using the K leene Fixed Point Theorem). The T P ⊕ T A op erator satisﬁes the condition of contin uity: Pr op osition 3.2 ( Continuity of T P ⊕ T A ) Let P b e a ﬁnite ground deﬁnite program, T a TBox and A a sequence of ABoxes. Then T P ⊕ T A is a cont inuous function on the complete partial order ( M , ⊇ ). Pr o of See Appendix Appendix D , page 34 . Now w e can deﬁne a minimal change dynamic stable mo del of P ⊕ T A , wher e P is a deﬁnite progra m, as the least ﬁxed p oint o f T P ⊕ T A : Deﬁnition 3.3 ( Minimal change dynamic stable m o del for deﬁnite pr o gr ams ) Let P b e a ﬁnite ground deﬁnite program, T a TBox and A a sequence of ABoxes. W e say a n MKNF interpretation M is a minimal change dynamic stable mo del of P ⊕ T A if it is the least ﬁxed p oint of T P ⊕ T A . Notice that for every de ﬁnite pr ogram P and each sequence of ABo xes A , P ⊕ T A has either no minimal change dynamic stable model (when the least ﬁxed p oint of T P ⊕ T A is empt y), or ex actly one minimal change dynamic stable mo del. In order to deal with default negatio n in the b o dies of rules , w e use the Gelfond- Lifschitz tr ansformation which w as us ed to deﬁne the stable mo dels of a no r mal logic progra m ( Gelfond and Lifschitz 1988 ). W e do this by deﬁning the deﬁnite pro gram P M which is the r esult of p erforming the Gelfond-Lifschitz transforma tion on P – rules from P with a negative bo dy that is in conﬂict with M ar e disca r ded, while 5 Recall that M | = B ( r ) holds if and only if M is an S5 m odel of every mo dal atom in B ( r ) (see also D ef . 2.6 ). T owar ds Close d World R e asoning in Dynamic Op en Worlds 13 for all the other rules, their negative b o die s are discar ded. Then P M is up dated by A using the above deﬁnition for deﬁnite logic programs and if the result is identical to M , then M is g iven the s ta tus of a minimal change dynamic stable mo del . Hence, the resulting op era to r can b e used to up date an a rbitrary normal logic pro gram b y a sequence of ABoxes. Deﬁnition 3.4 ( Minimal change dynamic stable m o del ) Let P b e a ﬁnite ground progr am, T a TB ox, A a sequence o f ABoxes and M an MKNF interpretation. W e say M is a minimal change dynamic stable mo del of P ⊕ T A if M is a minimal change dy na mic s table mo del of P M ⊕ T A where P M =  H ( r ) ⊂ B + ( r )   r ∈ P ∧ M | = B − ( r )  . The minimal change dynamic stable mo dels can b e used to deﬁne a consequence relation from P ⊕ T A where P is a ﬁnite g round pr ogram, T is a TBox and A a sequence of ABoxes. W e o ﬀer a deﬁnition which adopts a skeptical a pproach to inference, credulous and other deﬁnitions may b e o btained s imilarly . Deﬁnition 3.5 ( Conse quenc e r elation ) Let P b e a ﬁnite g round pro gram, T a TBox, A a sequence of ABoxes and φ an MKNF sentence. W e say that P ⊕ T A entails φ , written P ⊕ T A | = φ , if a nd only if M | = φ fo r a ll minimal change dynamic stable mo dels M of P ⊕ T A . W e no w demonstra te the deﬁned up date sema n tics on a simple example: Example 3.6 Consider the following TBox T a nd progra m P : T : A ≡ B ⊔ C (2) Ne gA ≡ ¬ A (3) D ≡ ¬ A ⊓ ∃ P − .A (4) P : Ne gA ( X ) ← not A ( X ) . (5) P ( X, Y ) ← A ( X ) , E ( Y ) , not E ( X ) . (6) TBox a ssertions ( 2 ) and ( 3 ) together with rule ( 5 ) deﬁne the concept A as a union of concepts B and C and they make this concept interpreted under CW A instead o f OW A, i.e. whenever for so me co nstant c we cannot c o nclude that A ( c ) is true, the rule ( 5 ) infers Ne gA ( c ) a nd by ( 3 ) w e obtain ¬ A ( c ). Assertion ( 4 ) deﬁnes concept D as tho se members d of ¬ A for which there exists s ome c from A with P ( c, d ). Rule ( 6 ) infers the rela tion P ( c, d ) whenev er c is in A but no t in E a nd d is in E . Given the initial deﬁnitions, a n up date by A 1 = { A ( c ) } now yields 6 P ⊕ T A 1 | = { A ( c ) , ¬ A ( d ) } . A further update by A 2 = { ¬ B ( c ) } intro duces a p os sibilit y o f A ( c ) no t b eing tr ue 6 In the example we assume that the rules are grounded using all constan ts explicitly ment ioned in the kno wledge base. In this case there are only t wo: c and d . 14 M. Slota and J. L eite in case B ( c ) was true b efore and C ( c ) was false. Since A is in terpr eted under the closed world assumption, we can now conclude that A ( c ) is false: P ⊕ T ( A 1 , A 2 ) | = { ¬ A ( c ) , ¬ B ( c ) , ¬ A ( d ) } Consider now the up date A 3 = { C ( c ) ∧ E ( d ) } . Given ( 2 ), this r einstates A ( c ). F ur thermore, rule ( 6 ) can now infer P ( c, d ) and by ( 3 ) we obtain D ( d ): P ⊕ T ( A 1 , A 2 , A 3 ) | = { A ( c ) , ¬ B ( c ) , C ( c ) , ¬ A ( d ) , E ( d ) , P ( c, d ) , D ( d ) } In the next up date A 4 = { E ( c ) } w e block the bo dy of rule ( 6 ), which also preven ts D ( d ) from being inferred: P ⊕ T ( A 1 , A 2 , A 3 , A 4 ) | = { A ( c ) , ¬ B ( c ) , C ( c ) , ¬ A ( d ) , E ( d ) , E ( c ) } The last upda te 7 A 5 = { ¬ E ( c ) ∧ ¬ P ( c, d ) } illustr ates ho w the conclusion of a r ule may be ov erridden through the ABox upda tes – thoug h the b o dy of rule ( 6 ) is true, its head do es not b ecome true since it is in direct conﬂict with A 5 : P ⊕ T ( A 1 , A 2 , A 3 , A 4 , A 5 ) | = { A ( c ) , ¬ B ( c ) , C ( c ) , ¬ A ( d ) , E ( d ) , ¬ E ( c ) , ¬ P ( c, d ) } 4 Properti es and Relatio ns In this s e ction we in vestigate a num ber of for mal prop erties of the deﬁned op era tor. The ﬁrst pro per t y g uarantees that e very minimal change dynamic sta ble mo del of P ⊕ T A is a mo del o f A . This is known as the principle of primacy of new i nformation ( Dalal 1988 ). Pr op osition 4.1 ( Primacy of n ew information ) Let P b e a ﬁnite gr ound pro g ram, T a TBox, A an ABox and M a minimal change dynamic stable mo del of P ⊕ T A . Then M | = A . Pr o of See Appendix Appendix D , page 35 . The se c ond prop erty g uarantees tha t our op erator is syntax-indep endent w.r.t. the TBox and the updating ABox. This is a desirable pr op erty as it shows tha t providing equiv alent TBoxes and up dating by equiv alent ABoxes always pro duces the s ame result. It is inherited from the classic al minimal change up date op erator. Pr op osition 4.2 ( Syntax indep endenc e ) Let P be a ﬁnite gr ound prog ram, T , T ′ be TBoxes such that mo d ( T ) = mo d ( T ′ ), A , A ′ be ABoxes suc h that mo d ( A ) = mo d ( A ′ ) and M be an MKNF interpretation. Then M is a minimal change dynamic stable mo del of P ⊕ T A if and only if M is a minimal change dynamic stable mo del of P ⊕ T ′ A ′ . 7 Updating ABo xes could, of course, b e more complex since arbitrary concept expressions may be used (e.g. ( ∃ P .C )( c )). H ere, due to limited space, we keep the example ve ry simple. T owar ds Close d World R e asoning in Dynamic Op en Worlds 15 Pr o of See Appendix Appendix D , pa ge 35 . The follo wing prop ositio n relates the h ybrid upda te op erator to the static MKNF semantics of h ybrid knowledge bas es. It gives suﬃcient conditions for the static and dynamic semantics to coincide. In particular, the suﬃcient condition r equires that for any set of consequences S of pr ogram P in the context of a mo del M , up dating S by A in the context of T has the same eﬀect as mak ing a n intersection o f the mo dels of S with the mo dels of A and T . Pr op osition 4.3 ( R elation t o Hybrid MKNF ) Let P b e a ﬁnite ground prog ram, O = T ∪ A a n ontology with TBox T and ABox A and M an MKNF interpretation such tha t for every subset S of the s et { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } the following condition is satisﬁed: mo d ( S ⊕ T A ) = mo d ( S ∪ O ) . Then M is an MKNF model of hO , P i if and only if M is a minimal c hange dynamic stable mo del of P ⊕ T A . Pr o of See Appendix Appendix D , pa ge 36 . The precondition of this prop osition is satisﬁed, for example, when predicates app earing in heads of P do not app ear in the ontology O . An imp ortant sub case of this is when O is empt y beca use then the prop osition implies that the minimal change dynamic sta ble mo dels of P ⊕ ∅ ∅ are exactly the MK NF mo dels of P . Since the MKNF s emantics g eneralises the sta ble mo del s emant ics ( Lifschitz 19 9 1 ), the minimal change dynamic stable models of P ⊕ ∅ ∅ a ls o co incide with the stable mo dels of P . In other words, our o per ator prop erly g eneralises stable mo dels. Cor ol lary 4.4 ( Gener alisatio n of stable mo dels ) Let P b e a ﬁnite g round prog ram. Then M is a stable mo del of P if and o nly if M is a minimal change dynamic stable mo del of P ⊕ ∅ ∅ . Pr o of See Appendix Appendix D , page 37 . T ur ning to relatio ns with the minimal change up date op erator, we s how that up- dating a n y logic progr a m that can be equiv alently tr a nslated into ﬁrs t-order lo gic has the same eﬀect as up dating the transla ted ﬁrst-o rder theory using the mini- mal change update op era to r. Hence, our update op era tor generalis es the classica l minimal c hang e up date op er a tor. Pr op osition 4.5 ( Gener alisation of the minimal change up date op er ator ) Let P b e a ﬁnite gro und prog ram containing only fac ts , T a TB ox, A a seq uence of ABoxes and M an MK NF in terpr etation. Then M is a minimal change dynamic stable mo del of P ⊕ T A if and only if M is a minimal change up date mo del of S P ⊕ T A where S P = { p | K p ∈ P } . 16 M. Slota and J. L eite Pr o of See Appendix Appendix D , page 38 . Another prop erty that o ur o pe r ator inher its fro m the classical minimal change upda te op erator is that empt y ABoxes in the up dating sequence do not inﬂuence the resulting mo dels. Simila rly , upda ting an empty progra m simply yields the set of all ﬁrst-order mo dels of T ∪ A . These last t w o prop erties ensure that empt y program and updates cannot inﬂuence the resulting mo dels under our up date op era tor 8 . Pr op osition 4.6 ( Indiﬀer en c e to empty up dates ) Let P b e a ﬁnite gro und pr ogram, T b e a TBox and A = ( A 1 , A 2 , . . . , A n ) a sequence of ABoxes (where n ≥ 1). Let A ′ = ( A 1 , A 2 , . . . , A i − 1 , A i , ∅ , A i +1 , . . . , A n ) for some i ∈ { 0 , 1 , 2 , . . . , n } . Then an MKNF interpretation M is a minimal change dynamic stable mo del o f P ⊕ T A if and only if M is a minimal change dynamic stable mo del of P ⊕ T A ′ . Pr o of See Appendix Appendix D , page 37 . Pr o of See Appendix Appendix D , page 40 . Pr op osition 4.7 ( Up dating an empty pr o gr am ) Let T be a TBox, A an ABox and M a n MKNF int er pretation. Then M is a minimal c hang e dyna mic stable model of ∅ ⊕ T A if and only if M = mo d ( T ∪ A ). R elation t o Katsu n o and Mendelzon ’s p ostulates In the following we brieﬂy discus s the rela tion of our op era tor to Katsuno a nd Mendelzon’s p ostulates for up dates of prop ositiona l knowledge ba s es formulated in ( Katsuno and Mendelzon 19 91 ). E a ch propo sitional knowledge base ov er a ﬁnite language can be represen ted by a s ing le prop ositiona l formula and the result o f the up date ca n also be represented as a prop ositional formula. The eight desira ble prop erties of an upda te ope r ator ⋄ are as follows: KM 1 : φ ⋄ ψ implies ψ . KM 2 : If φ implies ψ , then φ ⋄ ψ is equiv alent to φ . KM 3 : If b oth φ a nd ψ are satisﬁable, then φ ⋄ ψ is satisﬁa ble. KM 4 : If φ 1 is equiv alent to φ 2 and ψ 1 is equiv alent to ψ 2 , then φ 1 ⋄ ψ 1 is equiv a lent to φ 2 ⋄ ψ 2 . KM 5 : ( φ ⋄ ψ ) ∧ χ implies φ ⋄ ( ψ ∧ χ ). 8 Pe rhaps surpr isingly , as sho wn in ( Leite 2003 ), these t wo pr operties are vi ol ated b y man y up date operators in the cont ext of Logic Programming. T owar ds Close d World R e asoning in Dynamic Op en Worlds 17 KM 6 : If φ ⋄ ψ 1 implies ψ 2 and φ ⋄ ψ 2 implies ψ 1 , then φ ⋄ ψ 1 is equiv alent to φ ⋄ ψ 2 . KM 7 : If for eac h atom p either φ implies p or φ implies ¬ p , then ( φ ⋄ ψ 1 ) ∧ ( φ ⋄ ψ 2 ) implies φ ⋄ ( ψ 1 ∨ ψ 2 ). KM 8 : ( φ 1 ∨ φ 2 ) ⋄ ψ is equiv alent to ( φ 1 ⋄ ψ ) ∨ ( φ 2 ⋄ ψ ). In or der to examine these p ostulates in o ur setting, we restrict o ur attention to a ﬁnite prop ositional language. In order to interpret the po stulates in our setting, we need to deﬁne the se mant ics of a num b er of notions used in them. Let P , P 1 , P 2 be programs, T a TBox and α, α 1 , α 2 be prop ositional form ulae repr esenting ABox upda tes. W e need to discuss and deﬁne, at least: 1. When doe s P ⊕ T α 1 imply α 2 ? (used in KM 1 and KM 6) 2. When doe s P imply α ? (used in KM 2 and KM 7) 3. When is P 1 ⊕ T α equiv alent to P 2 ? (used in KM 2) 4. When is P sa tisﬁable? (used in KM 3) 5. When is P ⊕ T α satisﬁable? (used in KM 3) 6. When is P 1 equiv a lent to P 2 ? (used in KM 4) 7. When is P 1 ⊕ T α 1 equiv a lent to P 2 ⊕ T α 2 ? (used in KM 4 and KM 6) 8. What is the semantics of ( P ⊕ T α 1 ) ∧ α 2 ? (used in KM 5) 9. What is the semantics of ( P ⊕ T α 1 ) ∧ ( P ⊕ T α 2 )? (used in KM 7) 10. What is the semantics of P 1 ∨ P 2 ? (used in KM 8) Most of these ques tions can b e answered in mult iple diﬀerent w ays while some of them are hard to provide answers to at all. In the following, we s ug gest wa ys o f answering most o f these questions and then analyse whether our op e rator satisﬁes the corres po nding po stulates. Question 1. can b e ans w er ed using the conseq uence relation fr om Def. 3.5 . A similar consequence r elation can b e deﬁned using sta ble mo dels to ans w er questio n 2. A simple answer to que s tion 3. is to say that P 1 ⊕ T α is eq uiv alent to P 2 if the set of minimal change dynamic stable mo dels of P 1 ⊕ T α is equal to the set of stable mo dels of P 2 . Regar ding questions 4 . and 5., we ca n say that P is sa tisﬁable if it has a t least one stable mo del a nd P ⊕ T α is satisﬁable if it has at least one minimal change dynamic stable mo del. Que s tion 6. can b e answered similarly as questio n 3. by comparing the sets of minimal change dynamic stable mo dels o f P ⊕ T α 1 and P ⊕ T α 2 . Fina lly , ques tion 7. can be answered by comparing the s ets of stable mo dels of P 1 and P 2 or by us ing strong equiv alence ( Lifschitz et al. 20 01 ). Providing reasona ble answers to the remaining questions requires more inv estigation, so, for now, we do not further examine p ostulates K M 5 , KM 7 and KM 8. T ur ning to the res t of the p ostulates, we note that our op era tor adheres to KM 1 , which was pr oved in Pro p os ition 4.1 . The same is no t the case with p os tulate KM 2, as shown b y the following counterexample. Consider the pro gram P : p ← n ot q . r ← q , not r . q ← n ot p. r ← p. (7) and a n up date α = r . The only s table mo del of P is the maximal S5 mo del M of 18 M. Slota and J. L eite { p, r } . Clearly , M | = α . But P ⊕ T α has ano ther minimal change dynamic sta ble M ′ , which is the maxima l S5 mo del of { q , r } and so is not equiv alent to P . In fact, this behaviour is inherited from the stable se ma n tics for log ic pro grams which doe s not satisfy the very similar prop erty of cu m ulativity ( Makinson 198 8 ; Dix 1995 ). Hence, it is expecta ble that KM 2 is never sa tisﬁed by an y update semantics that prop erly gener a lises the stable mo del semantics. A similar situa tio n arises with p os tula te KM 3 b ecause the stable mo del semantics allows to express in tegrity constra ints, and these ma y easily b e brok en by an update. F o r ex a mple, the prog r am P = { p ← q , n ot p. } , upda ted by α = q , of whic h bo th are sa tisﬁable, do es not allow for any minimal c hang e dynamic s table mo del. It is not clear how an int eg rity constrain t should b e updated beca use, once it is a part of the knowledge base, which is assumed to b e a cor rect repres e n tation of the w o r ld, it should not be vio lated, and no new information should hav e the p ow er to ov er ride it. Or should it? That is another op en resear ch question w or th inv estig a ting. Postulate K M 4 is partially form ulated in Pro po sition 4.2 , whic h sho ws that upda ting by equiv a len t ABoxes pro duces the same r esult. The o ther half a mounts to pro ving that up dating equiv alent logic programs by the same ABo x also pro duce s equiv a lent re s ults. F or the tw o notions of progra m equiv alence that we pro po sed ab ov e, this prop erty do es not hold. As a co un terex a mple take P 1 = { p., q . } and P 2 = { p., q ← p. } which hav e the same ans wer se ts and ar e also s trongly equiv alent. An update by α = ¬ p , pro duces diﬀerent results for P 1 and P 2 , resp ectively , which we be lieve is in ac cord with in tuitions r egarding these tw o programs. It may b e the case that for diﬀerent no tions of prog r am equiv alence that better suit our scenar io, such as the up date equiv alence o f logic prog rams prop osed in ( Leite 2003 ), this prop erty holds. F ur ther inv estigation is needed to answer this q uestion. Finally , postula te KM 6 is also not satisﬁed by the op erato r. As a co un tere x - ample we can ta ke the progra m P deﬁned in ( 7 ), α 1 = r a nd α 2 = p ∨ q . Then P ⊕ T α 1 has tw o minimal change dyna mic stable mo dels: M 1 = mo d ( { p, r } ) and M 2 = mo d ( { q , r } ). Hence, P ⊕ T α 1 | = α 2 . F urthermore , P ⊕ T α 2 has only one min- imal change dynamic stable mo del which is M 1 and conseq ue ntly P ⊕ T α 2 | = α 1 . How ever, P ⊕ T α 1 is not equiv alent to P ⊕ T α 2 . 5 Conclusion and F uture W ork As seen, our o per ator pro per ly genera lises the tw o main ingr edien ts that it is mo- tiv ated by – the stable mo del se ma n tics of normal lo g ic prog rams (Corollary 4.4 ) and the minimal ch a ng e up date ope rator (Pro po s ition 4.5 ). The failure of our op- erator to satisfy many of Ka ts uno and Mendelzon’s p ostulates is not sur prising. A wide r ange of classic a l up date and revision po stulates was already studied in the context of rule up dates, only to ﬁnd that many o f them were inappro priate for characterising plausible rule up date o p er a tors ( Eiter et al. 200 2 ). F urthermor e, in ( Slota and Leite 2010 ) we show that even under the SE model semantics, which is strictly more expr essive than stable mo dels semantics, upda te op erator s satisfying only so me of the basic K atsuno a nd Mendelzon’s p ostulates necessar ily violate the prop erty of suppor t whic h is at the core of mos t logic pr ogramming semantics. The T owar ds Close d World R e asoning in Dynamic Op en Worlds 19 search for desirable pr op erties of hybrid upda te op era tors is an interesting future resear ch area. There are als o ma ny more prop erties still to b e examined, amo ng them decid- ability as well as complexity of re a soning. Since we cannot ex pect the op erato r to per form any better than the stable model semantics a nd the cla ssical up date o pe r - ator it is based on, its tra ctable approximations need to b e deﬁned and examined. The w ell-founded semant ics for log ic prog rams ( Gelder et al. 199 1 ) and its version for hybrid MKNF knowledge bases ( Alferes et al. 200 9 ) constitute crucia l starting po in ts. The re c e n t research on ontology evolution (s e e ( Flouris et al. 2008 ) for a survey) can help des ign tra ctable up date o per ators which, at the same time, oﬀer the necessar y functionalit y to b e interesting for use in practice . In this pap er, the TBox was considered static and was treated in the same wa y as integrity constrain ts in ( Winslett 1990 ). This appro ach to handling in- tegrity co nstraints in the co n text of up dates has b een criticized in the literature ( Herzig and Riﬁ 1999 ; Herzig 2 005 ), as in certain cases it do es not provide the exp ected results. How ever, the prop osed s o lutions are deﬁned only for the prop osi- tional c a se a nd a pr eliminary exa mination show ed that their treatment of equiv a- lences, such as the TBox deﬁnitions used in E xample 3.6 , is not alw ays the exp ected one. F ur ther inv estigation is needed to ﬁnd suitable s o lutions to these problems in the co nt ext of ontology up dates. F ur ther more, in truly dynamic environments, the TBox should also b e allow ed to b e up dated. W e b elieve that ﬁnding appropr iate upda te op erators for ont olo gies is still a la rgely op en research question. The lar ge b o dy o f work on rule up dates ( Leite 2003 ; Alferes et al. 2005 ), and more r e cen tly ( Delgrande et al. 200 8 ), also needs to b e exploited in the attempts to deﬁne an update op era tor that c a n deal with the evolution of bo th r ules and ontologies. Finally , while incorp orating new knowledge in a knowledge base is impo rtant, the complementary task of removing a certain piece of informatio n is also imp or- tant. Hence, h ybr id erasure op erato rs sho uld be studied a nd related to h ybrid up- date op e r ators. The w or k on eras ure ( Giacomo et al. 200 7 ) in description logics as well as forgetting in b oth descriptio n logics ( W a ng et al. 2009 ) and logic progra ms ( Eiter and W a ng 2008 ) should b e the starting p oints of this r esearch. T o conclude, in this pa per , to the b est o f our knowledge, we pr op osed the ﬁr st upda te op erator for hybrid knowledge bas es. W e deal with a constrained but inter- esting scena rio in whic h a TBox and no nmonotonic r ules repr esent s tatic knowledge, po licies, norms and default prefere nc e s , and the evolving ABox represents the op en and dynamic en vir onment. W e illustrated the b ehaviour o f our oper ator on a simple example. The op era tor can be used in realistic scenarios where the general notions and rules are r e latively ﬁxed, and individuals tend to change their state frequently . This is the case o f many real life institutions where stakeholders c hang e their state on a regular basis while the gener al rules and structur e s change only o ccasiona lly . W e prov ed a num b er of prop erties of our op erator , a mong which its relations with the theo r ies it was based on, such as the s ta ble mo del semantics for logic pr o- grams ( Gelfond and Lifschitz 19 88 ), the MKNF seman tics f or h ybrid knowledge 20 M. Slota and J. L eite bases ( Motik and Rosati 200 7 ) and Winslett’s minimal change update op erator ( Winslett 1990 ). W e b elieve that this new area of resear ch brings exciting new problems to solve and br idg es a num b er of exis ting res earch ar eas. 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Reasoning ab out action using a p ossible mod els approac h. In Pr o- c e e dings of the 7th National Confer enc e on Artiﬁcial I ntel ligenc e (AAAI 1988) . A AAI Press / The MIT Press, Saint P aul, MN, US A, 89–93. Winslett, M. 1990. Up dating L o gic al Datab ases . Cambridge Un ive rsity Press, New Y ork, NY, USA. Zhang, Y. and F oo, N. Y. 200 5. A uniﬁed framew ork for representing lo gic program up- dates. In Pr o c e e dings of the 20th National Confer enc e on Artiﬁcial Intel ligenc e (AAAI 2005) , M. M. V eloso and S . Kambhampati, Ed s. AAAI Press / The MIT Press, Pitts- burgh, Pennsylv ania, USA, 707–713. App endix A Klee ne Fixed Poin t Theorem Fixed p oints play a n imp ortant role in many of the inv estigations in the a rea of logic programming. Man y semantics o f lo gic progra ms are deﬁned b y a ﬁxed p oint equation, meaning that in order for an interpretation M to b e considered a “go o d” mo del of a logic progra m, it m ust satisfy some equation of the form M = f ( M ) where f is a mapping from in terpr etations to interpretations, also ca lled a n op er a- tor . Such ope rators were heavily studied in Or der Theory and Kleene Fixed Poin t Theorem is one of its basic results. Informally , it s ta tes that the least ﬁxed po in t of a contin uous op era to r ca n b e computed b y itera ting the op era to r. It is heavily used in logic progr amming. F o r the sake of self-containedness, this App endix intro duces the basic no tions of Or der Theo ry necessa r y to for mally state and pr ov e the Kleene Fixed P oint Theorem. F or an elab ora te s tudy of this to pic with many further references, we refer the reader to ( Dav ey and Priestley 1990 ). The ﬁrst deﬁnition is of a partially ordered set, under whic h w e mea n any set with an asso ciated r elation “ ≤ ” that ca n b e use d to compar e elements of this set. This relation is required to o be y cer tain prop erties that can b e na turally ex pected from any such order ing r e lation. Deﬁnition A.1 ( Partial Or der ) A p artial or der is a pair ( P , ≤ ) where P is a set and ≤ is a re ﬂex ive, an tisymetric and transitive relation o ver P , i.e. the following conditions a re satisﬁed for all a, b, c ∈ P : a ≤ a ( a ≤ b ∧ b ≤ a ) ⇒ a = b ( a ≤ b ∧ b ≤ c ) ⇒ a ≤ c W e also s ay that P is a pa r tially o r dered s e t (w.r.t. ≤ ). T owar ds Close d World R e asoning in Dynamic Op en Worlds 23 In log ic progr amming, the se t of int er pretations usually forms a partial or der that is usually or de r ed b y the subset relation. In ca se of MKNF interpretations, the partial order is determined by the sup erset rela tion. The following deﬁnitions intro duce the lea st and gr e a test elements and low er and upper b ounds o f a subset of a partially order ed set. Deﬁnition A.2 ( L e ast and Gr e atest Element ) Let P be a par tially order ed set, S ⊆ P and a ∈ S . Then a is the le ast element of S if for every b ∈ S it holds that a ≤ b , and a is the gr e atest element of S if for every b ∈ S it holds that b ≤ a . Deﬁnition A.3 ( L ower and U pp er Bound ) Let P be a partia lly ordered set, S ⊆ P a nd a ∈ P . Then a is a lower b ound of S if for every b ∈ S it holds that a ≤ b , a nd a is an upp er b oun d of S if for every b ∈ S it holds that b ≤ a . Combining the previous notions, w e obtain the notion of a lea st upper b ound (supremum) and grea test low er b ound (inﬁm um). Deﬁnition A.4 ( Supr emum and Inﬁmum ) Let P b e a pa rtially ordered set, S ⊆ P a nd a ∈ P . Then a is the supr emum of S , denoted b y a = sup( S ), if it is the leas t element of the set of upp er b ounds o f S , and a is the inﬁmum of S , de no ted by a = inf ( S ), if it is the gr eatest element of the set of low er bounds of S The next notio n of a dir e cte d set pla ys an impo rtant role in deﬁning when a function on a partial order is con tinuous. It is also re q uired in order to deﬁne a stricter structur e than a partial orde r , the c omplete p artial or der . W e need to int ro duce b oth these notions in o rder to formulate the Kleene Fixed Poin t Theorem which describ es one prop erty of contin uous functions on complete pa rtial o rders. Deﬁnition A.5 ( Dir e cte d Set ) A dir e cte d set is a pair ( D , ≤ ) wher e D is a non-empty set, ≤ is a reﬂexive a nd transitive re lation ov er D and for any elements a, b ∈ D there exists some c ∈ D such that a ≤ c and b ≤ c . As c a n b e seen, in a dire c ted set, every pair of elemen ts has an upp er b ound that also b elongs to the set. This pro per t y can b e natura lly extended to ﬁnite subsets of the directed set. Pr op osition A.6 Let ( D , ≤ ) b e a directed set and S a ﬁnite s ubset of D . Then D contains a n uppe r bo und of S . 24 M. Slota and J. L eite Pr o of Suppo se S = { s 1 , s 2 , . . . , s n } . Then we can construct a sequence { d i } n i =2 of elements of D suc h that s 1 ≤ d 2 and s 2 ≤ d 2 ; s i ≤ d i and d i − 1 ≤ d i for each i ∈ { 3 , 4 , . . . , n } . By induction on i it follows that d i ≤ d n for every i ∈ { 2 , 3 , . . . , n } and by applying transitivity we obtain s i ≤ d n for each i ∈ { 1 , 2 , . . . , n } . Hence d n is an upp er bound of S in D . As an impo rtant co nsequence, we obtain that every ﬁnite directed set contains its o wn supremum. Cor ol lary A.7 An y ﬁnite directed s et co n tains its s upr em um. Pr o of Let ( D , ≤ ) b e a ﬁnite directed set. Then by P rop. A.6 it co n tains its own upp er bo und d . C o nsider some o ther upp er bo und u of D . Then since d ∈ D , we hav e d ≤ u a nd s o d is the lea st upp er bo und of D , i.e. the supremum of D . W e can now in tro duce t wo pro per ties o f functions on pa rtial o rders. the weak er prop erty of mo no tonicity bas ically states that the function preserves the partial order: Deﬁnition A.8 ( Monotonic F un ct ion ) Let P, Q b e tw o partia lly o rdered sets and f : P → Q . W e say f is monotonic if for every a, b ∈ P suc h that a ≤ b we hav e f ( a ) ≤ f ( b ). The prop erty of contin uity is stricter and r equires that for a ll directed sets with a supremum in the domain, the imag e of that suprem um is the same a s the s upremum of images of elemen ts o f the dir ected set. Deﬁnition A.9 ( Continuous F unct ion ) Let P , Q b e tw o partially o rdered sets a nd f : P → Q . W e say f is c ontinuous if for every directed subset D of P with supremum in P it holds that sup( f ( D )) = f (sup( D )) where f ( A ) = { f ( a ) | a ∈ A } for any set A ⊆ P . The next prop osition for ma lly prov es that con tinuit y is a strong er prop erty thatn monotonicity . Pr op osition A.10 Every contin uous function is monotonic. T owar ds Close d World R e asoning in Dynamic Op en Worlds 25 Pr o of Consider a co n tinuous function f : P → Q and some a, b ∈ P s uc h that a ≤ b . Then the set D = { a, b } is a directed subse t of P and b y contin uity of f we obtain sup( f ( D )) = f (sup( D )) Since sup( D ) = b , we further obtain sup( { f ( a ) , f ( b ) } ) = f ( b ) and consequently f ( a ) ≤ f ( b ) a s desired. A complete partia l is simply a pa r tial order with a least element in which every directed se t has a supremum. Man y partially ordered str uctures, such as the space of in terpr e ta tions, satisfy this prop erty . Deﬁnition A.11 ( Complete Partial Or der ) A pa r tial or der ( P , ≤ ) is a c omplete p artial or der if P has a least elemen t and every directed subset S of P has a supremum in P . Finally , we are able to formulate and prove the main result of this app endix. It states that the least ﬁxed p oint of a cont inuous function on a complete partial order alwa ys exists and ca n b e approximated b y iterations of the function applied to the least element of the complete par tial or der. The or em A.12 ( Kle ene Fixe d Point The or em ) Let P b e a complete partial order with the le a st elemen t ⊥ and f b e a contin uous function on P . Then the least ﬁxed po in t o f f is sup { f n ( ⊥ ) | n ≥ 0 } . Pr o of This is a w ell-esta blished r esult, e ven so muc h that it is not easy to ﬁnd its original source. The oldest source we were able to ﬁnd and v erify is the b o ok ( Stoy 1977 ), pp. 1 12, Theor e m 6.64. The same pro of is a lso presented in the pap er ( Stoy 1979 ), pp. 55 (according to the num ber ing of the Pro ceedings ). A mo re recent bo o k on this topic is ( Dav ey and Prie s tley 19 9 0 ) where this result is formulated as Theorem 4.5 o n pp. 89. Now we start with the pres en tatio n of t he pro of. Suppose f is a contin uous function on the c o mplete partial order P . Then by Propo sition A.10 it is mono tonic from which it follows easily that the set D = { f n ( ⊥ ) | n ≥ 0 } is directed. Hence, its supremum sup D e x ists in P . W e will now show that sup D is a ﬁxed po in t of f : f (sup D ) = sup f ( D ) = sup f ( { f n ( ⊥ ) | n ≥ 0 } ) = sup { f n ( ⊥ ) | n ≥ 1 } = = sup( { ⊥ } ∪ { f n ( ⊥ ) | n ≥ 1 } ) = sup { f n ( ⊥ ) | n ≥ 0 } = = sup D F ur ther, supp ose a is some ﬁxe d po in t of f . In o rder to prov e that sup D is the lea st ﬁxed point o f f , we need to show that sup D ≤ a . By induction o n n we can easily obtain that f n ( ⊥ ) ≤ a fo r a ll n ≥ 0 : 1 ◦ f 0 ( ⊥ ) = ⊥ ≤ a 2 ◦ By inductive assumption f n − 1 ( ⊥ ) ≤ a , so by monotonicity of f we obtain f n ( ⊥ ) ≤ f ( a ) = a . So a is an upp er b ound of D and, by deﬁnition of a supremum, sup D ≤ a a s desired. 26 M. Slota and J. L eite App endix B Prop ertie s of MKNF B.1 Gener al Pr op erti es L emma B.1 ( Mo dels of Positive Sent enc es ) Let φ b e a po sitive MKNF sentence, I a prop ositional interpretation a nd M , N 0 ∈ M . If h I , M , N 0 i | = φ , then h I , M , N i | = φ for any N ∈ M . Pr o of F o llows directly from Deﬁnition 2.6 and the fact that the v aluation of a p ositive formula in a structure h I , M , N i is independent of N . Cor ol lary B.2 Let φ b e a po sitive MKNF sentence. Then the MK NF mo dels of φ are e xactly the subset-maximal S5 mo dels of φ . Pr o of F o llows from Deﬁnition 2.6 a nd Lemma B.1 . L emma B.3 Let ≤ b e a binary relation deﬁned on the s e t M of all sets of ﬁrst-o rder interpre- tations for any M , N ∈ M as follows: M ≤ N ⇐ ⇒ M ⊇ N Then ( M , ≤ ) is a co mplete partial order with the least element I . Pr o of F o llows from the s et-theoretic prop e rties of the subset r elation ⊆ a nd o f the set int er s ection ∩ . N o tice that even s ubsets of M that a re not directed have their supremum (intersection) in M . L emma B.4 Let φ be an ﬁrst-or der se ntence a nd M , N ∈ M b e suc h that M ≤ N . If M | = K φ , then also N | = K φ . Pr o of Suppo se M | = K φ and consider some interpretation I ∈ N . By the as sumption we obtain I ∈ M and so h I , M , M i | = K φ . Hence h I , M , M i | = φ and since φ is a ﬁrst-order formula, its v aluation in the structure h I , M , M i does n’t depend on M , so h I , N , N i | = φ . F urthermore, our choice of I was arbitrar y , s o w e can conclude that h I , N , N i | = φ for all I ∈ N . Conse q uen tly , N | = K φ as des ir ed. B.2 Mo dels of First-Or der The ori es T owar ds Close d World R e asoning in Dynamic Op en Worlds 27 L emma B.5 ( Gr e atest Mo del of a First-Or der The ory ) F o r any ﬁrst-order theory S it ho lds that mo d ( S ) = { I ∈ I | ( ∀ φ ∈ S )( I | = φ ) } Pr o of W e will prove that M S = { I ∈ I | ( ∀ φ ∈ S )( I | = φ ) } is the greatest set among the sets M ∈ M with the pro per t y M | = S . First we need to pr ov e that M S satisﬁes this pro per t y , i.e. that M S | = S . T ake some φ ∈ S and I ∈ M S . Then I | = φ a nd since φ is ﬁrs t-order, we also o btain h I , M S , M S i | = φ . This holds for any I ∈ M S , so M S | = φ . Now let M ∈ M be such that M | = S and supp ose I ∈ M . Then for every φ ∈ S we must hav e h I , M , M i | = φ and since φ is ﬁrst- o rder, this e n tails I | = φ . Hence, I ∈ M S , so M ⊆ M S . This fact ﬁnishes our pro o f. B.3 R elevant Part of an MKNF Interpr etati on Deﬁnition B.6 ( Pr e dic ate S ymb ols R elevant to a Gr ound F ormula ) Given a g round MK NF formula φ , we deﬁne the set P [ φ ] of pr e dic ate symb ols r ele- vant to φ inductively as follows: 1 ◦ If φ is a ﬁrst-or der ato m P ( c 1 , c 2 , . . . , c n ), then P [ φ ] = { P } ; 2 ◦ If φ is of the form ¬ ψ , then P [ φ ] = P [ ψ ] ; 3 ◦ If φ is of the form φ 1 ∧ φ 2 , then P [ φ ] = P [ φ 1 ] ∪ P [ φ 2 ] ; 4 ◦ If φ is of the form K ψ , then P [ φ ] = P [ ψ ] ; 5 ◦ If φ is of the form not ψ , then P [ φ ] = P [ ψ ] . Deﬁnition B.7 ( Constant Symb ols R elevant t o a Gr ound F ormula ) Given a gr ound MKNF formula φ , w e deﬁne the set C [ φ ] of c onstant symb ols r elevant to φ inductively as follows: 1 ◦ If φ is a ﬁrst-or der ato m P ( c 1 , c 2 , . . . , c n ), then C [ φ ] = { c 1 , c 2 , . . . , c n } ; 2 ◦ If φ is of the form ¬ ψ , then C [ φ ] = C [ ψ ] ; 3 ◦ If φ is of the form φ 1 ∧ φ 2 , then C [ φ ] = C [ φ 1 ] ∪ C [ φ 2 ] ; 4 ◦ If φ is of the form K ψ , then C [ φ ] = C [ ψ ] ; 5 ◦ If φ is of the form not ψ , then C [ φ ] = C [ ψ ] . Deﬁnition B.8 ( R estriction of an MKNF Interpr etation ) Let I ∈ I and M ∈ M . Given a ﬁnite set o f predicate symbols P ′ ⊆ P a nd a set of c o nstant symbols C ′ ⊆ ∆, we deﬁne the r estriction of I to P ′ and C ′ as the Her brand ﬁrs t-order interpretation I [ P ′ , C ′ ] ov er the Herbrand Univ erse C ′ that int er pr etes only the pre dicates from P ′ in such a wa y tha t ( c 1 , c 2 , . . . , c n ) ∈ P I [ P ′ , C ′ ] ⇐ ⇒ ( c 1 , c 2 , . . . , c n ) ∈ P I where P ∈ P ′ and c 1 , c 2 , . . . , c n ∈ C ′ . W e also deﬁne the r estriction of M to P ′ and C ′ as M [ P ′ , C ′ ] = n I [ P ′ , C ′ ]    I ∈ M o . 28 M. Slota and J. L eite L emma B.9 ( T ruth of Gr ound F ormulas u n der R estriction to Re levant Symb ols ) Let φ b e a gro und MKNF for m ula, P ′ ⊆ P a ﬁnite set of predicate symbols such that P ′ ⊇ P [ φ ] , C ′ ⊆ ∆ a ﬁnite set of constant sym b ols such that C ′ ⊇ C [ φ ] , I a prop ositional int er pretation a nd M , N ∈ M . Then h I , M , N i | = φ ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ . Pr o of W e will prove by structur al induction o n φ : 1 ◦ If φ is a gro und ﬁrst-orde r atom of the form P ( c 1 , c 2 , . . . , c n ), then P ∈ P [ φ ] and c 1 , c 2 , . . . , c n ∈ C [ φ ] , so P ∈ P ′ and c 1 , c 2 , . . . , c n ∈ C ′ . The following chain of equiv a lences now prov es the claim: h I , M , N i | = φ ⇐ ⇒ ( c 1 , c 2 , . . . , c n ) ∈ P I ⇐ ⇒ ( c 1 , c 2 , . . . , c n ) ∈ P I [ P ′ , C ′ ] ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ ; 2 ◦ If φ is of the form ¬ ψ , then P [ φ ] = P [ ψ ] and C [ φ ] = C [ ψ ] , so P ′ ⊇ P [ ψ ] and C ′ ⊇ C [ ψ ] . Hence, we can use the inductive hypo thesis for ψ as follows: h I , M , N i | = φ ⇐ ⇒ h I , M , N i 6| = ψ ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E 6| = ψ ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ ; 3 ◦ If φ is of the form φ 1 ∧ φ 2 , then P [ φ ] = P [ φ 1 ] ∪ P [ φ 2 ] and C [ φ ] = C [ φ 1 ] ∪ C [ φ 2 ] and we can eas ily verify that the inductive as sumption can be used on b oth φ 1 and φ 2 and the prop osition can b e prov ed for φ as follows: h I , M , N i | = φ ⇐ ⇒ h I , M , N i | = φ 1 ∧ h I , M , N i | = φ 2 ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ 1 ∧ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ 2 ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ ; 4 ◦ If φ is of the for m K ψ , then P [ φ ] = P [ ψ ] and C [ φ ] = C [ ψ ] , so P ′ ⊇ P [ ψ ] and C ′ ⊇ C [ ψ ] . The claim now follows from the inductiv e hypo thesis for ψ : h I , M , N i | = φ ⇐ ⇒ ( ∀ J ∈ M ) ( h J, M , N i | = ψ ) ⇐ ⇒ ( ∀ J ∈ M ) D J [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = ψ  ⇐ ⇒  ∀ J ∈ M [ P ′ , C ′ ]  D J, M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = ψ  ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ ; 5 ◦ If φ is of the form not ψ , then P [ φ ] = P [ ψ ] and C [ φ ] = C [ ψ ] , so P ′ ⊇ P [ ψ ] and T owar ds Close d World R e asoning in Dynamic Op en Worlds 29 C ′ ⊇ C [ ψ ] . The claim follows simila rly a s in the previous case: h I , M , N i | = φ ⇐ ⇒ ( ∃ J ∈ N ) ( h J, M , N i 6| = ψ ) ⇐ ⇒ ( ∃ J ∈ N ) D J [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E 6| = ψ  ⇐ ⇒  ∃ J ∈ N [ P ′ , C ′ ]  D J, M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E 6| = ψ  ⇐ ⇒ D I [ P ′ , C ′ ] , M [ P ′ , C ′ ] , N [ P ′ , C ′ ] E | = φ . App endix C Properti es of Hybrid Kno wledg e Bases L emma C.1 Let Z b e a set o f ﬁrs t-order theories. Then mo d  [ Z  = \ mo d ( Z ) where mo d ( Z ) = { mo d ( S ) | S ∈ Z } . Pr o of The following sequence of equiv alences prov es the claim: I ∈ mo d  [ Z  Lemma B.5 ⇐ = = = = = = = ⇒  ∀ φ ∈ [ Z  ( I | = φ ) ⇐ = = = = = = = ⇒ ( ∀S ∈ Z )( ∀ φ ∈ S )( I | = φ ) Lemma B.5 ⇐ = = = = = = = ⇒ ( ∀S ∈ Z )( I ∈ mo d ( S )) ⇐ = = = = = = = ⇒ I ∈ \ mo d ( Z ) Deﬁnition C.2 ( Hybrid Imme diate Conse qu enc e Op er ator ) The imme diate c onse quenc e op er ator asso cia ted with the deﬁnite P -ground hybrid knowledge base K = hO , P i is a mapping T K : M → M deﬁned for any M ∈ M as T K ( M ) = mo d ( O ∪ { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ) L emma C.3 Let K = hO , P i be deﬁnite P - ground hybrid k nowledge base. Then for every M ∈ M it holds that T K ( M ) = mo d ( O ) ∩ m od ( { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ) Pr o of Let S = { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } . W e need to show that mo d ( O ∪ S ) = mod ( O ) ∩ mo d ( S ) . This follows from Lemma C.1 . 30 M. Slota and J. L eite L emma C.4 Let D F be a ﬁnite directed set of ﬁrst-o rder interpretations and r be a gr ound deﬁnite rule. Then \ D F | = B ( r ) ⇐ ⇒ ( ∃ M ∈ D F )( M | = B ( r )) Pr o of By Co r ollary A.7 we hav e T D F ∈ D F , so if T D F | = B ( r ), then also ( ∃ M ∈ D F )( M | = B ( r )). Now supp ose that M | = B ( r ) for some M ∈ D F . Then M ≤ T D F and by a rep eated use of Lemma B.4 for ea ch conjunct o f B ( r ) we obtain T D F | = B ( r ). L emma C.5 Let D b e a directed s et o f MKNF interpretations, P ′ a se t of predicate symbo ls and C ′ a set of constant symbols and D [ P ′ , C ′ ] = n M [ P ′ , C ′ ]    M ∈ D o (C1) Then the following holds:  \ D  [ P ′ , C ′ ] = \ D [ P ′ , C ′ ] Pr o of  \ D  [ P ′ , C ′ ] =  \ { M | M ∈ D }  [ P ′ , C ′ ] = ( { I | ( ∀ M ∈ D )( I ∈ M ) } ) [ P ′ , C ′ ] = n I [ P ′ , C ′ ]    ( ∀ M ∈ D )( I ∈ M ) o = n I    ( ∀ M ∈ D )  I ∈ M [ P ′ , C ′ ]  o = n I     ∀ M ∈ D [ P ′ , C ′ ]  ( I ∈ M ) o = \ n M    M ∈ D [ P ′ , C ′ ] o = \ D [ P ′ , C ′ ] L emma C.6 Let D be a directed s e t of MKNF interpretations and r a ground deﬁnite rule. Then: ( ∃ M ∈ D )( M | = B ( r )) ⇐ ⇒ \ D | = B ( r ) Pr o of T owar ds Close d World R e asoning in Dynamic Op en Worlds 31 Let P ′ = P [ B ( r )] and C ′ = C [ B ( r )] and consider these equiv alences: ( ∃ M ∈ D )( M | = B ( r )) Lemma B.9 ⇐ = = = = = = = ⇒ ( ∃ M ∈ D )  M [ P ′ , C ′ ] | = B ( r )  ( C1 ) ⇐ = = = = = = = ⇒  ∃ M ∈ D [ P ′ , C ′ ]  ( M | = B ( r )) Lemma C.4 ⇐ = = = = = = = ⇒  \ D [ P ′ , C ′ ]  | = B ( r ) Lemma C.5 ⇐ = = = = = = = ⇒  \ D  [ P ′ , C ′ ] | = B ( r ) Lemma B.9 ⇐ = = = = = = = ⇒ \ D | = B ( r ) Pr op osition C.7 ( Continuity of T K ) Let K = hO , P i b e a deﬁnite P -gro und hybrid kno wledge base. Then T K is a contin uous function o n M . Pr o of Consider some directed subset D o f M . T o prove that T K is co nt inuous, w e need to show that sup( T K ( D )) = T K (sup( D )). By Lemma C.3 , we hav e: sup( T K ( D )) = mo d ( O ) ∩ \ M ∈D mo d ( { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ) . Let S de no te the set \ M ∈D mo d ( { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ) (C2) so that sup( T K ( D )) = mo d ( O ) ∩ S (C3) Consider the following identities: S Lemma C.1 = = = = = = = = = mo d [ M ∈D { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ! = = = = = = = = = mo d ( { H ∗ ( r ) | r ∈ P ∧ ( ∃ M ∈ D )( M | = B ( r )) } ) Lemma C.6 = = = = = = = = = mo d n H ∗ ( r )    r ∈ P ∧ \ D | = B ( r ) o T o gether with ( C3 ) and Lemma C.3 this implies that sup( T K ( D )) = mo d ( O ) ∩ S = mo d ( O ) ∩ mod n H ∗ ( r )    r ∈ P ∧ \ D | = B ( r ) o = T K (sup( D )) . Cor ol lary C.8 ( Monotonicity of T K ) Let K = hO , P i b e a deﬁnite P -gro und hybrid kno wledge base. Then T K is a monotonic function on M and for any n ≥ 0 it holds that T n K ( I ) ⊇ T n +1 K ( I ). 32 M. Slota and J. L eite Pr o of The mono to nicit y o f T K follows dire ctly from P rops. C.7 and A.10 . Now since I is the minimal element of ( M , ≤ ), we obtain T 0 K ( I ) = I ≤ T 1 K ( I ). By n times applying the monotonic ity of T K we obta in T n K ( I ) ≤ T n +1 K ( I ) which is equiv alent to T n K ( I ) ⊇ T n +1 K ( I ). The following prop osition shows that each deﬁnite P -ground h ybr id knowledge base either has no model at all, or, s imila rly as deﬁnite logic programs , it has the greatest S5 mo del that co incides with its unique MKNF mo del. It also shows how this mo del can be co mputed b y iterating the T K op erator s tarting from I . Pr op osition C.9 Let K = hO , P i be a deﬁnite P -g round hybrid knowledge base. Then either K has no S5 model or it has the grea test S5 mo del that a ls o coincides with its s ingle MKNF model. F urther mo re, the set mo d ( K ) = \ n ≥ 0 T n K ( I ) is empty if K ha s no S5 model and otherwise coincides with its u nique MK NF mo del. Pr o of First we will prov e a n aux iliary claim: M ⊆ T K ( M ) holds for any S5 mo del M of K . Supp ose M is an S5 mo del of K and r ecall that T K ( M ) = mo d ( O ∪ { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } ) Let’s take some formula φ ∈ O . W e know that M | = φ b ecause M is an S5 mo del of K . Now consider some rule r ∈ P suc h that M | = B ( r ). Since M is an S5 mo del of K , we obta in M | = H ∗ ( r ). Consequently , M | = H ∗ ( r ) for every such r . So M is an S5 mo del of O ∪ { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } and since T K ( M ) is by deﬁnition of mo d ( · ) the greatest S5 model of O ∪ { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } , w e can conclude that M ⊆ T K ( M ). Now we will pro ceed with the main part of the pr o of. L e t M K = \ n ≥ 0 T n K ( I ) Then, by Corolla ry C.7 and Theorem A.12 , M K is the least ﬁxed p oint of T K . First we will show that M K contains every S5 mo del of K . Assume, to the contrary , that M is an S5 model of K such tha t M * M K . Then b y deﬁnition M ⊆ I = T 0 K ( I ). It cannot b e the ca se that M ⊆ T n K ( I ) for all n ≥ 0 b ecause that would b e in conﬂict with M * M K . So let n 0 = max { n ≥ 0 | M ⊆ T n K ( I ) } . Now we hav e M ⊆ T n 0 K ( I ) and by the auxiliar y claim proved ab ov e, we o btain M ⊆ T K ( M ) whic h together with the mono to nicit y of T K (Corollar y C.8 ) yields M ⊆ T K ( M ) ⊆ T K ( T n 0 K ( I )) = T n 0 +1 K ( I ). Ho wev er, this is in conﬂict with the deﬁnition of n 0 , so no S5 mo del M o f K with M * M K can exist. T owar ds Close d World R e asoning in Dynamic Op en Worlds 33 Now w e will show tha t M K mo dels K . This can be easily v eriﬁed for every φ ∈ O . T a ke some r ∈ P . If M K 6| = B ( r ), then M K | = r and we a re do ne. So assume M K | = B ( r ). In this case we can use the ﬁxp oint prop er ty of M K : M K = T K ( M K ) = mod ( O ∪ { H ∗ ( r ) | r ∈ P ∧ M K | = B ( r ) } ) and conclude that M K | = H ∗ ( r ). Consequently also M K | = r . W e already prov ed that M K is the grea test set o f interpretations that mo dels K . So in ca se K has no S5 mo del, M K will b e empty . On the other hand, if K has some S5 mo del, this mo del is included in M K , so M K is non-empty and hence is the greatest S5 mo del o f K . F urther, by Cor ollary B.2 it follows that M K is also the unique MKNF mo del of K . F o r MKNF models of a r bitrary P - ground hybrid knowledge bases we also o bta in a characterisa tion tha t is similar to the ﬁxpoint deﬁnition of stable models of normal logic programs: Pr op osition C.10 An MKNF in terpr etation M is an MK NF mo del o f a P -ground h ybrid knowledge base K = hO , P i if and only if M = mo d (  O , P M  ) where P M =  H ( r ) ⊂ B + ( r )   r ∈ P ∧ M | = B − ( r )  Pr o of First notice that since P is ground, π ( r ) = r for every r ∈ P ∪ P M . Let K M =  O , P M  and supp ose M is an MKNF mo del of K . First we will show that M is an S5 mo del o f K M . O bviously , M mo dels all formu la s from π ( O ). Suppo se that r M = ( H ( r ) ⊂ B + ( r )) is a rule from P M . If M 6| = B + ( r ), then M | = r M . O n the other hand, if M | = B + ( r ), then M | = r implies als o M | = H ( r ). Consequently , M | = r M . As M is a n S5 model o f K M , it m ust hold that M is a subset o f mo d ( K M ) bec ause mo d ( K M ) is the g r eatest S5 mo del o f K M . By contradiction, we will show that M = mod ( K M ). Assume M ( mo d ( K M ). Since M is an MK NF mo del of K , there must b e some fo rmula φ ∈ π ( K ) and some I ′ ∈ mo d ( K M ) s uc h that  I ′ , mo d ( K M ) , M  6| = φ . But mo d ( K M ) mo dels π ( O ), s o φ must be some r ule r from P and the following m ust ho ld  I ′ , mo d ( K M ) , M  | = B − ( r ) ∧  I ′ , mo d ( K M ) , M  | = B + ( r ) ∧  I ′ , mo d ( K M ) , M  6| = H ( r ) which is equiv alent to M | = B − ( r ) ∧ mo d ( K M ) | = B + ( r ) ∧ mo d ( K M ) 6| = H ( r ) . How ever, this is in conﬂict with mo d ( K M ) b eing an S5 mo del of K M since H ( r ) ⊂ B + ( r ) ∈ P M . F o r the co n verse implication, a ssume M is a n MKNF interpretation such that M = mo d ( K M ). It m ust hold that M | = π ( O ), so co ns ider some rule r ∈ P . If M 6| = B − ( r ), then M is trivially a mo del of r . On the other hand, if M | = B − ( r ), 34 M. Slota and J. L eite then M is also a mo del of H ( r ) ⊂ B + ( r ), so M is a gain a mo del o f r . Co nsequently , M is an S5 model of K . Now take some M ′ ) M . Then s ince M is the greatest mo del of K M , there is s ome rule r M = ( H ( r ) ⊂ B + ( r )) ∈ P M such that M ′ 6| = r M , i.e. M | = B − ( r ) ∧ M ′ | = B + ( r ) ∧ M ′ 6| = H ( r ) F o r any I ′ ∈ M ′ , this is equiv alent to h I ′ , M ′ , M i | = B − ( r ) ∧ h I ′ , M ′ , M i | = B + ( r ) ∧ h I ′ , M ′ , M i 6| = H ( r ) which in turn is equiv alent to h I ′ , M ′ , M i 6| = r . So M is indeed an MKNF mode l of K . App endix D Prop erties of the Hybrid Up date Op erator Pr op osition 3.2 . L et P b e a ﬁnite gr ound deﬁnite pr o gr am, T a TBox and A a se quenc e of ABoxes. Then T P ⊕ T A is a c ontinuous fun ction on the c omplete p artial or der of al l subsets of I with the le ast element I . Pr o of of Pr op osition 3.2 Consider so me directed subset D o f M . T o pr ov e that T P ⊕ T A is co n tinuous, we need to show that sup( T P ⊕ T A ( D )) = T P ⊕ T A (sup( D )) . T o simplify notation in this pro o f, w e deﬁne for any set of ﬁrst-order interpretations M the following set: con ( M ) = { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } Notice that if M ⊇ N (or M ≤ N using the pa r tial or der o n sets o f ﬁr st-order int epre tations), then con ( M ) ⊆ c on ( N ). By deﬁnition we now hav e T P ⊕ T A (sup( D )) = mo d  con  \ D  ⊕ T A  = incorp orate T  A , mo d  T ∪ con  \ D  = incorp orate T  A , mo d ( T ) ∩ mo d  con  \ D  (D1) and sup( T P ⊕ T A ( D )) = \ M ∈D mo d  con ( M ) ⊕ T A  = \ M ∈D incorporate T ( A , mo d ( T ∪ con ( M ))) = \ M ∈D incorporate T ( A , mo d ( T ) ∩ mo d ( co n ( M ))) (D2) First supp ose that a ﬁrst-o rder int er pretation I is in T P ⊕ T A (sup( D )). Then b y the previous equatio n we hav e that there is some J ∈ mo d ( T ) ∩ mo d ( con ( T D )) s uch T owar ds Close d World R e asoning in Dynamic Op en Worlds 35 that I ∈ incorporate T ( A , J ) . F ur ther, for every M ∈ D it holds that con ( M ) ⊆ co n ( T D ), and, hence, als o that mo d ( con ( T D )) ⊆ mo d ( co n ( M )). Conseq uen tly , J ∈ mo d ( con ( M )) for every M ∈ D , and s o I ∈ incorporate T ( A , mo d ( T ) ∩ mo d ( co n ( M ))) also holds for every M ∈ D . By ( D2 ) we ca n now conclude that I ∈ sup( T P ⊕ T A ( D )) . F o r the converse inclusion, supp ose I / ∈ T P ⊕ T A (sup( D )) and le t S b e the set of all ﬁrst-order in terpr etations J ∈ mo d ( T ) such that I ∈ incorporate T ( A , J ) . By ( D1 ) we obtain that S ∩ mo d ( con ( T D ) ) = ∅ , i.e. that each J ∈ S is not a mo de l of some atom p J such that there is a rule r J ∈ P with H ( r J ) = K p J and T D | = B ( r J ). By Lemma C.6 , this implies that for s o me M ∈ D we als o hav e M | = B ( r J ). F urther, there are only ﬁnitely many rules in P , so by the directedness of D we ca n ﬁnd a n interpretation M S ∈ D such that M S | = B ( r J ) for all J ∈ S . F o r this interpretation it will hold that S ∩ mo d ( con ( M S )) = ∅ . Hence, I / ∈ incorp or ate T ( A , mo d ( T ) ∩ mo d ( con ( M S ))) and b y ( D2 ) we obta in that I / ∈ sup( T P ⊕ T A ( D )). Pr op osition 4.1 . L et P b e a ﬁnite gr ound pr o gr am, T a TBox, A an ABox and M a minimal change dynamic stable mo del of P ⊕ T A . Then M | = A . Pr o of of Pr op osition 4.1 If M is a minimal change dynamic sta ble mo del o f P ⊕ T A , then it is a ﬁxed point of T P M ⊕ T A , i.e . M = T P M ⊕ T A ( M ) = mo d  H ∗ ( r )   r ∈ P M ∧ M | = B ( r )  ⊕ T A  and by the deﬁnition o f the class ical minimal ch a ng e upda te o per ator it must hold that e very I ∈ M is a mo de l of A . In other w or ds, M | = A . Pr op osition 4.2 . L et P b e a ﬁnite gr ound pr o gr am, T , T ′ b e TB oxes su ch that mo d ( T ) = mo d ( T ′ ) , A , A ′ b e ABoxes such t hat mod ( A ) = mo d ( A ′ ) and M b e an MKNF interpr etation. Then M is a minimal change dynamic st able mo del of P ⊕ T A if and only if M is a minimal change dynamic stable mo del of P ⊕ T ′ A ′ . Pr o of of Pr op osition 4.2 F o llows from the fact that the operator s T P M ⊕ T A and T P M ⊕ T ′ A ′ are identical bec ause the clas sical minimal change upda te op e r ator only op erates with models of T , T ′ , A and A ′ , and not with their syntactic repr esentation. Pr op osition 4.3 . L et P b e a ﬁnite gr ound pr o gr am, O = T ∪ A an ontolo gy with 36 M. Slota and J. L eite TBox T and ABox A and M an MKNF interpr etation such that for every subset S of the set { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } the fol lowing c ondition is satisﬁe d: mo d ( S ⊕ T A ) = mo d ( S ∪ O ) . Then M is an MKNF m o del of hO , P i if and only if M is a minimal change dynamic stable m o del of P ⊕ T A . Pr o of of Pr op osition 4.3 By Prop ositions C.10 and C.9 , M is a n MK NF model of hO , P i if and o nly if M = \ n ≥ 0 T n hO , P M i ( I ) (D3) where for any set of ﬁrst-o rder interpretations N we hav e T hO , P M i ( N ) = mo d  O ∪  H ∗ ( r )   r ∈ P M ∧ N | = B ( r )  . On the o ther hand, by Pro po sition 3.2 a nd The o rem A.12 , M is a minimal c hange dynamic stable mo del of P ⊕ T A if and only if M = \ n ≥ 0 T n P M ⊕ T A ( I ) (D4) where for any set of ﬁrst-o rder interpretations N we hav e T P M ⊕ T A ( N ) = mo d  H ∗ ( r )   r ∈ P M ∧ N | = B ( r )  ⊕ T A  . Suppo se now that M is an MKNF mo del of hO , P i . Then fr o m ( D3 ) and Lemma B.4 we obtain that for every n ∈ N that n H ∗ ( r )    r ∈ P M ∧ T n hO , P M i ( I ) | = B ( r ) o ⊆ { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } . Hence, b y the a ssumption of the pr op osition, mo d n H ∗ ( r )    r ∈ P M ∧ T n hO , P M i ( I ) | = B ( r ) o ⊕ T A  = mo d  O ∪ n H ∗ ( r )    r ∈ P M ∧ T n hO , P M i ( I ) | = B ( r ) o (D5) By induction on n we will now pr ov e that T n hO , P M i ( I ) = T n P M ⊕ T A ( I ). 1 ◦ F o r n = 0 we hav e T 0 hO , P M i ( I ) = I = T 0 P M ⊕ T A ( I ) 2 ◦ W e assume the cla im holds for n − 1, i.e. T n − 1 hO , P M i ( I ) = T n − 1 P M ⊕ T A ( I ) (D6) T owar ds Close d World R e asoning in Dynamic Op en Worlds 37 and prov e that it holds for n . Indeed, we obtain: T n hO , P M i ( I ) = = = mo d  O ∪ n H ∗ ( r )    r ∈ P M ∧ T n − 1 hO , P M i ( I ) | = B ( r ) o ( D5 ) = = = = mo d n H ∗ ( r )    r ∈ P M ∧ T n − 1 hO , P M i ( I ) | = B ( r ) o ⊕ T A  ( D6 ) = = = = mo d n H ∗ ( r )    r ∈ P M ∧ T n − 1 P M ⊕ T A ( I ) | = B ( r ) o ⊕ T A  = = = T n P M ⊕ T A ( I ) So ( D4 ) is satisﬁed and consequently M is a minimal change dynamic stable mo del of P ⊕ T A . F o r the c o nv erse statement, supp o se M is a minimal change dy na mic stable model of P ⊕ T A . Then from ( D3 ) and Lemma B.4 we o btain for ev er y n ∈ N that  H ∗ ( r )   r ∈ P M ∧ T n P M ⊕ T A ( I ) | = B ( r )  ⊆ { H ∗ ( r ) | r ∈ P ∧ M | = B ( r ) } . Hence, b y the a ssumption of the pr op osition, mo d  H ∗ ( r )   r ∈ P M ∧ T n P M ⊕ T A ( I ) | = B ( r )  ⊕ T A  = mo d  O ∪  H ∗ ( r )   r ∈ P M ∧ T n P M ⊕ T A ( I ) | = B ( r )  (D7) By induction on n we will now pr ov e that T n P M ⊕ T A ( I ) = T n hO , P M i ( I ). 1 ◦ F o r n = 0 we hav e T 0 P M ⊕ T A ( I ) = I = T 0 hO , P M i ( I ) 2 ◦ W e assume the cla im holds for n − 1, i.e. T n − 1 P M ⊕ T A ( I ) = T n − 1 hO , P M i ( I ) (D8) and prov e that it holds for n . Indeed, we obtain: T n P M ⊕ T A ( I ) = = = mo d n H ∗ ( r )    r ∈ P M ∧ T n − 1 P M ⊕ T A ( I ) | = B ( r ) o ⊕ T A  ( D7 ) = = = = mo d  O ∪ n H ∗ ( r )    r ∈ P M ∧ T n − 1 P M ⊕ T A ( I ) | = B ( r ) o ( D8 ) = = = = mo d  O ∪ n H ∗ ( r )    r ∈ P M ∧ T n − 1 hO , P M i ( I ) | = B ( r ) o = = = T n hO , P M i ( I ) So ( D3 ) is satisﬁed and consequently M is an MKNF mo del of hO , P i . Cor ol lary 4.4 . L et P b e a ﬁnite gr ound pr o gr am. Then M is a stable mo del of P if and only if M is a minimal change dynamic stable mo del of P ⊕ ∅ ∅ . Pr o of of Cor ol lary 4.4 F o llows fro m the previous corollar y and the fact that MKNF mo dels coincide with stable mo dels on the class of normal logic progr ams ( Lifschitz 1 9 91 ). Pr op osition 4.5 . L et P b e a ﬁnite gr ound pr o gr am c ontaining only facts, T a TBox, 38 M. Slota and J. L eite A a se quenc e of A Boxes and M an MKNF interpr etation. Then M is a minimal change dynamic stable mo del of P ⊕ T A if and only if M is a minimal change up date mo del of S P ⊕ T A wher e S P = { p | K p ∈ P } . Pr o of of Pr op osition 4.5 Since P contains only facts , we can s ee that P = P M , so M is a minimal change dynamic sta ble mo del o f P ⊕ T A if and only if M = mo d ( P ⊕ T A ) which by deﬁnition holds if and only if M = \ n ≥ 0 T n P ⊕ T A ( I ) F ur ther, w e know that T 0 P ⊕ T A ( I ) = I T 1 P ⊕ T A ( I ) = mo d ( { H ∗ ( r ) | r ∈ P ∧ I | = B ( r ) } ⊕ T A ) = mo d ( { H ∗ ( r ) | r ∈ P } ⊕ T A ) = mo d ( S P ⊕ T A ) T n P ⊕ T A ( I ) = T 1 P ⊕ T A ( I ) for all n > 1 Hence, w e have \ n ≥ 0 T n P ⊕ T A ( I ) = mo d ( S P ⊕ T A ) . So M is a minimal change dynamic s table mo del of P ⊕ U if a nd only if M = mo d ( S P ⊕ T A ) which is by deﬁnitio n equiv alent to M b eing a minimal c hange upda te mo del o f S P ⊕ T A . L emma D.1 Let T be a TBox, A = ( A 1 , A 2 , . . . , A n ) a sequence of ABoxes (where n ≥ 1) and A ′ = ( A 1 , A 2 , . . . , A i − 1 , A i , ∅ , A i +1 , . . . , A n ) for some i ∈ { 0 , 1 , 2 , . . . , n } . Then for any M ⊆ mo d ( T ) it holds that incorporate T ( A , M ) = incorporate T ( A ′ , M ) Pr o of W e will prove by induction o n n : 1 ◦ If n = 1 , then i ∈ { 0 , 1 } , so we need to prove that incorporate T ( A 1 , M ) = incorp orate T ( A 1 , i ncorporate T ( ∅ , M )) and that incorporate T ( A 1 , M ) = incorp orate T ( ∅ , incor p orate T ( A 1 , M )) . This follows eas ily from the fact that i ncorporate T ( ∅ , N ) = N for any N ⊆ mo d ( T ). 2 ◦ W e assume the cla im holds for n − 1 and prove it for n . First let i = 0 . Then incorporate T ( A ′ , M ) = incorp orate T ( A , incorp orate T ( ∅ , M )) and the cla im ag a in fo llows fro m the fact that incorporate ( ∅ , N ) = N for any N ⊆ mo d ( T ). T owar ds Close d World R e asoning in Dynamic Op en Worlds 39 Now supp ose i > 0 a nd let B = ( A 2 , A 3 , . . . , A n ) B ′ = ( A 2 , A 3 , . . . , A i − 1 , A i , ∅ , A i +1 , . . . , A n ) By the inductive ass umption we k now that for any N ⊆ T it is ho lds that incorporate T ( B , N ) = incorp orate T ( B ′ , N ) Hence, incorporate T ( A ′ , M ) = incorp orate T ( B ′ , i ncorporate T ( A 1 , M )) = incorp orate T ( B , incorporate T ( A 1 , M )) = incorp orate T ( A , M ) . Cor ol lary D.2 Let T be a TBox, A = ( A 1 , A 2 , . . . , A n ) a sequence of ABoxes (where n ≥ 1) and A ′ = ( A 1 , A 2 , . . . , A i − 1 , A i , ∅ , A i +1 , . . . , A n ) for some i ∈ { 0 , 1 , 2 , . . . , n } . Then for any ﬁrst-order theo ry S it holds that mo d ( S ⊕ T A ) = mo d ( S ⊕ T A ′ ) Pr o of F o llows b y applying the pre v ious lemma to M = mo d ( T ∪ S ). Pr op osition 4.6 . L et P b e a ﬁnite gr ound pr o gr am, T a TBox and A a se quenc e of ABoxes = ( A 1 , A 2 , . . . , A n ) (wher e n ≥ 1 ). L et A ′ = ( A 1 , A 2 , . . . , A i − 1 , A i , ∅ , A i +1 , . . . , A n ) for some i ∈ { 0 , 1 , 2 , . . . , n } . Then an MKNF interpr etation M is a minimal change dynamic s table mo del of P ⊕ T A if and only if M is a minimal change dynamic stable m o del of P ⊕ T A ′ . Pr o of of Pr op osition 4.6 W e need to show that \ n ≥ 0 T n P M ⊕ T A ( I ) = \ n ≥ 0 T n P M ⊕ T A ′ ( I ) . By induction on n we will prove that for all n ∈ N it holds that T n P M ⊕ T A ( I ) = T n P M ⊕ T A ′ ( I ) . 1 ◦ F o r n = 0 we directly obtain T n P M ⊕ T A ( I ) = I = T n P M ⊕ T A ′ ( I ) . 2 ◦ W e assume the cla im holds for n − 1 and prove it for n . W e hav e T n P M ⊕ T A ( I ) = mo d n H ∗ ( r )    r ∈ P M ∧ T n − 1 P M ⊕ T A ( I ) | = B ( r ) o ⊕ T A  . 40 M. Slota and J. L eite By the inductive ass umption we o btain that T n − 1 P M ⊕ T A ( I ) = T n − 1 P M ⊕ T A ′ ( I ), so T n P M ⊕ T A ( I ) = mo d n H ∗ ( r )    r ∈ P M ∧ T n − 1 P M ⊕ T A ′ ( I ) | = B ( r ) o ⊕ T A  . Corollar y D.2 no w implies that T n P M ⊕ T A ( I ) = mo d n H ∗ ( r )    r ∈ P M ∧ T n − 1 P M ⊕ T A ′ ( I ) | = B ( r ) o ⊕ T A ′  = T n P M ⊕ T A ′ ( I ) . Pr op osition 4.7 . L et T b e a TBox, A an ABox and M an MKNF interpr etation. Then M is a m inimal change dynamic s t able mo del of ∅ ⊕ T A if and only if M = mo d ( T ∪ A ) . Pr o of of Pr op osition 4.7 By Pro p os ition 3.2 and Theore m A.1 2 , M is a minimal c hange dynamic stable mo del o f ∅ ⊕ T A if and only if M = \ n ≥ 0 T n P M ⊕ T A ( I ) where T 0 P M ⊕ T A ( I ) = I T 1 P M ⊕ T A ( I ) = mo d (  H ∗ ( r )   r ∈ P M ∧ I | = B ( r )  ⊕ T A ) = mo d ( ∅ ⊕ T A ) T n P M ⊕ T A ( I ) = T 1 P M ⊕ T A ( I ) for all n > 1 So M is a minimal change dynamic s table model of ∅ ⊕ T A if and only if M = mo d ( ∅ ⊕ T A ). F ur ther, mo d ( ∅ ⊕ T A ) = i ncorporate T ( A , mo d ( ∅ )) = i ncorporate T ( A , I ) = mo d ( T ∪ A ) . Hence, M is a minimal change dy na mic stable mo del of ∅ ⊕ T A if and only if M = mo d ( T ∪ A ).

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