A note on Schema Equivalence

Transcription

1 note on Schema Equvalence.H.M. ter Hofstede and H.. Proer and Th.P. van der Wede PUBLISHED S:.H.M. ter Hofstede, H.. Proer, and Th.P. van der Wede. Note on Schema Equvalence. Techncal Reort 9-30, Deartment of Informaton Systems, Unversty of Nmegen, Nmegen, The Netherlands, EU, 199. bstract In ths aer we ntroduce some termnology for comarng the exressveness of concetual data modellng technques, such as ER, NIM, and PM, that are fntely bounded by ther underlyng domans. Next we consder schema equvalence and dscuss the effects of the szes of the underlyng domans. Ths leads to the ntroducton of the concet of fnte equvalence. We gve some examles of fnte equvalence and nequvalence n the context of PM. The urose of the modellng rocess s to construct a formal descrton, (a secfcaton of U D, n terms of some underlyng formalsm. Ths secfcaton wll have a comonent S( that secfes S, a comonent τ( that secfes τ, and a state s 0 ( that s desgnated as the ntal state s 0. The man requrement for secfcaton s that t behaves lke U D. Ths can be shown by a (artal functon h, relatng the states from S( to the (real states S from U D such that h shows ths smlarty. Such a functon s called a (artal homomorhsm. If each state of U D s catured by the functon h, we call a correct secfcaton wth resect to U D. In that case, the functon h should be surectve, and s called an emorhsm (see also [Bor78]. s 0 h(x h(y U D 1 Schema Equvalence When modellng a Unverse of Dscourse ([ISO87], t s generally assumed that we can recognse stable states n ths Unverse of Dscourse, and that there are a number of actons that result n a change of state (state transtons. Ths s called the statetranston model. Furthermore we assume that the Unverse of Dscourse has a unque startng state. In mathematcal terms, a Unverse of Dscourse U D conssts of a set S of states, a bnary relaton τ over states, and an ntal state s 0 S: U D = S, τ, s 0 s 0( h h h x y Fgure 1: correct secfcaton Defnton 1.1 We call h a artal homomorhsm between and U D f Page 1

2 1. h s a (artal functon h : S( S. transtons commute under h: [ ] s, t τ( h(s, h(t τ s,t S( 3. h mas the ntal state of the secfcaton onto the ntal state of U D: h(s 0 ( = s 0 If h s surectve, we call h an emorhsm between and U D. We call an algebra (artally homomorhc wth algebra B, f there exsts a (artal homomorhsm from nto B. If schema s a descrton of, then we wll also call (artally homomorc wth B. The noton of emorsm s extended analogously. In the context of nformaton systems, the term nternal schema s generally used for a correct secfcaton. Note that n a correct secfcaton, a state of U D may have more than one corresondng state n S(. In that case we have a redundant reresentaton for the states of U D. Redundant reresentatons are useful as they rovde oortuntes for mrovement of effcency. The dsadvantage of a redundant reresentaton s that we do not have a descrton of U D that s free of mlementaton (reresentaton detals. descrton can only be mlementaton ndeendent f each state has a unque reresentant. Such a descrton s called a concetual schema n the context of nformaton systems. Ths s the case f the functon h that relates to U D s bectve. The exressveness of a formal method M s ntroduced as the set of U D s t can model. Ths can be descrbed by: { S(, τ(, s0 ( L(M } If we restrct ourselves n ths defnton to τ( =, we get the so-called base exressveness of method M. The base exressveness usually s the crteron that s used ntutvely when comarng dfferent methods. From the above defnton of concetual schema, the followng defnton of schema equvalence can be derved. Defnton 1. Two secfcatons and are equvalent, =, f there exsts a homomorhsm h from onto such that h s a becton. Schema Equvalence n PM In ths secton we consder the base exressveness of the PM ([BHW91], and focus at schema equvalence n that context. PM s a concetual data modellng technque survng as a common base for ER ([Che76] and NIM ([NH89]. Let be a PM schema, wth underlyng label tye set L, then ths schema secfes the followng set of states: S( = { IsPo L (, } oulaton s a functon assgnng a set of nstances to each obect tye n schema. The IsPo L redcate determnes whether s a roer oulaton. The oulaton of label values s restrcted to values of some doman D. We wll show that the base exressveness strongly deends on the actual choce of D. In ths restrcted sense the resultng state sace of schema s: S D ( = { } IsPo L (, x L [(x D] Usng ths defnton we ntroduce the noton of doman equvalence. Defnton.1 Two PM schemas and are doman equvalent over doman D, = D, f: frst result s: S D ( = S D ( Lemma.1 Let and be PM schemas (wthout enumeraton constrants, then: D countably nfnte = D Proof: We wll only gve a bref outlne of ths roof. The mortant ste s to rove that the number of oulatons n a schema wth a countable doman s countable tself (assumng fnte oulatons. Ths however, s true because every oulaton can be coded as a fnte strng by orderng the obect tyes n the schema at hand and lstng ther oulatons sequentally, accordng to ths orderng, searated by secal searator symbols. Each such fnte strng can unquely be translated to a fnte btstrng, whch can be consdered as a natural number n bnary reresentaton. Note that enumeraton constrants nvaldate the roerty as they enforce a lmted use of label values. Page

3 N Fgure : The most smle unversal schema We conclude that the exressveness of PM n the context of a countably nfnte doman s very low, as all schemata are equvalent n that case. Note that, n the context of countably nfnte domans, ths roerty holds for most other data models as well. Each schema thus can be consdered as a unversal schema, as t s exressve enough to smulate any other schema. The analogon of a unversal schema n the algorthmc world s the unversal Turng machne (see for examle [CB + 7]. The most smle unversal schema s shown n fgure. The role of the unary fact tye s to exclude all elements from N that do not corresond to a vald oulaton of the smulated schema. lthough all schemata are equvalent n ther exressve ower, one schema mght be much more convenent for ths urose than an other. The arorateness s measured by the comlexty of the oeratons that corresond wth the assocated transton relaton τ. In ths aer we wll not consder ths comlexty. We restrct ourselves to a fnte doman for label values. s a drect consequence, schema has a fnte state sace. We ntroduce the noton of fnte equvalence: Defnton. Two PM schemas and are fnte equvalent, = f, f for all D and D : D = D D < S D ( = S D ( Fnte equvalence can be roven by the constructon of a becton between the two state saces of the schemas. n Po(f and {r : { : a, q : b}, s : c} n Po(g to one nstance {t : a, u : b, v : c} n Po(h. Note the mortance of the total role (the black dot on redcator r n ths transformaton. Its semantcs s: x Po(f y Po(g [y(r = x] Therefore, the total role makes t unnecessary to consder nstances of fact tye f that do not contrbute n fact tye g. For a general defnton of the semantcs of constrants n NIM schemas, refer to [BHW91]. Fnte nequvalence can be roven by showng that the state saces of the underlyng schemas are not equal n sze. Examle. If we omt the total role from schema n fgure 3, the schemas are not fnte equvalent. Proof: Let a, b and c be the oulaton sze of, B and C resectvely. The number of oulatons of fact tye f amounts to: ab ( ab = ab =0 Now suose f s oulated wth tules, then for g we can have c dfferent oulatons. The number of oulatons of therefore amounts to: ab =0 ( ab c = =0 ( ab ( c = (1 + c ab On the other hand, the number of oulatons of equals abc = ( c ab. Examle.3 In fgure 4, another examle of fnte equvalence s shown. Examle.1 The schemas and from fgure 3, are fnte equvalent. Proof: The basc dea s to defne a translaton from nstances from to nstances from such that we have a becton between S( and S(. Ths s acheved by relatng dentcal nstances of obect tyes, B and C n both schemas and nstances { : a, q : b} Proof: The man observaton s that nstances occurrng n redcator of schema are to be maed onto dentcal nstances n the oulaton of fact tye g n schema. Instances of obect tyes and B n both schemas are agan related va an dentcal mang. Instances n fact tye f n schema are related to dentcal nstances n fact tye h n schema. Page 3

4 f q B =f r g s C h t v B u C Fgure 3: Examle of fnte equvalence f q B =f g r s h t B Fgure 4: nother examle of fnte equvalence Examle.4 In fgure 5 two schemas are dected, whch are not fnte equvalent. Proof: It s not hard to see that the number of oulatons n wth Po( = a and Po(B = b s ( b a, whle the number of oulatons n wth the same restrcton s ( b 1 a. 3 n uer bound for oulatablty data modellng technque s called fntely bounded by ts underlyng domans, f each schema from that technque allows for a fnte number of oulatons, n case of a fnte doman of label values. Defnton 3.1 The oulatablty of a schema s: m D ( = S D ( s each schema can be oulated by the emty oulaton ([BHW91], an mmedate consequence s: Lemma 3.1 D = 0 L(M [ md ( = 1 ] Defnton 3. Method M s called fntely bounded by ts underlyng domans D f: [ D < L(M md ( < ] In ths secton we derve an uer bound on the oulatablty of a schema. In order to smlfy the dervaton, we restrct ourselves to fact schemata,.e., schemata wthout entty tyes (.e., E( =. Lemma 3. [ E( = ] Proof: Relace each entty tye by a fact tye, corresondng to ts dentfcaton. If the dentfcaton of entty tye x conssts of the convoluton of k ath exressons (.e., mult(x = k, see [HPW93], then ths relacement leads to the ntroducton of a k-ary fact tye. The resultng schema s denoted as de(. Then obvously de( and E(de( =. The number (de( of redcators of schema de( s found by: Page 4

5 f q B =f f q B Fgure 5: Examle of fnte nequvalence Lemma 3.3 (de( = ( + Proof: Obvous! x E( (L q 1 q mult(x Fgure 6: Best oulatable schemata Next we ntroduce a seres {N } 0 of schemata (see fgure 6, consstng of a sngle -ary fact tye over some label tye L. These schemata are the best oulatable schemata among schemata wth the same number of redcators. Theorem 3.1 D > 1 [ m( m(n (de( ] n m( 1 m( m( E+19 Table 1: Growth of oulatablty Proof: Let D = n, then: m( 1 = m( = = m( 3 = ( ( n ( =0 =0 =0 ( ( n ( =0 ( n ( =0 =0 ( n ( =0 =0 =0 =0 ( n 3 ( ( n (3 = m( The result follows from the observaton: Proof: Frst we remark m( = m(de(. Next n > 1 (n3 > 3 (n we use the fact that a schema becomes better oulatable by undeeer nestng of (at least bnary fact tyes. Ths s shown n lemma 3.4. Furthermore, mergng fact tyes The oulatablty of schemata {N } 0 grows mroves oulatablty (see lemma 3.6. By extremely fast. reeatedtly alyng these stes, schema N (de( wll result. Lemma 3.4 Consder the schemata 1, and 3 from fgure 7, then: Lemma 3.5 Lemma 3.6 m(n = =0 ( n ( D > 1 m( 1 m( m( 3 m(n m(n q m(n +q Page 5

6 ( : ««: (3 n : n «1 n : n «n : n «3 Fgure 7: Transformaton stes From theorem 3.1 we conclude that ER, NIM and PM are fntely bounded by ther underlyng domans. References [BHW91] P. van Bommel,.H.M. ter Hofstede, and Th.P. van der Wede. Semantcs and verfcaton of obect-role models. Informaton Systems, 16(5: , October [Bor78] S.. Borkn. Data Model Equvalence. In Proceedngs of the Fourth Internatonal Conference on Very Large Data Bases, ages , nformaton models. Informaton Systems, 18(7:489 53, October [ISO87] Informaton rocessng systems Concets and Termnology for the Concetual Schema and the Informaton Base, ISO/TR 9007:1987. htt:// [NH89] G.M. Nssen and T.. Haln. Concetual Schema and Relatonal Database Desgn: a fact orented aroach. Prentce-Hall, Sydney, ustrala, SIN [CB + 7] J.N. Crossley, C.J. sh, C.J. Brckhll, J.C. Stllwell, and N.H. Wllams. What s mathematcal logc? Oxford Unversty Press, Oxford, Unted Kngdom, 197. [Che76] P.P. Chen. The entty-relatonsh model: Towards a unfed vew of data. CM Transactons on Database Systems, 1(1:9 36, March [HPW93].H.M. ter Hofstede, H.. Proer, and Th.P. van der Wede. Formal defnton of a concetual language for the descrton and manulaton of Page 6