Modeling and Analysis of Equivalences in Behavior Ontology

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1 Modelng and Analyss of Equvalences n ehavor Ontology MOONKUN LEE, YOUNGOK CHOE Department of Computer Engneerng Chonbu Natonal Unversty Jeonu, Jeonbu, Republc of Korea, Seoul, Republc of Korea moonun@bnu.ac.r Abstract Equvalences are classcal problems n computatonal scence snce the brth of computer scence. In 956, Norm Chomsy ned the noton of strong equvalence based on smlarty between two parse trees, as well as the noton of wea equvalence. In 989, Robn Mlner ned the noton of strong bsmulaton based on smlarty between labelled transton systems, as well as the noton of wea bsmulaton or observable equvalence. In practce, these notons are very tme and space consumng wor due to ther structural bases. Snce they are based on sem-formal structures, such as, parse tree and automata, analyss and verfcaton for the equvalences and bsmulatons tae algorthmc or model-checng processes. In other words, t s hard to apply mathematcal approach for the analyss and verfcaton. Is there any way of nng, analyzng and verfyng the equvalences, based on some mathematcal structure? The paper proposes an approach based on behavor ontology. The ontology nes dfference levels of abstractons based on the noton of behavors. Snce some actons between the behavors can be overlapped, they can be organzed n some ordered structure, namely, ehavor Lattce. ut the lattce has specal propertes whch are dfferent from the regular lattce. The man property s that the lattce can have multple ons and meets. It mples that the relatons between two behavors n the lattce can be nterpreted n a number of dfferent meanngs. Ths s very nterestng property n terms of equvalence, but can be seen as a natural one snce meets can be the abstract behavors of two behavors n dfferent abstract perspectves. Consequently, the lattce reveals polymorphc nterpretatons of equvalences among behavors, based on degree of abstracton. The comparatve study shows that the ontology s very effectve and effcent for representng such abstract behavors and verfyng strong and wea equvalnces n a lattce structure. The ontology can be consdered as one of the unque and nnovatve structure to represent such behavors n a herarchcal structure of abstracton. Key-Words: ehavor Ontology, Equvalence, smulaton, n:2-lattce, Abstracton. I. INTRODUCTION Equvalences are classcal problems n computatonal scence snce the brth of computer scence. In 956, Norm Chomsy ned the noton of strong equvalence based on smlarty between two parse trees, as well as the noton of wea equvalence[] n the theory of formal languages. In 989, Robn Mlner extended the noton of bsmulaton [2] and ned the noton of strong bsmulaton (~ based on smlarty between labeled transton systems, as well as the noton of wea bsmulaton or observable equvalence ( []. In practce, t s not easy to apply these notons snce they are based on sem-formal structures, that s, parse tree and automata. To analyze and verfy the equvalences and bsmulatons, some algorthmc or model-checng processes have to be appled, whch are very tme and space consumng wors. In case that composton of tress and automata are necessary, the wors become more complcated due to state-exploson problem. Further and( and or( operatons n CCS of Mlner mae the analyss and verfcaton more complcated or, n some cases, ncomprehensble. How can we handle ths sort of problems? Is there any way to handle the problems formally n mathematcal structure? Ths paper propose a mathematcal approach based on ehavor Ontology [], where was ned as a framewor to ne a lattce structure for process algebra, based on the notons of acton, behavor, abstract behavor, behavor lattce, and behavor ontology, as follows. Note that the man entty of acton and behavor s actor: Actons: A drected bnary relaton between two actors n a system. Smlar to Actve Ontology [6]. 2 ehavor: A sequence of the actons n. Abstract ehavor: Abstracton of behavor n 2 wth respect to cardnalty and capacty of actors of the behavor. Cardnalty and capacty mply the number of actors and the degree of nteractons wth other actors n behavor. ehavor Lattce: ehavors n can be nested, formng a lattce structure, especally n quantfed structure wth respect to the cardnalty and capacty. 5 ehavor Ontology: The lattces n can be ISN:

2 nter-connected to form a lattce of lattces, that s, ehavor Ontology. However [] ddn t present a mathematcal model for the ontology. In ths paper, the model, called n:2-lattce, s presented n mathematcal nton. Here n mples the degree of meets and ons n lattce, and 2 does the degree of upper and lower lmts n lattce. The noton of n s natural n terms of abstract behavors wth respect to cardnalty and capacty. It mples that polymorphc nterpretatons of n-ons between two behavors are possble at the dfferent levels of abstracton wth respect to quantfed actors n the behavors. It maes the nton, and analyss and the verfcaton of the abstract behavors more meanngful. In the ontology based on n:2-lattce, t s possble to ne the followng two equvalences: Strong equvalence: All the behavors at the certan levels n the lattce are nterpreted as same quanttatve behavors. 2 Wea equvalence: All the two behavors wth a same meet n the lattce are nterpreted as same abstract behavors n the quanttatve dmenson of abstracton. Ths s the structural dfference between the referenced equvalences and the ontology-based equvalences. There s no algorthmc or model-based checng for the equvalences. The locatons of behavors and relatons among them n the lattce determne the equvalences. Ths s the man advantage of the approach. Ths paper s organzed as follows. Secton II, III and IV ne the bref notons of n:2-lattce, ehavor Ontology and the equvalences. Secton V, VI and VII show the examples of the notons. Fnally, Secton VII presents concluson and future researches. II. N:2-LATTICE Ths secton presents a bref nton of n:2-lattce. [Def. ] n-lattce In POSET L,, f the followng two condtons are satsfed, the L, s ned as n-lattce: The exstence of bnary addton. If not, that s, f there s no least upper bound n {a, b}, more than one on exst between two elements a and b. 2 The exstence of bnary multplcaton. If not, that s, f there s no greast lower bound n {a, b}, more than one meet exst between two elements a and b. It s denoted by L,, n The man characterstc of the n-lattce s that t can have multple ons and meets, whch s the volaton of the general lattce. However t can be nterpreted polymorphcally between two elements wth respect to type of relatons between them. [Def. 2] Super-Greatest Element, Super-Least Element Super-Greatest Element (SGE: The greatest element of all the greatest elements n a lattce. 2 Super-Least Element (SLE: The least element of all the least elements n a lattce. In n-lattce, t s hard to control the fnal structure of the lattce due to the recursve exponental growth of ons. In some meanngful doman of elements, t could be necessary to control such a growng behavor, by nng some upper and lower lmts. Fg. shows example wth SGE and SLE. Fg. Examples wth SGE and SLE. [Def. ] n:2-lattce n-lattce L,, n wth SGE and SLE s ned as n:2-lattce, and s denoted as L,, n,2. Fg. 2 shows a trclnc example from n:2-lattce. Fgure 2 n:2-lattce Example In some cases, the n:2-lattce can be nterpreted as follows: There exts n-degree of non-determnsm between every two elements n the lattce. In order to control the non-determnsm at the ends, two upper and lower lmts are necessary as a boundary of the non-determnsm. [Def. ] n:2-lattce Operaton There are operatons possble for n:2-lattces as follows: Addton (: Merge operaton wth common elements. 2 Multplcaton ( : Cartesan producton. Subtracton ( : Inverse operaton of addton. Dvson ( : Inverse operaton of multplcaton. ISN:

3 Ths operatons wll be used to generate a new n:2-lattce from old ones. For example, Fg. and show smple addton and multplcaton operatons. III. EHAVIOR ONTOLOGY Ths secton presents a bref nton of the ontology. For detaled nton, refer []. Fgure Addton Example. Fgure Multplcaton Example. Here actor s consdered as the man entty of bebavors. Actors can be dstrbuted over some geographcal space and ncluded n another actor n geographcal space over tme. At the begnnng, each actor s ned wth ts type and capacty. [Def. 5] Actor Actor s the man entty of acton and behavor. [Def. 6] Acton Acton s a bnary ordered relaton between two actors. [Def. 7] ehavor ehavor s a seres of actons n order. Actons n behavour can be parallel, selectve, repeatable, and condtonal. ehavors can be entrely ncluded, or partally ncluded. [Def. 8] Abstract ehavor Abstract ehavor s the behavour that has been quanttatvely abstracted wth respect to cardnalty and capacty of actors. Abstract behavor s denoted by c (,, c n, where each c x s an actor, c p,, where x and p,,, px px are the cardnalty and capacty of c. [Defnton 9] ehavor Lattce ehavor Lattce s the n:2-lattce L,, n,2, where L s a set of abstract behavors and s ncluson relaton. Here n mples a number of ons possble for ncluson among dfferent dmenson of actors n the behavor. In other words, There are n possble ways of abstracton n the lattce. 2 mples two lower SLE and upper bound SGE lmts, that s, and 0 behavor, whch mae the lattce n:2-complete n T sense. [Defnton 0] ehavor Ontology ehavor Ontology s a n:2-lattce generated from behavour lattces wth the n:2-lattce operatons. All behavour lattces are connected to each other by the common abstract behavors among them to form a lattce of lattces, that s, behavor ontology. IV. EHAVIORAL EQUIVALENCES Ths secton nes two behavoural equvalences from behavour ontology as follows. [Defnton ] Strong ehavor Equvalence If two behavors occur at one locaton n behavour lattce, two behavors are ned to be strongly equvalent. In the strong equvalence, two behavors have the same actors wth the same cardnalty and capacty. It mples that two behavors have same quanttatve abstractons wth respect to the number of actors and ther capactes. Snce t s n the lattce, the only nterpretaton avalable for the behavors s restrcted to the structure of the lattce. [Defnton 2] Wea ehavor Equvalence If there exsts a meet of two behavors n behavour lattce, two behavors are ned to be wealy equvalent. In the wea equvalence, two behavors do not have the same actors wth the same cardnalty and capacty. ut once they are abstracted nto the hgher level n the lattce, they can be nterpreted as the same behavors at the level of the abstracton n the lattce. In mples that these are not equal at that level of quanttatve behavor, but become to be equal at the hgher level of quanttatve abstracton n the certan semantc order of the lattce. V. RAD EXAMPLE Ths secton demonstrates the ontology-based approach wth Resource Allocaton Problem (RAP example, as shown n Fg. 5. ISN:

4 Fg. 5 Overvew of Defnng RAP ehavor Ontology RAP s a problem doman that a carrer delvers a resource from source to target. The elements of RAP behavour doman can be ned as follows: Actors: There are dfferent nds of actors. Resource: obect to be delvered. 2 Carrer: Actor to delver obect. Source: Place for obect to s delvered from. Target: Place for obect to s delvered to. 2 Actons: There are 5 nds of actons. Note that R, C, S and T represent Resource, Carrer, Source and Target, respectvely. a = CS, : A Carrer goes to a Source. n 2 a2 = RC, : A Resource gets on a Carrer. n a = CT, : A Carrer goes to a Target. n a = CR, : A Resource gets off a Carrer. out 5 a5 = RT, : A Resource goes to a Target. n ehavors: There are 9 behavors ned. = a, a2, a, a, a5 : A Carrer goes to a Source, gets a Resource on, goes to a Target, and gets the Resource off, whch goes nto the Target. 2 2 = a, a2, a, a, a 5 : A repeatng behavor of. = a, a2, a, a, a5 : A Carrer goes to a Source, gets Resources on, goes to a Target, and gets the Resources off, whch go nto the Target. And t repeats tself. = a, a2, a, a, a5 : A Carrer goes to a Source, gets Resources on, and goes to Targets to get some of the Resources off untl all the Resources off, each group of whch goes nto the Target. And t repeats tself. 5 5 = a, a2, a, a, a5 a, a, a5 : A repeatng behavor of,, that s, through. 6 6 = a, a2, a, a, a5 : A Carrer goes to Sources to get Resources on, goes to a Target, and gets the Resources off, whch go to the Target. And t repeats tself. 7 7 = a, a2, a, a, a5 : A repeatng behavor that a Carrer goes to Sources to get Resources on, goes to Targets, and gets some of the Resources off to each Target untl all the Resources get off, each group of whch goes to the Target. And t repeats tself. 8 8 = a, a2, a, a, a5 a, a, a5 : A T repeatng behavor of 6, 7, that s, through 7 except,, 5. a, a, a5 a, a2, a, a, a5 9 9 = a, a, a5 a, a2, a, a, a5 : A repeatng behavor of 2, 5 or 8, that s, through 8. Abstract ehavors for Carrer. They are organzed at the bottom lattce n Fg. 6: = R, C, T ( 2 2 = ( R, C, T x,, x z,, z = ( R, C, T x y z = ( R, C, T x y,, 5 5 = ( R, C, T x y z,, z 6 6 = ( R, C, T,, y 7 7 = ( R, C, T,,,, 8 8 = ( R, C, T,, y z,, z 9 9 = ( R, C, T x,, x y z,, z 5 Abstract ehavors for n Carrer. They are organzed at the top lattce n Fg. 6: = ( R, C, T x y,, y z 2 2 = ( R, C, T x y,,,, y z z = ( R, C, T x,, x y,, y z ISN:

5 = ( R, C, T x,, x y,,,, y z z Note that R 2,, follows: C and,, T mply, respectvely, as 2 R : There are 2 sets of and resources n sources., C : There are carrers wth capactes of, and.,, T : There s target wth capacty of. 2 Here, R (, C, T can be nterpreted as an abstract,,, behavor that two sets of and resources are to be delvered by carrers wth capactes of, and to a target wth capacty of. Fg. 6 shows a behavor ontology for RAD example. A., A.2 2 Ambulance ( behavors:.,.,.5 Ambulance C ( behavors: C. For example, C. s seen n Fg. 8. A A Heart Dsease A = P, Heart Dsease = P, Dabetes A = P, H H D A C Hgh lood pressure A = P, Hgh lood pressure A = P, Hgh lood pressure C = P, A Food posonng A = P, Food posonng = P, F House of Patent = PH, School = PS, Center = Center = Ambulance = A Ambulance = A C Ambulance = A A 9 9,,,, = = = A C 5 Hosptal A H, Hosptal H, Hosptal C H, Rest Room of Doctors = HRRD, Doctor = HD, Srugery = HS A A A PH P H PH P PH P D PH P PS PH P PF P F C C A Center A C 9 A A EMS = EMS H A A HRRD [ HD ] HS H HRRD[ HD] HS C H5 HRRD[ HD] HS F PH = PH open msg0.( z. msg z P = P msg0 m. open msg. ( z8, z9. n z8. out z8. n HOR Center = Center! open msg. ( z2. msg2 agent, agent2, agent. msg m2 open msg2. ( m. out z28. n z22.! msg, agent z out z A = A n z2! n z2. out z2. n z2. out z2 n. n H = H open msg( z29. msg5 z29 HD = HD open msg5. ( z0. out HDDR. n HS m: = ( z28, z22, z2,, ( z28, z22, z2,,( z28 n, z22 n, z2n Fg. 7 The example s descrbed n Moble Ambents Calculus. Fg. 6 RAD ehavor Ontology. VI. EMS EXAMPLE Ths secton shows the Emergency Medcal Servce (EMS example, as an nstance of RAD. There are nstance actors as follows: Resource: 8 Patents. 2 Carrer: Ambulances wth capacty of,,. Source: Houses and School. Target: Hosptals wth capacty of,, 5. Note that ault capacty for actors s. Ther actons are specfed n Moble Ambent[6] as shown n Fg 7. From the specfcaton, the followng basc behavors can be obtaned, based on the RAD behavor ontology: Ambulance A (2 behavors: Fg. 8 Confguraton for EMS Example. The nstance behavors for each behavor types from RAD behavor ontology are shown at the bottom of Fg. 9. Note ISN:

6 that all the and are abstracted up to at the top of the fgure. VII. ANAYLSYS OF EQUIVALENCES From the EMS example, we can extract the followng strong and wea equvalences as shown n Fg. 9 as follows: Strong equvalences: A. A.2 (--a n Fg. 9..,, (--b n Fg ( P A H. A. A. A.2.,,. A. 5 (-2 n Fg A 2 2.,, A 2.2,,,, (2- n Fg. 9. P A H 2 A 2 9.,, ,, C 9. (9- n Fg Wea equvalences A. C. 2 (The above. C. 2 A 2 2.,, C 5. (The above. C 5. Fg. 9 EMS ehavors and Ther Equvalences VIII. CONCLUSION AND FUTURE RESEARCHES. Ths paper presented a new noton of equvalences based on behavour ontology. Compared wth other equvalences of Norm Chomsy and Robn Mler, the ontology-based equvalence has an advantage of havng some systematc mechansm to specfy, analyse and verfy the equvalences based on a formal structure. Snce the ontology s based on the n:2-lattce, the specfcaton, analyss and the verfcaton can be performed n the sound and complete noton of the lattce. Future research wll nclude applcaton to real ndustral examples, developng a real tool for the applcaton, etc. ACKNOWLEDGMENT Ths wor was supported by asc Scence Research Program through the NRF(Natonal Research Foundaton of Korea funded by the Mnstry of Educaton, Scence and Technology( and the MSIP(Mnstry of Scence, ICT and Future Plannng, Korea, under the ITRC(Informaton Technology Research Center support program (NIPA-20-H supervsed by the NIPA(Natonal IT Industry Promoton Agency. REFERENCES [] Chomsy, N. Three models for the descrpton of language, IEEE Transactons on Informaton Theory, Vol. 2, No., 956, pp. -2. [2] Par, Davd (98. "Concurrency and Automata on Infnte Sequences". In Deussen, Peter. Theoretcal Computer Scence. Proceedngs of the 5th GI-Conference, Karlsruhe. Lecture Notes n Computer Scence 0. Sprnger-Verlag. pp do:0.007/fb ISN [] Robn Mlner, Communcaton and Concurrency, Prentce Hall, Internatonal Seres n Computer Scence, ISN [] S. Woo, J. On, M. Lee, An Abstracton Method for Moblty and Interacton n Process Algebra Usng ehavor Ontology, n Computer Software and Applcatons Conference (COMPSAC, 20 IEEE 5th Annual, pp.28-, July 20. [5] W. Xng, O. Corcho, C. Goble, and M. Daaos, Actve Ontology: An Informaton Integraton Approach for Hghly Dynamc Informaton Sources, n Europe Semantc Web Conference 2007 (ESWC-2007, Innsbruc, Austra, [6] L. Cardell, and A. D. Gordon, Moble ambents, n Foundatons of Software Scence and Computatonal Structures, Maurce Nvat(Ed, No. 78 n Lecture Node n Computer Scence, Sprnger, pp. 0-55, 998. ISN: