RDF as Graph-Based, Diagrammatic Logic: Syntax, Semantics, and Calculus

Transcription

1 RDF a Grah-Baed, Diagrammatic Lgic: Syntax, Semantic, and Calculu Frithjf Dau 1 and Jnathan Haye 2 1 Det. f Mathematic, Darmtadt Univerity f Technlgy, Germany 2 Center fr Web Reearch, Det. f Cmuter Science, Univeridad de Chile dau@mathematik.tu-darmtadt.de jnathan.haye@gmx.de Abtract. The Reurce Decritin Framewrk (RDF) i a tandard fr rereenting infrmatin n the Web t make it machine-undertandable and -rceable. Thi aer exlre a cmlementary view in which we invetigate RDF a a lgic ytem fr human. RDF i develed a a grah-baed lgic ytem, i.e., a diagrammatic reaning ytem. Mrever, a und and cmlete et f tranfrmatin rule i rvided which act n the diagrammatic rereentatin f RDF. 1 Intrductin: RDF and Grah The Reurce Decritin Framewrk (RDF) i a tandard fr rereenting infrmatin n the Web. The infrmatin i encded a tatement, which are ubject-redicate-bject trile f infrmatin reurce. RDF i a data frmat deigned t enable emantic intererability between cmuter, and the larget art f it ecificatin rvide the mean fr thi in defining the cre vcabulary, an XML erializatin, inferencing, etc. Hwever, RDF trile yntax (and much le RDF/XML cde) i nt eaily read and undertd by human. A RDF wa develed t be a machine-acceible metadata frmat, why care fr rereentatin fr human? Tim Berner-Lee ay in [2] that the Web will reach it full tential when it becme an envirnment where data can be hared and rceed by autmated tl a well a by ele. (emhaziing by Dau/Haye). Particularly fr RDF, there are everal rean t invetigate a human-centered arach. Firt, there i the natural need f RDF authr and tl develer t revie RDF ecificatin they created. Secnd, it aear that RDF becme attractive a a general data frmat where hetergenu infrmatin mut be integrated (e.g., [3]), which imlie the need f viualizatin fr the ame rean a in the dmain the data riginate frm. Third, RDF rvide the ibility f reaning n data hwever, it i nt atifactry t leave lgic entirely t cmuter: human mut be able t fllw machine-generated rf. An arach t make RDF acceible fr human i given in the RDF ecificatin: the RDF Primer [10] intrduce a viualizatin f RDF mdel by rereenting tatement ubject and bject by nde, which are cnnected by redicate-labeled directed edge which crrend t tatement. Hwever, thi i nly an infrmal rereentatin, becaue it de nt accunt fr eculiaritie

2 f RDF uch a the fact that a tatement redicate may aear a the ubject f anther tatement. A critique f thi default arach i given in [5, 6], alng with an invetigatin f alternate RDF grah rereentatin. While the RDF Biartite Grah reented there atify the requirement f a frmal grah rereentatin f RDF, iue uch a entailment and diagrammatic reaning have nt been addreed. Thi i the gal f thi aer. RDF ue i nt limited t nly encde infrmatin, it urt al reaning. [7] rvide emantic fr RDF tatement baed n mdel thery, and cmlete ytem f inference rule ecify hw RDF mdel may evlve. In thi undertanding, RDF i a lgic ytem with a yntax, a emantic baed n mathematical mdel thery, and a calculu. The main difference between RDF and ymblic lgic ytem i the diagrammatic tential f RDF. There are already diagrammatic lgic ytem: fr examle, Semantic Netwrk and Tic Ma, which are ued fr knwledge rereentatin and rceing in artificial intelligence, r Exitential Grah a intrduced by C. S. Peirce. The mt imrtant DRS, articularly fr RDF, are J. Swa Cncetual Grah (CG) (ee [11, 13]), which are a ytem fr exreing meaning in a lgically recie, humanreadable, and cmutatinally tractable frm [12]. In fact, the RDF Primer tate that RDF draw un variu idea frm knwledge rereentatin, including CG fr examle, T. Berner-Lee tudied the relatin between RDF and CG [1]. Unfrtunately, CG lack reciene frm a lgical and mathematical view: a cmrehenive critique can be fund in the PhD thei f the firt authr [4]. The gal f thi aer i t devel RDF a a Diagrammatic Reaning Sytem (DRS), t rvide the mean fr bth human-acceible and mathematically recie RDF rereentatin and reaning n RDF mdel. We will call thi framewrk RDF/DRS. Cmaring t the RDF XML erializatin, ne culd emhaize the antagnim: 1. RDF/DRS i a kind f frnt-end f RDF fr human 2. RDF/XML i a kind f back-end fr tring RDF by machine. A already tated, fr elabrating RDF a a DRS, it i nt ufficient t cnider nly the claic, tatic viualizatin f RDF data in frm f grah. We al have t cnider a viual rereentatin f tranfrmatin f RDF grah a rt f dynamic cmnent f viualizatin. The mt imrtant cla f tranfrmatin fr RDF grah are the which derive an entailed RDF grah (ee [7]) frm anther. Althugh [6] rvide by mean f RDF Biartite Grah a mathematically und grah rereentatin, the tudy f a grah-yntactical calculu cnducted in thi aer ha a different ce. While [6] argued fr the imrtance f an unambiguu and redundancy-free grah fundatin f RDF fr ue a a data tructure in ftware and t enure the alicability f grah thery reult, the ue f RDF a DRS a we reent it here i geared tward a human audience. Particularly, fr human it i cnvenient t allw different kind f redundancy in the rereentatin, becaue thi enable the grah t carry me 2

3 rt f rhetrical tructure (nnethele, the trile ntatin fr grah allw redundancie a well in that the ame reurce may aear reeatedly in variu trile). Thi will be dicued in mre detail later n. In thi aer we cncentrate n the viualizatin f RDF grah tranfrmatin. The gal f the frmalim t be reented here i t make the tranfrmatin undertandable by breaking them u int a equence f elementary te, very much in the way frmula are ubequently derived in mathematical rf. Let u cnider a imle examle f entailment between tw grah. (Simle) entailment i defined in [7] a fllw: Sue that M i a maing frm a et f blank nde t me et f literal, blank nde and URI reference; then any grah btained frm a grah G by relacing me r all f the blank nde N in G by M(N) i an intance f G. Interlatin Lemma. S entail a grah E if and nly if a ubet f S i an intance f E. Cnider the fllwing examle: The RDF grah G 1 cniting f the trile (Ernt, aint, hitire naturelle), (Ernt, aint, flying geee), (Ernt, aint, le leil), (hitire naturelle, ha tyle, urrealim), (flying geee, ha tyle, dadaim), (le leil, ha tyle, dadaim) entail the fllwing RDF grah 1 G 2 : (Ernt, aint, hitire naturelle), (Ernt, aint, flying geee), (Ernt, aint, le leil), ( 1, aint, 2), ( 1, aint, 3), ( 2, ha tyle, dadaim), ( 3, ha tyle, urrealim) By nly cnidering the entailment definitin in [7] it i difficult fr human t decide whether an RDF grah entail the ther r nt: it i arduu t check all ible intance-decribing maing frm G 2 t G 1 t ee whether an intance f G 2 i a ubgrah f G 1. Intead, uing a grahical rereentatin it i much eaier t arehend the entailment between thee grah becaue ne can ee that G 2 exree le infrmatin than G 1. Thi hall be exemlified. The firt grah can be rereented a fllw: Ernt The ecnd grah i: aint ha_tyle hitire naturelle urealim aint ha_tyle flying geee dadaim aint le leil ha_tyle dadaim _1 aint aint _2 _3 ha_tyle urealim ha_tyle dadaim Ernt aint hitire naturelle aint flying geee aint le leil 1 Nte that the entailed grah ha mre trile than the grah we tarted with. 3

4 Nw we ee that G 2 i cmed f tw cie f G 1 uch that in each cy, me ecific infrmatin f G 1 i remved. T be mre recie: The uer art f G 2 i a cy f G 1 where the infrmatin abut the ainting flying geee i remved, and where me name are relaced by blank. Thu, it i entailed by G 1, a it cntain le infrmatin than G 1. Similar, the lwer art f G 2 i entailed by G 1 a well, a the infrmatin abut the aint tyle ha been remved. Thu, bth art f G 2 can be undertd a cie f the firt grah, where me infrmatin i remved, hence G 2 i by entailed G 1. The tranfrmatin rule which will be rvided in thi aer are baed n the idea reented in the examle. We will have rule which allw t 1. change a grah withut changing the rereented infrmatin (like making a cy f a tatement r a ubgrah), r 2. remve mall art f the infrmatin in a grah (like eraing a tatement r relacing name by blank). Thi tranfrmatin rule will be decribed mathematically, but having ur gal in mind t devel RDF a a DRS, they are deigned t be erfrmed n the grahical rereentatin f RDF grah. In hrt: They are diagrammatic tranfrmatin rule. T rvide an examle fr the rule, we will later hw hw G 2 can be yntactically be derived frm G 1 with thee rule. In the fllwing ectin the utlined elabratin f RDF/DRS i carried ut. In Sec. 2, the yntax i rvided, and me nrmal frm f RDF grah are defined. In Sec. 3, the emantic f RDF/DRS, which i mainly the emantic rvided in [7] r [8], i reented. In Sec. 4, the calculu, i.e., the tranfrmatin rule, are rvided, and it i hwn that the calculu i und and cmlete. Finally, in Sec. 5, an utlk fr further reearch i given. In thi aer, due t ace limitatin, n rf are rvided. The rf can be fund in an extended verin f thi aer (ee the remark after the bibligrahy). 2 Syntax Like in every lgic ytem, we firt have t fix the yntax f the ytem, tarting with the vcabulary we will ue. Definitin 1 (Vcabulary). Let Blank be a cuntably infinite et. The element f Blank are called blank. A trile Vc := (URI, TyedLit, PlainLit), cniting f three et URI, TyedLit, PlainLit which are airwie dijint and dijint frm Blank i called vcabulary. The member f the three et are called univeral reurce identifyer, tyed literal, and lain literal, re. We et Lit := TyedLit PlainLit. The element f Lit are called literal. The element f URI TyedLit PlainLit are called name. The well-knwn trile ntatin f RDF i defined a fllw: Definitin 2 (RDF grah, Trile Ntatin). An RDF grah ver Vc := (URI, TyedLit, PlainLit) in trile ntatin i a et f trile (,, ) with (,, ) (URI Blank) URI (URI Blank Lit) 4

5 fr each trile. Given a trile (,, ), we call the ubject, the redicate and the bject f the trile. RDF grah have a eculiarity which de uually nt ccur in cmmn mathematical lgic: Sme entitie can ccur a redicate in me tatement and a ubject r bject in ther tatement. A the et f rertie f an RDF grah i nt dijint frm it et f ubject and bject, it i nt ible t tranfrm an RDF grah in trile ntatin t an rdinary mathematical grah (V, E) by aigning t each bject r ubject a nde f the grah and aigning t each redicate an edge which link an ubject nde t an bject nde. In the infrmal grahical rereentatin fr RDF grah, there are different wrkarund t ce with thi rblem (fr examle, edge which link vertice and edge, r the aearance f the ame name bth n vertice and edge). A cmrehenive dicuin f thi rblem can be fund in [5], the dilma thei f ne f the authr. A reanable grah frmalizatin f RDF grah i t aign t each ubject, rerty and bject a nde in a mathematical grah, and t aign t each tatement, i.e. a trile (,, ), a hyer-edge which link the nde f,, and. Each RDF grah in trile ntatin can be tranlated t a grah frmalizatin in thi ene, thu the grah frmalizatin can be undertd a exlicit rereentatin f RDF grah in trile ntatin a mathematical grah. Thi i the underlying undertanding in [5], where ne ible tranlatin f RDF grah in trile ntatin t mathematical grah i mathematically elabrated. In thi wrk, in cntrat t [5], we undertand and invetigate the grah frmalizatin f RDF grah a tructure n it wn. In the next definitin, the grah frmalizatin f RDF grah are defined. Definitin 3 (Grah Frmalizatin f RDF grah). A tructure G := (V, E, ν, κ) i called grah frmalizatin f an RDF grah ver Vc := (URI, TyedLit, PlainLit), iff 2 V and E are dijint, finite et whe element are called vertice and edge. ν : E V 3 i a maing uch that each vertex v ccur in a trile ν(e) fr an edge (i.e., each vertex i incident with at leat ne edge) κ : V Vc i a maing uch that fr each e E, we have κ(e 1 ) (URI Blank), κ(e 2 ) URI, κ(e 3 ) (URI Blank Lit) Fr edge e, we will ften write e = (v 1, v 2, v 3 ) intead f ν(e) = (v 1, v 2, v 3 ), and we will ften dente the ith vertex v i, i = 1, 2, 3, with e i. Fr v V let E v := {e E i.ν(e) i = v} be the et f all edge which are incident with v. Analguly, fr e E let V e := {v V i.ν(e) i = v}. The et {bl Blank there exit a v V with κ b (v) = bl} i called the et f blank f G. The et f URI f G, etc. i defined analguly. 2 A uual in mathematic, we ue iff fr if and nly if. 5

6 In the fllwing, we will mainly wrk with the grah frmalizatin f RDF grah. A it will be clear frm the cntext when RDF grah in trile ntatin r grah frmalizatin f RDF grah are ued we will mit the hrae grah frmalizatin r trile ntatin and imly eak f RDF grah. We will diagrammatically rereent the grah frmalizatin f RDF grah a fllw: Each vertex i drawn a an val with the label κ(v) incribed in it. An edge (v 1, v 2, v 2 ) i drawn a mall circle which i cnnected with three line, labeled with,, and, re., t the val f v 1, v 2, and v 3. T ee an examle fr thi cnventin, cnider the firt fur aximatic trile rvided in RDF Semantic belw. A grah frmalizatin and it grahical rereentatin f the fur trile i given in Fig. 1. (rdf:tye rdf:tye rdf:prerty) (rdf:ubject rdf:tye rdf:prerty) (rdf:redicate rdf:tye rdf:prerty) (rdf:bject rdf:tye rdf:prerty) G := ({v 1, v 2, v 3, v 4, v 5, v 6}, {e 1, e 2, e 3, e 4}, {(e 1, (v 1, v 5, v 6)), (e 2, (v 2, v 5, v 6)), (e 3, (v 3, v 5, v 6)), (e 4, (v 4, v 5, v 6))}, {(v 1, rdf :ubject), (v 2, rdf :redicate), (v 3, rdf :bject), (v 4, rdf :tye), (v 5, rdf :tye), (v 6, rdf :P rerty)}) rdf:ubject rdf:redicate rdf:bject rdf:tye rdf:tye rdf:prerty Fig. 1. Fur RDF trile, a grah frmalizatin f them and it diagrammatic rereentatin Next, we have t fix me elementary technical ntatin. A ubgrah f an RDF grah in trile ntatin i imly a ubet f the trile. Fr grah frmalizatin f RDF grah, ubgrah are defined a fllw: Definitin 4 (Subgrah). Let (V, E, ν, κ) be a grah frmalizatin f an RDF grah ver Vc := (URI, TyedLit, PlainLit). A ubgrah f (V, E, ν, κ) i an RDF grah (V, E, ν, κ ) with V V, E E, ν = ν E and κ = κ V. If we have mrever E v E fr each v V, the ubgrah i called cled. It i f n ignificance which blank in an RDF grah are ued t exre exitential quantificatin. Fr thi rean, ne cnider RDF grah t be identical if they differ nly in the name f the ccurring blank (thi crrend t the well-knwn alha cnverin f frmula in mathematical lgic). In [7], thi cnventin i fixed by defining me RDF grah t be equivalent. Thi definitin i nw tranferred t grah frmalizatin f RDF grah. Definitin 5 (Equivalence f RDF grah). Let G 1 := (V 1, E 1, ν 1, κ 1 ), G 2 := (V 2, E 2, ν 2, κ 2 ) be tw RDF grah. We ay that G 1 and G 2 are equivalent, if there are a bijective maing f V : V 1 V 2, F E : E 1 E 2 and f Blank : Blank Blank with: 6

7 Each e = (v 1, v 2, v 3 ) E 1 atifie f E (v 1, v 2, v 3 ) = (f V (v 1 ), f V (v 2 ), f V (v 3 )), κ(f V (v)) = κ(v) fr each v V with κ(v) URI TyedLit PlainLit, and κ(f V (v)) = f Blank (κ(v)) fr each v V with κ(v) Blank. Cmared t RDF grah in trile ntatin, their grah frmalizatin rvide richer mean t exre me given amunt f infrmatin. Thi i mainly due t the fact that a vertex in an RDF grah can be incident with everal edge. Aume that we have everal RDF tatement where an entity ccur (a ubject, redicate, r bject). If we ue RDF grah in trile ntatin, then we will have a crrending number f ccurrence f thi entity in me trile. In cntrat t that, if we ue grah frmalizatin f RDF grah, we can che the number f vertice which tand fr thi entity. The richer mean t exre infrmatin, tgether with the variu way a grah frmalizatin f an RDF grah can be grahically rereented, rvide the ibility t emhaize me amunt f the infrmatin in an RDF grah. In thi undertanding, RDF grah rvide a rhetrical tructure. Fr machine rceing f RDF, thi feature i, f cure, unneceary. Hwever, thi i an adequate arach fr human. T exemlify the rhetrical tructure f RDF grah, we cnider again the firt fur aximatic trile frm RDF Semantic. The RDF grah in Fig. 2 cver nly the meaning f the firt trile rdf:tye rdf:tye rdf:prerty, but three time in different way. rdf:tye rdf:tye rdf:prerty rdf:prerty rdf:tye rdf:tye rdf:prerty Fig. 2. An RDF grah with redundant edge he grah f Fig. 1, a well a the tw grah in Fig. 3 and 4 are ible tranlatin f all fur trile. Their different underlying tructure can be ued t emhaize me ecific view n the rereented infrmatin. The grah f Fig. 3 i fragmented int elementary grah (i.e., grah which carry atmic infrmatin) and i therefre eay t read, but n infrmatin i emhaized. It crrend directly t the RDF Grah in trile ntatin. The grah f Fig. 1 emhaize that ubject, redicate and bject tand in the ame relatin t tye and Prerty, but it i al eay t ee that tye lay a ecial rle, a it aear bth a ubject and redicate f an edge. The grah f Fig. 4 emhaize that all fur entitie ubject, redicate, bject and tye tand in the ame relatin t tye and Prerty, but t ee that tye differ mehw 7

8 frm the ther three name ne ha t kee track f all label and realize that tw different vertice are labeled with tye. rdf:tye rdf:tye rdf:ubject rdf:prerty rdf:redicate rdf:prerty rdf:tye rdf:tye rdf:bject rdf:prerty rdf:tye rdf:prerty Fig. 3. A RDF grah fr the fur trile in anti-nrmal frm rdf:ubject rdf:redicate rdf:bject rdf:tye rdf:prerty Fig. 4. A RDF grah fr the fur trile in nrmal frm The grah in Fig. 3 and 4 are in me ene nrmalized. Mre reciely: In the firt grah, each vertex i cnnected exactly nce t an edge. U t additin f redundant cie f edge, it ha the maximum number f vertice. Fr the third grah, we have an ite ituatin: Each vertex ha a different label, i.e., thi grah cntain the minimal number f vertice. Thee tw way f nrmalizatin hall be frmalized in the next definitin. Definitin 6 (Nrmal Frm and Anti-Nrmal Frm). Let G := (V, E, ν, κ) be an RDF grah which ha n redundant cie f edge, i.e., if e = (v 1, v 2, v 3 ) and f = (w 1, w 2, w 3 ) are tw edge with κ(v i ) = κ(w i ) fr i = 1, 2, 3, then we have e = f. We ay that G i in nrmal frm, if different vertice alway have different label, i.e., if it atifie the fllwing cnditin: Fr all v 1, v 2 V with κ(v 1 ) = κ(v 2 ), we have v 1 = v 2. We ay that G i in anti-nrmal frm, if each vertex i incident with exactly ne edge, in exactly ne lace, and if it ha furthermre n redundant cie f edge. That i, it ha t atify the fllwing cnditin: If v i an vertex, and if e = (v 1, v 2, v 3 ) and f = (w 1, w 2, w 3 ) are tw edge with v i = v = w j fr me i, j {1, 2, 3}, then we have e = f and i = j. Next, we have t define tw eratin n RDF grah which are already defined in [7] fr grah in trile ntatin: The jin and merge f grah. In the diagrammatical rereentatin fr grah frmalizatin f RDF grah, the jin 8

9 f me grah i t juxtae them, i.e., t imly write them ide by ide. But the jin f grah can uually nt be undertd a their lgical cnjunctin. Thi hall be exemlified: cnider the ne-trile RDF grah ( 1 ha clr blue) (which mean mething ha the clr blue) and ( 1 ha clr red) (meaning mething i red) but their jin huld be mething i blue and mething i red rather than mething i blue and red. S, nly if the jined grah have n blank in cmmn, their jin crrend t lgical cnjunctin. In thi cae, we will eak f the merge f grah. Frmally, the jin and merge are defined a fllw: Definitin 7 (Jin and Merge f grah). Let n N 0 and fr i = 1,..., n let G i := (V i, E i, ν i, κ i ) be a RDF grah. W.l..g. we aume that all V i and E i, i = 1,..., n, are airwie dijint. The jin f the grah G i i defined t be the fllwing RDF grah G := (V, E, ν, κ): V := V i, E := E i, ν := ν i, and κ := κ i. i=1,...,n i=1,...,n i=1,...,n i=1,...,n In the grahical rereentatin, the jin f the G i i imly nted by writing the grah next t each ther, i.e. we write: G 1 G 2... G n. A jin f the grah G i i al called merge f the grah G i, if the grah G i d nt have blank in cmmn, i.e., fr each i, j {1,..., n}, v V i, w V j with κ i (v) = κ j (w) Blank it hld i = j. Nte that the jin r merge f an emty et f grah i the emty grah. Mrever, recall that RDF grah are nly cnidered u t equivalence. Thu, given me RDF grah G i, i = 1,..., n, we can alway merge them by relacing each grah G i with an equivalent grah G i uch that the grah G i d nt have any blank in cmmn, and frm the jin f the grah G i. 3 Semantic In thi ectin, the emantic fr RDF grah i defined. It i baed n mathematical mdel thery and adted frm [7, 8] fr RDF. The mdel fr RDF are defined a fllw: Definitin 8 (Interretatin). An interretatin r mdel I fr a vcabulary Vc := (URI, TyedLit, PlainLit) i a tructure (IR, IP, IEXT, IS, IL) where IR i a et f reurce, called the dmain r univere f I. IP i a et f rertie f I. IEXT : IP P(IR IR) i a maing. IS : URI IR IP i a maing. IL : Lit IR i a maing. 9

10 Thi definitin f an interretatin i nearly the definitin f an interretatin given in [7, 8], with ne light change: In [7, 8], the maing IL ma nly tyed literal t reurce, and an interretatin cntain a ubet LV IR with LV Lit (i.e., the et f untyed literal i aumed t be a ubet f IR). Frm the RDF viewint, if an interretatin i imlemented in an RDF alicatin, it i reanable t aume that the untyed literal are member f the interretatin. But frm a mathematical int f view, a clear ditinctin between the name f the vcabulary and the bject (the reurce) f an interretatin i arriate. Thi change in the definitin f an interretatin i nt ubtantial: An interretatin accrding t Def. 8 can eaily be tranfrmed t an interretatin accrding t [7] by etting LV := IL[PlainLit]. Mrever, leae nte that the relatinhi between IR and IP i nt ecified. Frm the viewint f claical mathematical lgic, ne wuld aume that IR and IP are dijint. Frm the RDF viewint, it i quite nrmal t aume that IP IR hld. Thi i becaue rertie may ccur in RDF grah a ubject r bject f ther trile a well, i.e., a rerty may be alied t anther rerty. Bth cae, IR IP = and IP IR, are allwed. In the next definitin, RDF grah are evaluated in tructure. If a grah and an interretatin are given, we d nt aume that they are baed n the ame vcabulary. But if a given name f the grah i nt interreted in the mdel, the emantic will alway yield that the grah i fale in the mdel. Fr thi rean, we will uually aume that the vcabularie f a grah and an interretatin are the ame. Definitin 9 (Interreting RDF grah in Mdel, Entailment). Let G := (V, E, ν, κ) be an RDF grah ver a vcabulary Vc G := (URI G, TyedLit G, PlainLit G ) and let I := (IR, IP, IEXT, IS, IL) be an interretatin ver a vcabulary Vc I := (URI I, TyedLit I, PlainLit I ). A functin I : V IR i an interretatin functin, if it atifie 1. If v 1, v 2 V are vertice with κ(v 1 ) = κ(v 2 ), then I(v 1 ) = I(v 2 ), and 2. I(v) = (IL IR)(n) fr each each name n URI I PlainLit I TyedLit I and each vertex v V with κ(v) = n. We ay the grah G hld in the mdel I, which i written I = G, iff 1. fr each vertex v V, we have κ(v) URI I TyedLit I PlainLit I, and 2. there exit an interretatin functin I : V IR uch that fr each edge e = (v 1, v 2, v 3 ) E and := κ(v 1 ), := κ(v 2 ), := κ(v 3 ), we have I() IP and (I(), I()) IEXT(I()) (thi cnditin i called edge cnditin). If G a, G b are RDF grah uch that I = G b hld fr each interretatin I with I = G a, we ay that G a (emantically) entail G b, and write G a = G b. 4 Calculu In the fllwing, a calculu fr grah frmalizatin f RDF grah i rvided. Thi calculu cnit f five tranfrmatin rule. In the fllwing, each f thee 10

11 rule i decribed in a emi-frmal manner, befre a recie definitin i rvided. In the definitin, the term freh i ued fr vertice r edge, imilar t it cmmn ue in lgic: given a frmula, a freh variable i a variable which de nt ccur in the frmula. Analguly, given a grah (V, E, ν, κ), a vertex r edge x i called freh i we have x / V E. Nw, let G := (V, E, ν, κ) be given. The fllwing tranfrmatin are allwed: 1. eraing an edge Let e be an edge. Then e may be eraed (and each vertex v which i nly incident with e ha t be eraed a well). That i, we cntruct the grah G e := (V e, E e, ν e, κ e ) where we et V (e) := V \{v V e v / V f fr any f E with f e}, E (e) := E\{e}, ν (e) := ν E e, and κ (e) := κ E e. 2. generalizing a label Let bl Blank be a blank which de nt aear in G, i.e., fr each vertex v V we have κ(v) bl. Let V V be a et f vertice which are identically labeled, i.e., we have κ(v 1 ) = κ(v 2 ) fr all v 1, v 2 V. Then, fr each v V, κ(v) may be relaced by κ(v) := bl. That i, we cntruct the grah G g := (V, E, ν, κ g ) with κ g (w) := κ(w) fr all w / V and κ g (v) := bl fr all w V. 3. merging tw vertice Let v 1, v 2 be tw different vertice with κ(v 1 ) = κ(v 2 ). Then v 2 may be merged int v 1 (i.e., v 2 i eraed and, fr every edge e E, e i = v 2 i relaced by e i = v 1 ). That i, we cntruct the grah G m := (V m, E, ν m, κ) with V m := V \{v 2 }, and fr all e E and i = 1, 2, 3 we have ν m (e)(i) := ν(e)(i), if ν(e)(i) v 2, and ν m (e)(i) := v 1 ele. 4. litting a vertex Let v 1 be a vertex, incident with edge e 1,..., e n. Then the fllwing may be dne: A freh vertex v 2 with κ(v 2 ) := κ(v 1 ) i inerted. On e 1,..., e n, arbitrary ccurrence f v 1 may be relaced by v 2. That i, we cntruct a 3 grah G i := (V i, E, ν i, κ) with V m := V. {v 2 }, fr all e E and i = 1, 2, 3 we have ν m (e)(i) := ν(e)(i), if ν(e)(i) v 1 and ν m (e)(i) {v 1, v 2 } ele, and which atifie E v1 = E v2 (therwie the tructure i nt a well-frmed RDF grah). 5. iterating an edge Let e be an edge with ν(e) = (v 1, v 2, v 3 ). Then a freh edge e with ν(e ) := (v 1, v 2, v 3 ) and κ(e ) := κ(e ) may be inerted. That i, we cntruct the grah G i := (V, E i, ν i, κ) with E (i) := E. {e } and ν i := ν. {(e, (v 1, v 2, v 3 ))}. Baed n thee tranfrmatin, we can nw define what a frmal rf with RDF grah i. 3 Even fr a fixed vertex, thi rule allw t derive different grah. 11

12 Definitin 10 (Syntactical Entailment Relatin). Let G a, G b be tw RDF grah. Then G b can be derived frm G a (which i written G a G b ), if there i a finite equence (G 1, G 2,..., G n ) with G 1 = G a and G b = G n uch that each G i+1 i derived frm G i by alying ne f the rule abve. The equence i called a rf fr G a G b. Tw grah G 1, G 2 with G 1 G 2 and G 2 G 1 are aid t be rvably equivalent. If H := {G i i I} i a (ibly emty) et f RDF grah, then a grah G can be derived frm H if there i a finite ubet {G 1,..., G n } H uch that G can be derived frm the merge f G 1... G n. We want t make me remark t thee rule. In nne f the rule, new name are added t a grah. Thu, if a grah G ver a vcabulary Vc i given, and G i derived frm G, then G i an RDF grah ver Vc a well. In the rule generalizing a label, it i imrtant that the label f a ubet f vertice and nt nly frm a ingle vertex can be generalized. Otherwie, the fllwing derivatin wuld nt be ible. _1 _1 _1 q _1 _2 _2 q Obviuly, the rule merging tw vertice and litting a vertex are invere t each ther. That i, if G can be derived frm G with the rule merging tw vertice, then G can be derived frm G with the rule litting a vertex, and vice vera. T gra the cmlete ibilitie f the rule litting a vertex and merging tw vertice, it i imrtant t nte that a vertex v can be incident u t three time with an edge e (edge e = (v, v, v) are allwed). Thee are the ccurrence f v n e, and it ible t relace me f the ccurrence by the cy v f v. T ee an examle: The grah f Fig. 1 i btained frm the grah f Fig. 4 by litting the vertex labeled with rdf:tye, and vice vera, the grah f Fig. 4 i btained frm the grah f Fig. 1 by merging the vertice labeled with rdf:tye. The rule iterating an edge can be revered with the rule eraing an edge. On the ther hand, let a redundant edge e = (v 1, v 2, v 3 ) be given, i.e., there i anther edge f e, f = (w 1, w 2, w 3 ) with κ(v i ) = κ(w i ) fr i = 1, 2, 3. If e i eraed with the rule eraing an edge, then thi tranfrmatin can be revered by iterating and ilating e and merging the freh vertice v 1, v 2, v 3 int w 1, w 2, w 3, reectively. If the rule iterating an edge i alied t an edge e = (v 1, v 2, v 3 ), then the new edge e i incident with v 1, v 2, v 3 a well (we have e = (v 1, v 2, v 3 )). But, alying the rule litting a vertex t v 1, v 2 and v 3 allw t ilate the incident vertice f e (we will call thi tranfrmatin iterating and ilating the edge e). That i, it i ible t build a grah G i := (V i, E i, ν i, κ i ) with V (i) := V. {v 1, v 2, v 3} fr freh vertice v 1, v 2, v 3, E (i) := E. {e }, E (i) := E. {e }, ν i := ν. {(e, (v 1, v 2, v 3)} and κ i := κ. {(v 1, κ(v 1 )), (v 2, κ(v 2 )), (v 3, κ(v 3 ))}. 12

13 The lat idea can eaily be extended t arbitrary ubgrah. T ee thi, let G := (V, E, ν, κ) be an RDF grah with a ubgrah G := (V, E, ν, κ ). Fr each e E, we can make a cy e f e with the rule iterating an edge. Afterward, we can lit each v V uch that fr each edge e E where v i incident with, the cy v f v i incident with the cy e f e. Let u call thi tranfrmatin iterating the ubgrah G. We have een that if G i btained frm G by iterating a ubgrah, we have G G, i.e., G can be derived frm G. Similar t the rule iterating an edge, the rule eraing an edge can be extended t ubgrah a well. But we have t take care f ne imrtant detail. Sue again we have an RDF grah G := (V, E, ν, κ) with a ubgrah G := (V, E, ν, κ ), and we want t remve all vertice v V and all edge e E frm G. But if V cntain a vertex v which i incident with an edge e E\E (i.e., E v E ), then remving v, but nt e, de nt yield a well-frmed RDF grah. Aume n the ther hand that we have E v E fr each v V, i.e., aume that G i a cled ubgrah. Then remving all edge e E with the rule eraing an edge autmatically yield a grah G where all vertice v V are remved a well. Let u call thi tranfrmatin eraing the cled ubgrah G. Obviuly, if G i btained frm G by eraing a cled ubgrah, we have G G. In [7] i i hwn that given tw RDF grah G, G in trile ntatin, then G entail G iff a ubgrah f G i an intance f G. If we call G a generalizatin f G iff G i an intance f G, we ee that G entail G iff G i a generalizatin f a ubgrah f G. Fr RDF grah in trile ntatin, an intance f a grah may cntain le trile, thu a generalizatin f a grah may cntain mre trile. Fr the grah rereentatin f grah, the rule generalizing a label de nt change the number f edge, i.e., the number f tatement. Fr increaing the number f tatement, the rule iterating an edge i needed. Hence the idea f cntructing a generalizatin fr grah frmalizatin f RDF grah i reflected by the rule iterating an edge and generalizing a label. Mrever, cntructing a ubgrah f an RDF Grah in trile ntatin fr grah frmalizatin f RDF Grah i reflected by the rule eraing an edge. Thu, rughly ken, thee three rule fr grah rereentatin f RFD grah reflect the calculu fr RDF grah in trile ntatin. The remaining tw rule merging tw vertice and litting a vertex are needed becaue grah frmalizatin f RDF Grah rvide richer mean t exre rhetric tructure (a dicued abve). T exemlify the calculu, we return t the Max Ernt examle f the intrductin: tw grah, ne entailing the ther, were reented in trile ntatin. A diagram f a ible grah frmalizatin fr the tarting grah i deicted belw. Furthermre, a dtted rectangle indicate a ubgrah f thi grah. Ernt aint hitire naturelle flying geee le leil ha tyle urealim dadaim dadaim 13

14 Nw we d the fllwing: Firt, we iterate the ubgrah, i.e., we generate a redundant cy f the infrmatin given by the ubgrah. Ernt aint hitire naturelle flying geee le leil ha tyle urealim dadaim dadaim Ernt aint hitire naturelle flying geee le leil Next, we can remve me infrmatin by remving tw edge. aint hitire naturelle ha tyle urealim Ernt le leil dadaim Ernt aint hitire naturelle flying geee le leil Finally we relace me f the name in the grah by blank. aint _2 ha tyle urealim _1 _3 dadaim Ernt aint hitire naturelle flying geee le leil Thi i a ible grah frmalizatin fr the ecnd grah f the examle in the intrductin. The next lemma hw that, uing the rule merging tw vertice, litting a vertex and eraing an edge (where nly redundant cie f edge are remved), each RDF grah can be tranfrmed int nrmal frm r anti-nrmal frm. Thi will turn ut ueful in the rf f the cmletene. 14

15 Lemma 1 (Nrmalizing grah). Uing the rule litting an vertex, merging tw vertice and eraing an edge, each RDF grah G i rvably equivalent t a RDF grah G n in nrmal frm, and t a RDF grah G an in anti-nrmal frm. Prf: We tart t hw hw G i tranfrmed int nrmal frm. Firt, we aly the rule litting an vertex until each vertex i incident with exactly ne edge, in exactly ne lace. Then, redundant cie f edge are eraed with the eraure rule. Becaue all thee tranfrmatin may b e revered (with the rule merging tw vertice and iterating and ilating an edge ) the new cntructed grah G an i equivalent t G. Obviuly, the new grah i in anti-nrmal frm. Analguly, if we aly the rule merging tw vertice until different vertice alway have different label, and if we again erae afterward redundant cie f edge are eraed with the eraure rule, we get a grah G n which i equivalent t G and which i in nrmal frm. In the fllwing, we will hw that the calculu i adequate, i.e., und and cmlete. We tart with the undne f the rule. Therem 1 (Sundne f the Rule). Tw RDF grah G a, G b G a G b = G a = G b. atify Prf: We have t hw the fllwing: If I i an interretatin with I = G a, and G b i derived frm G a with ne f the rule, then we have I = G b. Thi will be dne fr each rule earately. A an RDF grah G can nly hld in an interretatin I if each name which aear in G belng t the vcabulary f I, we can aume w.l..g. that G a, G b and I are baed n the ame vcabulary. Nw a RDF grah G hld in an interretatin I, if there i a interretatin functin I which atifie the edge cnditin fr G. In thi rf, thi will be dented by (I, I) = G. A we have I = G, we have an Interretatin functin I a with (I, I a ) = G a. Fr each rule, we have t cntruct an interretatin functin I b with (I, I b ) = G b. 1. eraing an edge Fr I b := I a we bviuly have (I, I b ) = G b, ince fr G b, we have t check the ame edge cnditin, excet the cnditin fr e. 2. generalizing a label A bl de nt aear in G and all vertice v V are labeled the ame, I b := I a i an interretatin functin fr G b (the firt cnditin fr interretatin functin i atified). Again it i bviu that we have (I, I b ) = G b. 3. merging tw vertice We et I b := I a \{(v 1, I a (v 1 ))}. A I a (v 1 ) = I a (v 2 ), I b atifie (I, I b ) = G b. 4. litting a vertex We et I b := I a {(v 2, I a (v 1 ))}. I b i an interretatin functin (articularly, the firt cnditin fr interretatin functin i atified) with (I, I b ) = G b. 5. iterating an edge We et I b := I a. Fr checking (I, I b ) = G b, we have t check all edge cnditin fr G b, which are exactly the edge cnditin fr G a lu the additinal edge cnditin fr the freh edge e. Since e ha the ame 15

16 vertice a e, thi edge cnditin i atified, becaue the edge cnditin fr e i atified. Thu we have (I, I b ) = G b. Befre we hw that the rule are cmlete f well, we have t intrduce the ntin f Herbrand mdel. Herbrand mdel are cmmnly ued in cmletene rf fr mathematical lgic, articularly fr firt rder redicate lgic. Uually, they are nt uniquely defined, but thi i different fr RDF grah. A we have in RDF grah nly the ibility t exre exitential quantificatin and lgical cnjunctin, we can aign t each RDF grah a unique Herbrand mdel which encde exactly the ame infrmatin a the RDF grah. Fr thi rean, we call thee mdel tandard mdel a well. Definitin 11 (Herbrand Mdel). Let G := (V, E, ν, κ) be a RDF grah ver a vcabulary Vc. Then I G i defined a fllw: IR := URI Lit {bl Blank It exit a vertex v V with κ(v) = bl} IP := {κ(v) v V and there exit an edge e with e 2 = v} IEXT() = {(, ) it exit a e E with κ(e 1 ) =, κ(e 2 ) =, κ(e 3 ) = } IS and IL are the identity maing n URI re. Lit. The mdel I G i called Herbrand mdel r tandard mdel fr G. Uing tandard mdel f RDF grah, we can nw hw that the tranfrmatin rule are cmlete. A we have already hwn the undne a well, we therefre have that the calculu i adequate. Therem 2 (Cmletene f the Rule). Let G a and G b be tw RDF grah ver Vc with G a = G b. Then we have G a G b. Prf: Due t Lem. 1 we aume that bth G a := (V a, E a, ν a, κ a ) and G b := (V b, E b, ν b, κ b ) are in anti-nrmal frm. Mrever, a we cnider grah nly u t equivalence, we aume that G a and G b have n blank in cmmn. Let I a := (IR a, IP a, IEXT a, IS a, IL a ) be the Herbrand mdel f G a. We have I a = G b, i.e. there i an interretatin functin I b : V b IR a uch that fr each edge e = (v 1, v 2, v 3 ) E b and := κ(v 1 ), := κ(v 2 ), := κ(v 3 ), we have I b () IP and (I b (), I b ()) IEXT(I b ()). Due t the definitin f the Herbrand mdel, the maing λ N := I b i a maing frm the et f blank and name f G b int the et f blank and name f G a which i the identity maing n the name f Vc, and we have a maing λ E : E b E a which atifie Fr each e := (v 1, v 2, v 3 ) E b and λ E (e) := (w 1, w 2, w 3 ) E a ( ) we have λ N (κ(v i )) = κ(w i ) fr i = 1, 2, 3 The maing λ E i ibly neither injective nr urjective, but we can tranfrm G a in rder t make λ E bijective. Thi will be dne nw. Aume firt that we have λ E (e 1 ) = λ E (e 2 ) fr tw edge e 1, e 2 E b. Set f := λ E (e 1 ). We can aly the rule iterating and ilating an edge t f in rder t btain a cy f f f, and we can change λ E t λ E a fllw: We et λ E (e 1) := f and λ E (e 2) := f, therwie we et λ E (e) := λ E(e). A we d 16

17 nt add r remve name frm G b, the maing λ N remain unchanged. Thi te i reeated a ften a neceary until we btain a grah G a (1) frm G a (articularly, we have G a G a (1) ) with a maing λ (1) N := λ N and an injective maing λ (1) E : E b E a (1) which atifie ( ) a well. Next, we remve all edge f E a (1) which are nt in the range f λ (1) E. The reulting grah i dented by G a (2), the retrictin f λ (1) N and λ(1) E t G(2) a by λ (2) N and λ(2) E. Again, nw fr G(2) a, λ (1) N and λ(1) E, the cnditin ( ) i fulfilled. Obviuly, the maing λ (2) N i bijective. Mrever, a in G(2) a and G b each vertex i incident with exactly ne edge, in exactly ne lace, we btain mrever a bijective maing λ (2) V : V b V a (2) which reect the incidence relatin, i.e. if e E b i an edge and v = e i fr an i = 1, 2, 3; we have λ (2) V (v) = λ(2) E (e) i. That i, the underlying grah tructure f G a (2) and G b are imrhic, nly the label f the vertice differ. Fr each blank bl f G b we et V bl := {w V (2) a it exit a v V b with κ b (v) = bl and λ V (v) = w}. Nw, fr each uch blank bl, we can aly the rule generalizing a label t G a (2) in rder t label all vertice in V bl with the blank bl. Nte that thi i ible becaue we aumed that the et f blank in G a (thu the et f blank in G a (2) ) i dijint frm the et f blank in G b. The reulting grah G (2) b bviuly i imrhic t G a (wherea imrhim between RDF grah are defined cannically), thu we are dne. 5 Outlk Thi aer i nly a firt te t devel RDF a a diagrammatic reaning ytem. The authr are currently wrking n verin f thi lgic ytem where the grah frmalizatin f RDF grah are retricted t be in nrmal frm re. in anti-nrmal frm. Thi crrend t the trile mdel f RDF withut any rhetrical tructure in the grah frmalizatin. Fr thee cae, the calculu i nt cmlete anymre, and ha t be redeigned. The next lgical te i t extend thi ytem t cature RDF Schema emantic, articularly t the rerty and cla hierarchie. Mrever, it ha t be invetigated hw thi ytem can be extended t further lgical eratin, like r and nt. Fr thi, we can ue the reult f [4], where the cut f Peirce exitential grah, which are ued t lgically negate me ubgrah f the grah, are added t a fragment f cncetual grah which i very imilar t the RDF grah invetigated here. In [4], adding cut t cncetual grah can be cmared t make the te frm the (trivially decidable) exitential-cnjunctive fragment f firt rder redicate lgic t full firt rder redicate lgic. Reference 1. T. Berner-Lee: Cncetual Grah and the Semantic Web. 17

18 htt:// 2. T. Berner-Lee, E. Miller: The Semantic Web lift ff htt:// 3. T. Bray: Onging: DC Intel htt:// 4. F. Dau: The Lgic Sytem f Cncet Grah with Negatin (And It Relatinhi t Predicate Lgic). Lecture Nte in Artificial Intelligence, Vl. 2892, ISBN Sringer, Berlin Heidelberg New Yrk, J. Haye: A Grah Mdel fr RDF. Dilma thei, Techniche Univerität Darmtadt, Deartment f Cmuter Science, Germany, htt://url.rg/net/ jhaye/rdfgrahmdel.html (2004) 6. J. Haye, C. Gutierrez: Biartite Grah a Intermediate Mdel fr RDF. In: Prceeding f ISWC 2004, The Third Internatinal Semantic Web Cnference, Hirhima, Jaan, Nvember 7-11, Lecture Nte in Cmuter Science, Sringer Verlag (2004). 7. P. Haye: RDF Semantic: W3C Recmmendatin. 10 February htt:// 8. P. Haye, C. Menzel: A Semantic fr the Knwledge Interchange Frmat. Prceeding f 2001 Wrkh n the IEEE Standard Uer Ontlgy, Augut htt://reliant.teknwledge.cm/ijcai01/hayemenzel-skif-ijcai2001.df 9. G. Klyne, J.J. Carrll: Reurce Decritin Framewrk (RDF): Cncet and Abtract Syntax. W3C Recmmendatin. htt:// (10 February 2004) 10. F. Manla, E. Miller: RDF Primer. htt:// 11. J. F. Swa: Cncetual Structure: Infrmatin Prceing in Mind and Machine. Addin Weley Publihing Cmany Reading, J. F. Swa: Cncetual Grah Summary. In: T. E. Nagle, J. A. Nagle, L. L. Gerhlz, P. W. Eklund (Ed.): Cncetual Structure: current reearch and ractice, Elli Hrwd, 1992, J. F. Swa: Knwledge Rereentatin: Lgical, Philhical, and Cmutatinal Fundatin. Brk Cle Publihing C., Pacific Grve, CA,