RDF as Graph-Based, Diagrammatic Logic

Transcription

1 RDF a Grah-Baed, Diagrammatic Lgic Frithjf Dau Det. f Mathematic, Dreden Technical Univerity, Germany Abtract. The Reurce Decritin Framewrk (RDF) i the baic tandard fr rereenting infrmatin in the Semantic Web. It i mainly deigned t be machine-readable and -rceable. Thi aer take the ite ide f view: RDF i invetigated a a lgic ytem deigned fr the need f human. RDF i develed a a lgic ytem baed n mathematical grah, i.e., a diagrammatic reaning ytem. A uch, i ha humanly-readable, diagrammatic rereentatin. Mrever, a und and cmlete calculu i rvided. It rule are uited t act n the diagrammatic rereentatin. Finally, me nrmalfrm fr the grah are intrduced, and the calculu i mdified t uit them. 1 Intrductin: RDF and Grah The Reurce Decritin Framewrk (RDF) i the baic tandard fr rereenting infrmatin in the Semantic Web. The infrmatin i encded in tatement, which are ubject-redicate-bject trile f infrmatin. RDF can be een frm tw erective: cmuter and human. Fr cmuter, it i imrtant that RDF i a machine rceable data-frmat. Thu, much effrt i ent n ecificatin which uit thi ure (e.g. RDF/XML). An arach t make RDF acceible fr human can already be fund in the RDF-rimer [8], where the et f trile i cnverted int a mathematical grah: The ubject and bject f all tatement are maed t vertice, and each tatement i maed t a directed edge between it ubject and bject, labeled by it redicate. Cnider the fllwing et f trile T 1 and it rereentatin a grah: (Ernt,aint,hit. naturelle),(ernt,aint,flying geee), (Ernt,aint,le leil),(hit. naturelle,ha tyle,urrealim), (flying geee,ha tyle,dadaim),(le leil,ha tyle,dadaim) Ernt aint ha_tyle hitire naturelle urealim aint ha_tyle flying geee dadaim aint le leil ha_tyle dadaim The tranlatin f trile et t thee grah i aealing, but i ha tw rblem. RDF allw a tatement redicate t aear a the ubject f anther tatement. E.g., {(tye tye r), (ubject tye r)} i a well-frmed et f trile, but it cannt be tranfrmed t a grah with the cnventin f [8]. The firt gal f the aer i cle the ga that nt all trile et can be rereented a grah. Fr thi, we adt the arach f [1, 2] intead: RDF trile et are tranlated t hyer grah, where ubject, bject, and redicate are

2 maed t vertice, and each tatement (,, ) i maed t an 3-ary hyeredge linking thee vertice. RDF i ued fr infrmatin-rceing. In [3] an entailment relatin between trile et i intrduced. But thi relatin de nt the diagrammatic rereentatin f RDF. It i well acceted that diagram are ften eaier t cmrehend than ymblic ntatin ([7, 6]). The ecnd gal f the aer i, after adting the emantic f [3], t rvide an adequate diagrammatic calculu. In thi undertanding, RDF i develed a diagrammatic reaning ytem (DRS), in a humanly readable and mathematically recie manner. In the next ectin, an verview f the yntax, emantic, a und and cmlete calculu, and me nrmalfrm f RDF Grah fr RDF a DRS i rvided. Due t ace limitatin, me frmal definitin an all rf are mitted. They can be fund in a technical rert n 2 Syntax In thi ectin we define the yntax f the ytem, tarting with the vcabulary. Definitin 1. Let Blank := { 1, 2, 3,..., } be a et f -called blank. A trile Vc := (URI, TyedLit, PlainLit) i called vcabulary. The element f thee et are called univeral reurce identificatr, 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. We aume that Blank, URI, TyedLit, PlainLit are airwie dijint. Next we define the well-knwn trile ntatin f RDF a well a the crrending mathematical grah. T avid cnfuin, we ue the term RDF trile et intead f the cmmn term RDF grah fr the trile ntatin. Definitin 2. An RDF trile et ver Vc i a et f trile (,, ) with URI Blank, URI, URI Blank Lit. We call the ubject, the rerty and the bject f the trile. A tructure (V, E, ν, κ) i an RDF Grah ver Vc iff V and E are finite et f vertice and edge, ν : E V 3 i a maing uch that each vertex i incident with a leat ne edge, and κ : V Vc i a maing uch that fr each e E, we have κ(e 1 ) (URI Blank), κ(e 2 ) URI, and κ(e 3 ) (URI Blank Lit). Fr e E with ν(e) = (v 1, v 2, v 3 ) we will ften write e = (v 1, v 2, v 3 ), and each v i, i = 1, 2, 3 i dented by e i. Fr v V we et E v := {e E i.ν(e) i = v}, and fr e E we et V e := {v V i.ν(e) i = v}. The et {bl Blank v V : κ b (v) = bl} i called the et f blank f G. Cmared t RDF trile et, RDF grah rvide a richer mean t exre in different way me given amunt f infrmatin, and redicate which aear a the ubject f ther tatement d nt caue rblem. Thi hall be exemlified with the firt fur aximatic trile frm the RDF-emantic (ee [3]). They are: (rdf:tye rdf:tye rdf:prerty) (rdf:ubject rdf:tye rdf:prerty) (rdf:redicate rdf:tye rdf:prerty) (rdf:bject rdf:tye rdf:prerty) 2

3 Nte that tye ccur bth a ubject and a redicate in thee trile, thee fur trile cannt be dilayed due t the grah-drawing cnventin f [5]. But with the herein defined RDF grah, there are nw different ibilitie t tranfrm thi et f trile t a grah. Three f them are deicted belw. rdf:tye rdf:tye 1) rdf:ubject rdf:prerty rdf:redicate rdf:prerty rdf:tye rdf:tye rdf:bject rdf:prerty rdf:tye rdf:prerty 2) rdf:ubject rdf:redicate rdf:bject rdf:tye rdf:prerty 3) rdf:ubject rdf:redicate rdf:bject rdf:tye rdf:tye rdf:prerty In the firt grah, each vertex i incident exactly nce with an edge: It ha t-eak a maximum number f vertice. Grah which atify thi cnditin will be aid t be in anti-nrmal-frm (ANF). Fr the third grah, we have an ite ituatin: N different vertice are labeled the ame, i.e., thi grah cntain the minimal number f vertice. Such grah are aid t be in nrmalfrm (NF). Fr grah in ANF r NF, we aume mrever that they d nt have redundant edge, i.e., fr all edge e = (v 1, v 2, v 3 ) and f = (w 1, w 2, w 3 ) with κ(v i ) = κ(w i ) fr i = 1, 2, 3, we have e = f. Let T be an RDF trile et ver Vc. Then let Φ N (T ) := (V, E, ν, κ) be the fllwing RDF grah in NF: Fr each name r blank x in T, let v x be freh 1 vertex. Let V be the et f thee vertice. Let E := T. Fr e = (,, ) E, let ν(e) = (v, v, v ). Fr v = v x V, let κ(v x ) := x. Similarly, let Φ AN (T ) := (V, E, ν, κ) be the fllwing RDF grah in ANF: Fr each edge e = (,, ) E, let e, e, e be freh vertice. Let V be the et f thee vertice, let E := T, and fr e = (,, ) E, let ν(e) := (e, e, e ). Fr v = e x V, we et κ(v x ) := x. Nw let G := (V, E, ν, κ) be an RDF Grah. Let Ψ(G) be the fllwing RDF trile et: Ψ(G) := { ( κ(e 1 ), κ(e 2 ), κ(e 3 ) ) e E}. Each RDF grah in NF re. in ANF i already been ufficiently decribed by the et f trile ( κ(e 1 ), κ(e 2 ), κ(e 3 ) ) with e E. If T i an RDF trile et, we have T = Ψ(Φ N (T )) and T = Ψ(Φ AN (T )). Vive vera: If G i an RDF grah in NF, we have G = Φ N (Ψ(G)), and if G i an RDF grah in ANF, we have G = Φ AN (Ψ(G)). S the maing Φ N, Φ AN and Ψ can be undertd a tranlatin between grah in NF r ANF and trile et. A ubgrah f an RDF trile et i imly a ubet. A ubgrah f an RDF grah (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. 1 A vertex r edge x i called freh iff it i nt already an element f the et f vertice and edge we cnider. 3

4 We will imlicitly identify RDF Grah if they differ nly in the name f the ccurring blank. A in [3], grah like thee are called equivalent. Finally, we have t define the jin and merge f RDF grah. The jin f RDF trile et in defined in [3] t be the et-theretical unin. But nly if the jined RDF trile et have n blank in cmmn, their jin crrend t their lgical cnjunctin. Then ne eak f the merge f the RDF trile et. Analguly, if G i := (V i, E i, ν i, κ i ) i fr i = 1,..., n an RDF grah uch that 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 RDF grah G := (V, E, ν, κ) with V := i V i, E := i E i, ν := i ν i and κ := i κ i. If the et f blank f the grah G i are airwie dijint, the jin f the grah G i i al called merge 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 [3, 4] fr RDF trile et. Definitin 3. A mdel I fr 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, and IL : Lit IR i a maing. Nte that the relatinhi between IR and IP i nt ecified. Frm the viewint f 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. Bth cae and arbitrary ther relatin between IR and IP are allwed. In the next definitin, RDF grah are evaluated in mdel. We d nt aume that the grah and mdel are baed n the ame vcabulary. But if a name in a grah i nt interreted in the mdel, the emantic will alway yield that the grah evaluate t fale. Thu we will uually aume that the vcabularie f a grah and an interretatin are the ame. Definitin 4. Let G := (V, E, ν, κ) be an RDF grah ver Vc G and let I := (IR, IP, IEXT, IS, IL) be an interretatin ver Vc I. A functin I : V IR i an interretatin functin (fr G in I), iff fr all v 1, v 2 V with κ(v 1 ) = κ(v 2 ), we have I(v 1 ) = I(v 2 ), and fr each each name n URI I PlainLit I TyedLit I and each vertex v V with κ(v) = n, we have I(v) = (IL IR)(n). We ay the grah G hld in the mdel I and write I = G, iff fr each v V, κ(v) i a name f Vc I r a blank, and 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(), I()) IEXT(I()) and I() IP. If G a, G b are RDF grah uch that I = G b hld fr each I with I = G a, we ay that G a (emantically) entail G b, and write G a = G b. 4 Calculu We rvide a calculu with five rule (which huld be undertd t maniulate the diagram f RDF grah) fr G. Let G := (V, E, ν, κ) be given. 4

5 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 ) with 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. Generalizing a label: Let bl Blank be a blank which de nt aear in G. 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. Merging tw vertice: Let v 1 v 2 be tw 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. Slitting a vertex: Let v 1 be a vertex, incident with edge e 1,..., e n. Then we can inert a freh vertex v 2 with κ(v 2 ) := κ(v 1 ), and n e 1,..., e n, arbitrary ccurrence f v 1 may be relaced by v 2. That i, we cntruct a 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 well-frmed). 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 ))}. Imrhim: Tw grah which are imrhic are imlicitly identified. A finite equence (G 1, G 2,..., G n ) uch that each G i+1 i derived frm G i by alying ne f the rule abve i a rf. We ay that G n can be derived frm G 1 and write G a G b. Tw grah G a, G b with G a G b and G b G a are aid t be rvably equivalent. The calculu i adequate, i.e. we have: Therem 1. Tw grah G a, G b atify G a G b G a = G b 5 Retricting the Grah t Nrmal Frm We have een that trile et directly crrend t grah in NF (r in ANF). Thu the quetin arie whether we can ue the reult f the lat ectin, articularly the und- and cmletene f the calculu, fr the ytem f RDF grah in NF and the ytem f RDF grah in ANF. Clearly, the rule f the calculu are till und if the ytem f grah i yntactically retricted. But aume we retrict urelve t grah in NF: It i nt allwed t lit a vertex, a thi yield a grah which i nt in NF. Vice vera, it i nt ible t aly the rule merging tw vertice : We never find tw different vertice which are identically labeled. Let u cnider the fllwing valid entailment between tw RDF trile et: ( 1 2 ) = ( 1 2 ), ( 1 2 ). Thi entailment can directly tranlated t a valid entailment fr grah in NF r in ANF, uing the maing Φ N r Φ AN. But in nne f the retricted ytem, 5

6 we can find a frmal rf within the given calculu. S fr each retricted ytem, we have t alter the calculu t btain cmletene. The rule eraing an edge and generalizing a label tranfrm RDF grah in ANF int RDF grah in ANF again. A third rule i btained a fllw: By iterating an edge e and then litting each f it three incident vertice, we can btain a cy f e with freh vertice. The cmbinatin f thee rule i called iterating and ilating the edge e. Thi rule tranfrm RDF grah in ANF int RDF grah in ANF, t. We will write G a an G b, if G b can be derived frm G a with thee three rule. The rule eraing an edge and generalizing a label can be ued fr RDF grah in NF a well. If a grah G b i derived frm G a by firtly litting a vertex v 1 int v 1 and a freh vertex v 2, and then by generalizing the label f v 2, then we ay that G b i derived frm G a by litting and generalizing a vertex. Thee three rule tranfrm RDF grah in NF int RDF grah in NF. We will write G a n G b, if G b can be derived frm G a with thee three rule. Therem 2. Let G a and G b be tw RDF grah with G a = G b. If bth grah are in NF, we have G a n G b. If bth grah are in ANF, we have G a an G b. 6 Outlk Thi aer i nly a firt te fr develing the underlying lgic f the Semantic Web a mathematical DRS. The exreivity f RDF i rather weak. S, f cure, it ha t be invetigated hw well-knwn extenin f RDF can be develed a DRS. A aer which invetigate hw Decritin Lgic, which are the underying lgic f the tate-f-the-art SW-language OWL, i in rearatin. Reference [1] 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) [2] J. Haye, C. Gutierrez: Biartite Grah a Intermediate Mdel fr RDF. In: Prceeding f ISWC 2004, LNCS 3298, Sringer, [3] P. Haye: RDF Semantic: W3C Recmmendatin. 10 February htt:// [4] 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 [5] G. Klyne, J.J. Carrll: Reurce Decritin Framewrk (RDF): Cncet and Abtract Syntax. W3C Recmmendatin. htt:// [6] R. Kremer: Viual Language fr Knwledge Rereentatin. Prc. f (KAW 98) Banff, Mrgan Kaufmann, [7] J. H. Larkin H. A. Simn: Why a diagram i (metime) wrth ten thuand wrd. Cgnitive Science, 11, [8] F. Manla, E. Miller: RDF Primer. htt:// 6