A METHOD TO REPRESENT THE SEMANTIC DESCRIPTION OF A WEB SERVICE BASED ON COMPLEXITY FUNCTIONS

Transcription

1 UPB Sc Bull, Sere A, Vol 77, I, 5 ISSN 3-77 A METHOD TO REPRESENT THE SEMANTIC DESCRIPTION OF A WEB SERVICE BASED ON COMPLEXITY FUNCTIONS Andre-Hora MOGOS, Adna Magda FLOREA Semantc web ervce repreent an mportant and very actve reearch area n computer cence The emantc decrpton of a web ervce ha a crucal role when workng wth emantc web ervce In th paper we propoe a method for repreentng the emantc decrpton of a web ervce ung complexty functon The man reult of our paper a theorem that lnk the emantc decrpton, repreented a complexty functon, wth everal mportant aymptotc notaton Keyword: emantc decrpton of a web ervce, complexty functon, aymptotc notaton Introducton A pecfed n [], a web ervce a oftware ytem degned to upport nteroperable machne-to-machne nteracton over a network A web ervce a oftware ytem that run on a computer n a network (for example, n the Internet) and offer a ervce that can be acceed by another computer from the ame network The Semantc Web a machne-readable and machne-proceable web A pecfed n [], the Semantc Web not a eparate Web but an extenon of the current one, n whch nformaton gven well-defned meanng, better enablng computer and people to work n cooperaton The Semantc Web a web that allow machne (computer) to read, undertand and proce the nformaton; th made by addng machne-proceable emantc to the document Semantc web ervce are web ervce wth a formal emantc decrpton [3] Semantc web ervce repreent a technology that combne the web ervce and the Semantc Web The work on emantc web ervce very actve and a gnfcant number of reult were obtaned n the lat year In the area of emantc web ervce, the emantc decrpton (the emantc) of a web ervce ha a crucal role The man approache propoed n the lterature for the emantc of web ervce are the followng [4]: OWL-S [5], WSDL-S [6], and WSMO [7] Accordng to [8], ome newer approache are SAWSDL [9] and WSMO-Lte [] All thee approache repreent the emantc Lecturer, Computer Scence Department, Unverty POLITEHNICA of Bucharet, Romana, e-mal: andremogo@cpubro Profeor, Computer Scence Department, Unverty POLITEHNICA of Bucharet, Romana, e-mal: adnaflorea@cpubro

2 4 Andre-Hora Mogo, Adna Magda Florea decrpton of a web ervce by ung ontologe expreed n XML-compatble language Thu, thee approache repreent a emantc decrpton a a et of word In th paper we propoe a method of repreentng the emantc decrpton of a web ervce ung a pecal type of functon, called complexty functon Th way, each emantc decrpton wll be a complexty functon, n fact a equence of potve real number The mot mportant dfference between our approach and the man approache n the lterature that we ue real number ntead of word for repreentng emantc decrpton The man reult of our paper a theorem that lnk the emantc decrpton repreented by complexty functon wth the aymptotc notaton The paper organzed a follow Secton contan a bref dcuon about complexty functon and aymptotc notaton In Secton 3 we propoe a new approach wth repect to the repreentaton of the emantc decrpton of web ervce Secton 4 contan the man reult of the paper: we propoe and prove a theorem that lnk the emantc decrpton, repreented ung our approach, wth the aymptotc notaton In the end, Secton 5 preent the concluon of the paper Complexty functon and aymptotc notaton We denote by N the et of potve nteger and by R + the et of potve real number A complexty functon a functon N R + (ee, for example, [], []) Let be g : N R+ an arbtrary, fxed complexty functon We preent fve well known aymptotc notaton (ee, for example, [], [], [3], [4]): Θ ( { f : N R+ c, c R+, n N uch that () c f c, n } O( { f : N R+ c R+, n N f c, n } ( { f c : N f, n } c R Ω R+ +, n N o( { f : N R+ c R+, n N f < c, n } uch that uch that uch that ω( { f : N R+ c R+, n N uch that c < f, n} We alo preent other two aymptotc notaton (ee, []): () (3) (4) (5)

3 A method to repreent the emantc decrpton of a web ervce baed on complexty functon43 oθ ( O( \ ( o( Θ( ) (6) Θ ω( Ω( \ ( Θ( ω( ) (7) 3 Repreentng emantc decrpton ung complexty functon In th ecton we preent our approach wth repect to the repreentaton of the emantc decrpton of web ervce 3 Dctonary Conder a et of word W We make the followng aumpton: ) Each word from W ha a ngle meanng; ) From our context we can ealy obtan the anwer for the followng queton: a) Have two word the ame meanng? b) I a word more general than another one? Defnton The emantc equalty of two word w, w W, w w w and w have the ame meanng Theorem The bnary relaton on W an equvalence relaton Proof The proof follow from Defnton Theorem All the word wth the ame meanng from the et W form an equvalence cla Proof The proof follow from Defnton and Theorem Let be ( C ) NC the famly of equvalence clae determned by on W Theorem 3 The famly of equvalence clae ( C form a partton of W ) NC Proof The proof follow from Defnton, Theorem and Theorem Defnton Dctonary The dctonary aocated to W and D ( ) C NC Defnton 3 The meanng functon The meanng functon m aocated to the dctonary D : m : W {,,, NC}, m ( w), where w C (8) We ay that m (w) the meanng of the word w Theorem 4 Properte of the meanng functon a) The meanng functon m urectve b) If there ext {,,, NC} uch that card( C ) > then the meanng functon m not nectve c) If card( C ), {,,, NC} then the meanng functon m bectve Proof The proof follow from Defnton 3 3 An order relaton between the equvalence clae Defnton 4 The emantc nequalty of two word

4 44 Andre-Hora Mogo, Adna Magda Florea w, w W, w w w more general than w n term of meanng or w w Example If we conder the word car and vehcle, then we have the relaton: car vehcle and car vehcle Theorem 5 a) The bnary relaton on W a preorder relaton b) If there ext {,,, NC} uch that card( ) > then the bnary relaton on W not an order relaton c) If card( C ), {,,, NC} then the bnary relaton on W an order relaton Proof The proof follow from Defnton 4 Defnton 5 The repreentatve word of an equvalence cla For each equvalence cla C, {,,, NC} we chooe a word w C Th word wll be called the repreentatve word of cla C Defnton 6 The emantc equalty of two equvalence clae C, C D, C C c Theorem 6 The bnary relaton Proof The proof follow from Defnton 6 Remark Each equvalence cla determned by C c on D an equvalence relaton c on D ha only one member Defnton 7 The emantc nequalty of two equvalence clae C, C D, C C w w c Theorem 7 The bnary relaton c on D an order relaton Proof The proof follow from Defnton 7 Remark One can oberve that, f C, C D wth the properte C c C and then C c C 33 Repreentaton of Semantc Decrpton Conder the followng emantc decrpton of a web ervce: w w w l, where w, w,, w W l We want to expre th emantc decrpton a a complexty functon For th purpoe, we wll preent everal tep for tranformng the ntal emantc decrpton nto a complexty functon A a frt tep the emantc decrpton wll be repreented a follow:

5 A method to repreent the emantc decrpton of a web ervce baed on complexty functon45 :{,,, NC} {,}, f w { w, w,, } w uch that w C l n (9), otherwe A a econd tep, we extend the doman of the functon to N A new emantc decrpton obtaned: : N {,}, ( ( n mod NC) () where we defne mod a followng: n mod NC n mod NC, f n mod NC and n mod NC NC, f n mod NC Oberve that the functon may return the value and a complexty functon ha only potve real value Alo, we want that the reult of the functon to depend on n The new emantc decrpton the followng: Defnton 8 The emantc decrpton of a web ervce The emantc decrpton of a web ervce : n +, f ( : N R, ( + /( n + ), f ( () Example Conder the dctonary D Suppoe we have the followng emantc decrpton expreed n word: ell whte car Alo uppoe that ell C n, whte C n, and car C n The emantc decrpton n form () : 3 n +, f ( n mod NC) { n, n, n3} and /( n + ), otherwe Defnton 9 Two approxmaton of a emantc decrpton Let be Q ( n, p) and Q ( n, p) two predcate wth the followng form: Q ( n, p) n, p {,,, NC}, n p, uch that /( n+ ), ( p) p+ andcn c Cp () Q ( n, p) n, p {,,, NC}, n p, uch that n+, ( p) /( p+ ) andc C n c p Let be n, p {,,, NC} uch that Q ( n, p ) An approxmaton of type of the emantc decrpton wth repect to n and p the followng: ( n + ), f n n () () n, p : N R +, n, p /( n + ), f n p (3) (, otherwe Let be n, p {,,, NC} uch that Q ( n, p ) An approxmaton of type of the emantc decrpton wth repect to n and p the followng:

6 46 Andre-Hora Mogo, Adna Magda Florea /( n + ), f n n () () n, p : N R +, n, p n +, f n p (4) (, otherwe Example 3 a) Suppoe we have a emantc decrpton and two ndexe, {,,, NC}, Conder that ( ) /( + ) and ( ) + Alo conder that the cla C correpond to the meanng car and the cla C correpond to the meanng vehcle An approxmaton of type of repreent a weaker emantc decrpton (the one that ha car ntead of vehcle) b) Suppoe we have a emantc decrpton and two ndexe, {,,, NC}, Conder that ( ) + and ( ) /( + ) Alo conder that the cla C correpond to the meanng car and the cla C correpond to the meanng vehcle An approxmaton of type of repreent a tronger emantc decrpton (the one that ha vehcle ntead of car) 34 The et of all emantc decrpton Defnton The et of all emantc decrpton We denote by ASD the et of all emantc decrpton, both exact and approxmate Defnton The equalty of two emantc decrpton, Theorem 8 The bnary relaton on ASD an equvalence relaton Proof The proof follow from Defnton Defnton The nequalty of two emantc decrpton,, N Theorem 9 The bnary relaton on ASD an order relaton Proof The proof follow from Defnton Remark 4 Let be, ASD We make the followng conventon: a) We wrte < f <, N b) Another form of the relaton c) Another form of the relaton < > 4 Semantc decrpton and aymptotc notaton Th ecton contan the man reult of the paper: a theorem that lnk the emantc decrpton and the aymptotc notaton Theorem Semantc decrpton and aymptotc notaton a), Θ( )

7 A method to repreent the emantc decrpton of a web ervce baed on complexty functon47 b), O( ) c), Ω( ) d), < o( ) e), > ω( ), f), </ and oθ(, g), >/ and Θω( ) ) Proof a) " " If then ; from the reflexvty of Θ we have that Θ( ) " " Let be Θ( ) From Defnton 8 and Defnton 9 the value of the functon and for the argument n can be ( n + ), n+ or /(n+) Suppoe (for a contradcto that n N uch that Conder that ( n mod NC) k, where k {,,, NC} a) If /( n + ) and n + then, from Defnton 8 and Defnton 9, we have that ( p) /( p + ) and ( p) p +, p N uch that ( p mod NC) k From Θ( ) we have: c, c R+, n N uch that (5) c c, n Let be p N wth ( p mod NC) k, p n and c ( p + ) > From (5) we have c ( p + ) /( p + ) c ( p + ) It follow that, c ( p + ) ; th contradct the nequalty c ( p + ) > a) If /( n + ) and ( n + ) then the reult follow ung the ame dea ued for a) a3) If n + and ( n + ) then the reult follow ung the ame dea ued for a) a4) If n + and /( n + ), we ue the ymmetry of Θ : Θ( ) Θ( ) and then the ame dea ued for a) a5) If ( n + ) and /( n + ) then the reult follow ung the ame dea ued for a4) a6) If ( n + ) and n + then the reult follow ung the ame dea ued for a4)

8 48 Andre-Hora Mogo, Adna Magda Florea For all the x cae we found a contradcton Conequently,, N It follow that b) " " If then, N It follow that c R+, n N uch that (6) c, n Conequently, O( ) " " Let be O( ) From Defnton 8 and Defnton 9 the value of the functon and for the argument n can be ( n + ), n+ or /(n+) Suppoe (for a contradcto that n N uch that > Conder that ( n mod NC) k, where k {,,, NC} b) If n + and /( n + ) then, from Defnton 8 and Defnton 9, we have that ( p) p + and ( p) /( p + ), p N uch that ( p mod NC) k From O( ) we have: c R+, n N uch that (7) c, n Let be p N wth ( p mod NC) k, p n and ( p + ) > c From (7) we have p + c /( p + ) It follow that ( p + ) c ; th contradct the nequalty ( p + ) > c b) If ( n + ) and /( n + ) then the reult follow ung the ame dea ued for b) b3) If ( n + ) and n + then the reult follow ung the ame dea ued for b) Conequently,, N It follow that c) The proof follow from b) and the tranpoed ymmetry of O and Ω : O( ) Ω( ) d) " " If < then <, N From Defnton 8 and Defnton 9 the value of the functon and for the argument n can be ( n + ), n+ or /(n+) d) If /( n + ), N and n +, N then o( )

9 A method to repreent the emantc decrpton of a web ervce baed on complexty functon49 d) If /( n + ), N and ( ) ( ) n n +, N then o( ) d3) If n +, N and ( ) ( ) n n +, N then o( ) Conequently, o( ) " " Let be o( ) Snce o( ) O( ) and ung b) we have that ; conequently,, N Suppoe (for a contradcto that n N uch that Conder that ( n mod NC) k, where k {,,, NC} From Defnton 8 and Defnton 9, we have that ( p) ( p), p N uch that ( p mod NC) k From o( ) we have: c R+, n N uch that (8) < c, n Let be p N wth ( p mod NC) k, p n and let be c R+ Ung (8) we have ( p) < ( p) and th a contradcton Conequently, <, It follow that N < e) The proof follow from d) and the tranpoed ymmetry of o and ω : o( ) ω( ) f) Ung a), b), d) and (6) we have: O( ) oθ( ) o( ) </ (9) Θ( ) g) The proof follow from f) and the tranpoed ymmetry of o Θ and Θ ω : oθ( ) Θω( ) 5 Concluon In th paper we propoe a method for repreentng the emantc decrpton of a web ervce baed on complexty functon Whle the man approache n the lterature repreent a emantc decrpton a a et of word, we repreent a emantc decrpton a a complexty functon, and thu a a equence of real number The man reult of the paper cont of a theorem that lnk the emantc decrpton expreed a complexty functon wth everal mportant aymptotc

10 5 Andre-Hora Mogo, Adna Magda Florea notaton Th theorem how that any two emantc decrpton (repreented wth our approach) can be compared ung aymptotc notaton Our method mple and eay to ue Alo, a emantc decrpton repreented a a complexty functon can be ealy read and nterpreted by computer (oftware program), nce t cont of real number Conequently, our approach utable to be ued for varou reearch topc n the area of emantc web ervce Acknowledgement The work ha been funded by the Sectoral Operatonal Programme Human Reource Development 7-3 of the Mntry of European Fund through the Fnancal Agreement POSDRU/59/5/S/3397 R E F E R E N C E S [] Web Servce Archtecture W3C Workng Group Note, February 4, ee lat acceed June 4 [] T Berner-Lee, J Hendler and O Lala, The Semantc Web, n Scentfc Amercan, May [3] F Lécué, A Deltel, Makng the Dfference n Semantc Web Servce Compoton, n Proceedng of the nd Conference on Artfcal Intellgence (AAAI-7), -6 July 7, Vancouver, Brth Columba, Canada, vol, pp [4] J Cardoo, Approache to Developng Semantc Web Servce, n Internatonal Journal of Computer Scence, vol, no, 6, pp 8- [5] OWL-S: Semantc Markup for Web Servce, W3C Member Submon, November 4, ee lat acceed June 4 [6] Web Servce Semantc - WSDL-S, W3C Member Submon, November 5, ee lat acceed June 4 [7] Web Servce Modelng Ontology (WSMO), W3C Member Submon, June 5, ee lat acceed June 4 [8] M Lanthaler, M Grantzer and C Gutl, Semantc Web Servce: State of the Art, n Proceedng of IADIS Internatonal Conference on Internet Technologe & Socety, 9 November December, Perth, Autrala, pp 7-4 [9] Semantc Annotaton for WSDL and XML Schema, W3C Recommendaton, Augut 7, ee lat acceed June 4 [] WSMO-Lte: Lghtweght Semantc Decrpton for Servce on the Web, W3C Member Submon, Augut, ee lat acceed June 4 [] A H Mogo, A M Florea, A Method to Compare Two Complexty Functon Ung Complexty Clae, n UPB Scentfc Bulletn, Sere A: Appled Mathematc and Phyc, vol 7,,, pp [] C A Gumale, Introducere n Analza Algortmlor Teore Aplcate (Introducton to the Analy of Algorthm Theory and Applcato, Polrom, Bucharet, Romana, 4, pp 3-38 [3] T H Cormen, C E Leeron, R L Rvet and C Sten, Introducton to Algorthm, Second edton, MIT Pre, Cambrdge Maachuett, London England, USA,, pp 3-6, pp 4-5 [4] A H Mogo, Three Varant of the Mater Theorem, n IEEE Proceedng of CSCS 3, the 9th Internatonal Conference on Control Sytem and Computer Scence, 9-3 May 3, Bucharet, Romana, pp 6-66