Data integration: A theoretical perspective

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1 Data integation: A theoetical esective Mauizio Lenzeini Diatimento di Infomatica e Sistemistica Antonio Rubeti Univesità di Roma La Saienza Tutoial at PODS 2002 Madison, Wisconsin, USA, June 2002

2 Data integation Quey Global schema Maing R 1 C 1 D 1 T 1 c 1 d 1 t 1 c 2 d 2 t 2 Souce schema Souce schema Souce 1 Souce 2 Mauizio Lenzeini 1

3 Outline Fomal famewok fo data integation Aoaches to data integation Quey answeing in diffeent aoaches Dealing with inconsistency Reasoning on queies in data integation Conclusions Mauizio Lenzeini 2

4 Fomal famewok A data integation system I is a tile G, S, M, whee G is the global schema (ove an alhabet A G ) S is the souce schema (ove an alhabet A S ) M is the maing between G and S Semantics of I: which ae the databases that satisfy I (models of I)? We efe only to databases ove a fixed infinite domain Γ, and we stat with a souce database C, (data available at the souces, also called souce model) ove Γ. The set of databases that satisfy I elative to C is: sem C (I) = { B B is legal wt G and satisfies M wt C } Mauizio Lenzeini 3

5 Semantics of queies to I A quey q of aity n is a FOL fomula with n fee vaiables. If D is a database, then q D denotes the extension of q in D (i.e., the set of valuations in Γ fo the fee vaiables of q that make q tue in D). If q is a quey of aity n osed to a data integation system I (i.e., a quey ove A G ), then the set of cetain answes to q wt I and C is q I,C = {(c 1,..., c n ) q B B sem C (I)} Mauizio Lenzeini 4

6 Databases with incomlete infomation Taditional database: one model of a fist-ode theoy Quey answeing means evaluating a fomula in the model. Database with incomlete infomation: set of models (secified, fo examle, as a esticted fist-ode theoy) Quey answeing means comuting the tules that satisfy the quey in all the models in the set. Thee is a stong connection between quey answeing in data integation and quey answeing in database with incomlete infomation unde constaints. Mauizio Lenzeini 5

7 Outline Fomal famewok fo data integation Aoaches to data integation Quey answeing in diffeent aoaches Dealing with inconsistency Reasoning on queies in data integation Conclusions Mauizio Lenzeini 6

8 The maing How is the maing M between G and S secified? Ae the souces defined in tems of the global schema? Aoach called souce-centic, o local-as-view, o LAV. Is the global schema defined in tems of the souces? Aoach called global-schema-centic, o global-as-view, o GAV. A mixed aoach? Aoach called GLAV. Maing between souces, without global schema? Aoach called P2P. Mauizio Lenzeini 7

9 GAV vs LAV examle Global schema: movie(title, Yea, Diecto) euoean(diecto) eview(title, Citique) Souce 1: 1 (Title, Yea, Diecto) since 1960, euoean diectos Souce 2: 2 (T itle, Citique) since 1990 Quey: Title and citique of movies in 1998 D. movie(t, 1998, D) eview(t, R), witten { (T, R) movie(t, 1998, D) eview(t, R) } Mauizio Lenzeini 8

10 Fomalization of LAV In LAV, the maing M is constituted by a set of assetions: s φ G (sound souce) x (s( x) φ G ( x)) s φ G (exact souce) x (s( x) φ G ( x)) one fo each souce element s in A S, whee φ G is a quey ove G. Given souce database C, a database B fo G satisfies M wt C if fo each s S: s C φ G B (sound souce) s C = φ G B (exact souce) The maing M and the souce database C do not ovide diect infomation about which data satisfy the global schema. Souces ae views, and we have to answe queies on the basis of the available data in the views. Mauizio Lenzeini 9

11 LAV examle Global schema: movie(title, Yea, Diecto) euoean(diecto) eview(title, Citique) LAV: associated to souce elations we have views ove the global schema 1 (T, Y, D) { (T, Y, D) movie(t, Y, D) euoean(d) Y 1960 } 2 (T, R) { (T, R) movie(t, Y, D) eview(t, R) Y 1990 } The quey { (T, R) movie(t, 1998, D) eview(t, R) } is ocessed by means of an infeence mechanism that aims at e-exessing the atoms of the global schema in tems of atoms at the souces. In this case: { (T, R) 2 (T, R) 1 (T, 1998, D) } Mauizio Lenzeini 10

12 Fomalization of GAV In GAV, the maing M is constituted by a set of assetions: g φ S (sound souce) x (φ S ( x) g( x)) g φ S (exact souce) x (φ S ( x) g( x)) one fo each element g in A G, whee φ S is a quey ove S. Given souce database C, a database B fo G satisfies M wt C if fo each g G: g B φ S C g B = φ S C (sound souce) (exact souce) Given a souce database, M ovides diect infomation about which data satisfy the elements of the global schema. Relations in G ae views, and queies ae exessed ove the views. Thus, it seems that we can simly evaluate the quey ove the data satisfying the global elations (as if we had a single database at hand). Mauizio Lenzeini 11

13 GAV examle Global schema: movie(title, Yea, Diecto) euoean(diecto) eview(title, Citique) GAV: associated to elations in the global schema we have views ove the souces movie(t, Y, D) { (T, Y, D) 1 (T, Y, D) } euoean(d) { (D) 1 (T, Y, D) } eview(t, R) { (T, R) 2 (T, R) } Mauizio Lenzeini 12

14 GAV examle of quey ocessing The quey { (T, R) movie(t, 1998, D) eview(t, R) } is ocessed by means of unfolding, i.e., by exanding the atoms accoding to thei definitions, so as to come u with souce elations. In this case: movie(t,1998,d) eview(t,r) unfolding 1 (T,1998,D) 2 (T,R) Mauizio Lenzeini 13

15 GAV and LAV comaison LAV: (Infomation Manifold, DWQ, Picsel) Quality deends on how well we have chaacteized the souces High modulaity and extensibility (if the global schema is well designed, when a souce changes, only its definition is affected) Quey ocessing needs easoning (quey efomulation comlex) GAV: (Canot, SIMS, Tsimmis, IBIS, Picsel,... ) Quality deends on how well we have comiled the souces into the global schema though the maing Wheneve a souce changes o a new one is added, the global schema needs to be econsideed Quey ocessing can be based on some sot of unfolding (quey efomulation looks easie) Fo moe details, see [Ullman, TCS 00], [Halevy, VLDBJ 01]. Mauizio Lenzeini 14

16 Beyond GAV and LAV: GLAV In GLAV, the maing M is constituted by a set of assetions: φ S φ G (sound souce) x (φ S ( x) φ G ( x)) φ S φ G (exact souce) x (φ S ( x) φ G ( x)) whee φ S is a quey ove S, and φ G is a quey ove G. Given souce database C, a database B that is legal wt G satisfies M wt C if fo each assetion in M: φ S C φ S C φ G B (sound souce) = φ G B (exact souce) The maing M does not ovide diect infomation about which data satisfy the global schema: to answe a quey q ove G, we have to infe how to use M in ode to access the souce database C. Mauizio Lenzeini 15

17 Examle of GLAV Global schema: W ok(p eson, P oject), Aea(P oject, F ield) Souce 1: Souce 2: Souce 3: HasJob(P eson, F ield) T each(p of esso, Couse), In(Couse, F ield) Get(Reseache, Gant), F o(gant, P oject) GLAV maing: { (, f) HasJob(, f) } { (, f) W ok(, ) Aea(, f) } { (, f) T each(, c) In(c, f) } { (, f) W ok(, ) Aea(, f) } { (, ) Get(, g) F o(g, ) } { (, ) W ok(, ) } Mauizio Lenzeini 16

18 Beyond GLAV: P2P data integation In P2P, the global schema does not exist. Constaints (that we can still call G) ae defined ove A G = A S1 A Sn and the maing M is constituted by a set of assetions (φ S i 1, φ S j 2 the alhabets A Si and A Sj, esectively): φ S i 1 φ S j 2. ae queies ove A S is a distinguished subset of edicates in A G, called base edicates (whee data ae). A souce database is a database fo the base edicates. Given souce database C, a database W that satisfies I elative to C is a database fo S such that, fo each assetion φ 1 φ 2 in M, φ W 1 φ W 2. Queies ae now exessed ove alhabet A Si, and the notion of cetain answes is the usual one. Mauizio Lenzeini 17

19 A unified view Alhabet: A = A G A S Integity constaints: constaints G, and maing M Patial database: souce database Database: data fo all symbols in A that ae both coheent with the atial database and satisfy the integity constaints Quey answeing: comuting the tules that satisfies the quey in evey database Unde this view, the diffeence between LAV, GAV, GLAV, P2P is eflected in the kinds of integity constaints that ae exessible. Mauizio Lenzeini 18

20 Quey answeing with incomlete infomation [Reite 84]: elational setting, databases with incomlete infomation modeled as a fist ode theoy [Vadi 86]: elational setting, comlexity of easoning in closed wold databases with unknown values Seveal aoaches both fom the DB and the KR community [van de Meyden 98]: suvey on logical aoaches to incomlete infomation Mauizio Lenzeini 19

21 Connection to quey containment Quey containment (unde constaints T ) is the oblem of checking whethe q B 1 is contained in q B 2 fo evey database B (satisfying T ), whee q 1, q 2 ae queies with the same aity. A souce database C can be eesented as a conjunction q C of gound liteals ove A S (e.g., if x is in s C, then the coesonding liteal is s( x)) If q is a quey, and t is a tule, then we denote by q t the quey obtained by substituting the fee vaiables of q with t The oblem of checking whethe t q I,C can be educed to the oblem of checking whethe q C is contained in q t unde the constaints G M The combined comlexity of checking cetain answes is identical to the comlexity of quey containment unde constaints, and the data comlexity is at most the comlexity of quey containment unde constaints. Mauizio Lenzeini 20

22 Outline Fomal famewok fo data integation Aoaches to data integation Quey answeing in diffeent aoaches Dealing with inconsistency Reasoning on queies in data integation Conclusions Mauizio Lenzeini 21

23 Dealing with incomleteness and inconsistency We analyze the oblem of quey answeing in diffeent cases, deending on two aametes: Global schema: - without constaints, - with constaints Maing: - GAV o LAV, - sound o comlete Given a souce database C, we call etieved global database any database fo G that satisfies the maing wt C. Mauizio Lenzeini 22

24 Incomleteness and inconsistency Constaints Tye of Incomle- Inconsiin G maing teness stency no GAV/exact no no no GAV/sound yes/no no no LAV/sound yes no no LAV/exact yes yes yes GAV/exact no yes yes GAV/sound yes yes yes LAV/sound yes yes yes LAV/exact yes yes Mauizio Lenzeini 23

25 Incomleteness and inconsistency Constaints Tye of Incomle- Inconsiin G maing teness stency no GAV/exact no no no GAV/sound yes/no no no LAV/sound yes no no LAV/exact yes yes yes GAV/exact no yes yes GAV/sound yes yes yes LAV/sound yes yes yes LAV/exact yes yes Mauizio Lenzeini 24

26 INT[noconst, GAV/exact]: examle Conside I = G, S, M, with Global schema G: student(scode, Sname, Scity) univesity(ucode, Uname) enolled(scode, Ucode) Souce schema S: database elations s 1, s 2, s 3 Maing M: student(x, Y, Z) { (X, Y, Z) s 1 (X, Y, Z, W ) } univesity(x, Y ) { (X, Y ) s 2 (X, Y ) } enolled(x, W ) { (X, W ) s 3 (X, W ) } Mauizio Lenzeini 25

27 INT[noconst, GAV/exact]: examle Univesity Student Enolled code AF BN name bocconi ucla code name bill anne city oslo floence Scode Ucode AF BN 12 anne floence 21 AF bocconi 12 AF s C 1 15 bill oslo 24 s C 2 BN ucla s C 3 16 BN Examle of souce database and coesonding etieved global database Mauizio Lenzeini 26

28 INT[noconst, GAV/exact] Model of I Global schema = Retieved GDB Maing Souces Souce model Mauizio Lenzeini 27

29 INT[noconst, GAV/exact]: quey answeing Use M fo comuting fom C the etieved global database, whee each element g of G satisfies exactly the tules of C satisfying the φ S that M associates to g Since G does not have constaints, the etieved global database is legal wt G Actually, it is the only database that is legal wt G, and that satisfies M wt C Thus, we can simly evaluate the quey q ove the etieved global database, which is equivalent to unfolding the quey accoding to M, in ode to obtain a quey on A S to be evaluated ove C Answeing queies to I means answeing queies to a single database. Mauizio Lenzeini 28

30 INT[noconst, GAV/exact]: examle of quey answeing Maing M: student(x, Y, Z) { (X, Y, Z) s 1 (X, Y, Z, W ) } univesity(x, Y ) { (X, Y ) s 2 (X, Y ) } enolled(x, W ) { (X, W ) s 3 (X, W ) } s C 1 12 anne floence bill oslo 24 s C 2 AF BN bocconi ucla s C 3 12 AF 16 BN Quey: { (X) student(x, Y, Z), enolled(x, W ) } Unfolding wt M: { (X) s 1 (X, Y, Z, V ), s 3 (X, W ) } etieves the answe {12} fom C. A simle unfolding stategy is sufficient in this context. Mauizio Lenzeini 29

31 Incomleteness and inconsistency Constaints Tye of Incomle- Inconsiin G maing teness stency no GAV/exact no no no GAV/sound yes/no no no LAV/sound yes no no LAV/exact yes yes yes GAV/exact no yes yes GAV/sound yes yes yes LAV/sound yes yes yes LAV/exact yes yes Mauizio Lenzeini 30

32 INT[noconst, GAV/sound]: examle Univesity Student Enolled code AF UR BN name bocconi unioma ucla code name bill anne city oslo floence Scode Ucode AF BN s C 1 12 anne floence bill oslo 24 s C 2 AF BN bocconi ucla s C 3 12 AF 16 BN Examle of souce database and coesonding etieved global database Mauizio Lenzeini 31

33 INT[noconst, GAV/sound] The GAV maing assetions have the logical fom: x φ s ( x) g( x) The intesection of all etieved global databases (which can be comuted by letting each element g of G satisfy exactly the tules of C satisfiying the φ S that M associates to g) still satisfies M wt C, and theefoe, is the only minimal model of I. Incomleteness is of secial fom. Fo queies without negation, unfolding is sufficient. Mauizio Lenzeini 32

34 INT[noconst, GAV/sound] Global schema Maing = Minimal Model of I Intesection of etieved GDBs Souces Souce model Mauizio Lenzeini 33

35 Incomleteness and inconsistency Constaints Tye of Incomle- Inconsiin G maing teness stency no GAV/exact no no no GAV/sound yes/no no no LAV/sound yes no no LAV/exact yes yes yes GAV/exact no yes yes GAV/sound yes yes yes LAV/sound yes yes yes LAV/exact yes yes Mauizio Lenzeini 34

36 INT[noconst, LAV/sound]: incomleteness The LAV maing assetions have the logical fom: x s( x) φ G ( x) In geneal, given a souce database C thee ae seveal solutions of the above assetions (i.e., diffeent databases that ae legal wt G that satisfies M wt C). Incomleteness comes fom the maing. This holds even fo the case of simle queies φ G : s 1 (x) { (x) y g(x, y) } s 2 (x) { (x) g 1 (x) g 2 (x) } Mauizio Lenzeini 35

37 INT[noconst, LAV/sound] Global schema = = Models of I Maing Retieved GDBs Souces Souce model Mauizio Lenzeini 36

38 INT[noconst, LAV/sound]: dealing with incomleteness View-based quey ocessing: Answe a quey based on a set of mateialized views, athe than on the aw data in the database. Relevant oblem in Data waehousing Quey otimization Poviding hysical indeendence Mauizio Lenzeini 37

39 INT[noconst, LAV/sound]: dealing with incomleteness In LAV/sound data integation, the views ae the souces. Two aoaches to view-based quey ocessing: View-based quey ewiting: quey ocessing is divided in two stes 1. e-exess the quey in tems of a given quey language ove the alhabet of A S 2. evaluate the ewiting ove the souce database C View-based quey answeing: no limitation is osed on how queies ae ocessed, and the only goal is to exloit all ossible infomation, in aticula the souce database, to comute the cetain answes to the quey Mauizio Lenzeini 38

40 INT[noconst, LAV/sound]: connection to quey containment If queies in M ae conjunctive queies, then we can substitute the quey that M associates to s fo evey s-liteal in q C, and theefoe, checking cetain answes can be educed to checking ue containment (without constaints) of two queies in the alhabet A G The data comlexity is at most the comlexity of quey containment Mauizio Lenzeini 39

41 INT[noconst, LAV/sound]: some esults fo quey answeing Conjunctive queies using conjunctive views [Levy&al. PODS 95] Recusive queies (datalog ogams) using conjunctive views [Duschka&Geneseeth PODS 97], [Afati&al. ICDT 99] Comlexity analysis [Abiteboul&Duschka PODS 98] [Gahne&Mendelzon ICDT 99] Vaiants of Regula Path Queies [Calvanese&al. ICDE 00, PODS 00] [Deutsch&Tannen DBPL 01], [Calvanese&al. DBPL 01] Mauizio Lenzeini 40

42 INT[noconst, LAV/sound]: data comlexity Fom [Abiteboul&Duschka PODS 98]: Sound souces CQ CQ PQ datalog FOL CQ PTIME conp PTIME PTIME undec. CQ PTIME conp PTIME PTIME undec. PQ conp conp conp conp undec. datalog conp undec. conp undec. undec. FOL undec. undec. undec. undec. undec. Mauizio Lenzeini 41

43 INT[noconst, LAV/sound]: basic technique Conside conjunctive queies and conjunctive views. 1 (T ) { (T ) movie(t, Y, D) euoean(d) } 2 (T, V ) { (T, V ) movie(t, Y, D) eview(t, V ) } T 1 (T ) Y D movie(t, Y, D) euoean(d) T V 2 (T, V ) Y D movie(t, Y, D) eview(t, V ) movie(t, f 1 (T ), f 2 (T )) 1 (T ) euoean(f 2 (T )) 1 (T ) movie(t, f 4 (T, V ), f 5 (T, V )) 2 (T, V ) eview(t, V )) 2 (T, V ) Answeing a quey means evaluating a goal wt to this nonecusive logic ogam (PTIME data comlexity). Mauizio Lenzeini 42

44 INT[noconst, LAV/sound]: olynomial intactability Given a gah G = (N, E), we define I = G, S, M, and souce database C: V b R b V f R f V t R g R g R b R b R gb R bg V b C = {(c, a) a N, c N} V f C = {(a, d) a N, d N} V t C = {(a, b), (b, a) (a, b) E} Q R b M R f whee M descibes all mismatched edge ais (e.g., R g R b ). If G is 3-coloable, then B whee M (and Q) is emty, i.e. (c, d) Q I,C. If G is not 3-coloable, then M is nonemty B, i.e. (c, d) Q I,C. = conp-had data comlexity fo ositive queies and ositive views. Mauizio Lenzeini 43

45 INT[noconst, LAV/sound]: in conp Conside the case of Datalog queies and ositive views. t is not a cetain answe to Q wt I and C, if and only if thee is a database B fo I such that t Q B, and B satisfies M wt C Because of the fom of M x (s( x) y 1 α 1 ( x, y 1 )... y h α h ( x, y h )) each tule in C foces the existence of k tules in any database that satisfies M wt C, whee k is the maximal length of conjuncts in M If C has n tules, then thee is a database B B fo I that satisfies M wt C with at most n k tules. Since Q is monotone, t Q B. Checking whethe B satisfies M wt C can be done in PTIME wt the size of B. = conp data comlexity fo Datalog queies and ositive views. Mauizio Lenzeini 44

46 INT[noconst, LAV/sound]: the case of RPQ We deal with the oblem of answeing queies to data integation systems of the fom G, S, M, whee G simly fixes the labels (alhabet Σ) of a semi-stuctued database the souces in S ae elational the maing M is of tye LAV queies ae tyical of semi-stuctued data (vaiants of egula ath queies) Mauizio Lenzeini 45

47 Global semi-stuctued database sub sub calls sub va sub sub sub calls sub va calls sub va sub va va Mauizio Lenzeini 46

48 Global semi-stuctued databases and queies sub sub a calls sub va sub sub sub calls b sub va calls sub va sub va va Regula Path Quey (RPQ): (sub) (sub (calls sub)) va Mauizio Lenzeini 47

49 Global semi-stuctued databases and queies sub sub calls sub va sub sub sub calls b sub va calls sub va sub va va a 2RPQ: (sub ) (va sub) Mauizio Lenzeini 48

50 INT[noconst, LAV/sound]: the case of RPQ Given I = G, S, M, whee G simly fixes the labels (alhabet Σ) of a semi-stuctued database the souces in S ae binay elations the maing M is of tye LAV, and associates to each souce s a 2RPQ w ove Σ x, y s(x, y) x w y a souce database C a 2RPQ Q ove Σ a ai of objects t we want to detemine whethe t Q I,C. Mauizio Lenzeini 49

51 Quey answeing: Technique We seach fo a counteexamle to t Q I,C, i.e., a database B legal fo I wt C such that t Q B Cucial oint: it is sufficient to estict ou attention to canonical databases, i.e., databases B that can be eesented by a wod w B $ d 1 w 1 d 2 $ d 3 w 2 d 4 $ $ d 2m 1 w m d 2m $ whee d 1,..., d 2m ae constants in C, w i Σ +, and $ acts as a seaato Use wod-automata theoetic techniques! Mauizio Lenzeini 50

52 We need techniques fo... checking whethe a ai of objects satisfies a 2RPQ quey in the case of a wod eesenting a ath a wod eesenting semiath a wod eesenting a canonical database Mauizio Lenzeini 51

53 Finite-state automata and RPQs. a b c q d. Q = ( q) q q Automaton fo Q s 1 δ(s 0, ), s 2 δ(s 1, ), s 2 δ(s 1, q), s 3 δ(s 2, q), s 3 δ(s 3, q) The comutation fo RPQs is catued by finite-state automata. Mauizio Lenzeini 52

54 2way Regula Path Queies 2way Regula Path Queies (2RPQ) ae exessed by means of finite-state automata ove Σ { Σ }. ( q) ( ) q q q _ q q Mauizio Lenzeini 53

55 Finite-state automata and 2RPQs. a b c q d. Wod: Quey: q Q = ( q) q q Automaton fo Q s 1 δ(s 0, ), s 2 δ(s 1, ), s 2 δ(s 1, q), s 3 δ(s 2, ), s 4 δ(s 3, ), s 5 δ(s 4, q), s 5 δ(s 5, q) State: s 0 Tansition: s 1 δ(s 0, ) Mauizio Lenzeini 54

56 Finite-state automata and 2RPQs. a b c q d. Wod: Quey: q Q = ( q) q q Automaton fo Q s 1 δ(s 0, ), s 2 δ(s 1, ), s 2 δ(s 1, q), s 3 δ(s 2, ), s 4 δ(s 3, ), s 5 δ(s 4, q), s 5 δ(s 5, q) State: s 1 Tansition: s 2 δ(s 1, ) Mauizio Lenzeini 55

57 Finite-state automata and 2RPQs. a b c q d. Wod: Quey: q Q = ( q) q q Automaton fo Q s 1 δ(s 0, ), s 2 δ(s 1, ), s 2 δ(s 1, q), s 3 δ(s 2, ), s 4 δ(s 3, ), s 5 δ(s 4, q), s 5 δ(s 5, q) State: s 2 Tansition: none Mauizio Lenzeini 56

58 Finite-state automata and 2RPQs. a b c q d. Wod: Quey: q Q = ( q) q q State: s 2 Tansition: none (a, d) satisfies quey Q, but the ath fom a to d is not acceted by the 1NFA coesonding to Q: the comutation fo 2RPQs is not catued by finite-state automata. Mauizio Lenzeini 57

59 2way automata (2NFA) A 2way automaton A = (Γ, S, S 0, ρ, F ) consists of an alhabet Γ, a finite set of states S, a set of initial states S 0 S, a tansition function ρ : S Σ 2 S { 1,0,1} and a set of acceting states F S. Given a 2way automaton A with n states, one can constuct a one-way automaton B 1 with O(2 n log n ) states such that L(B 1 ) = L(A), and a one-way automaton B 2 with O(2 n ) states such that L(B 2 ) = Γ L(A). Mauizio Lenzeini 58

60 2way automata and 2RPQs Given a 2RPQ E = (Σ, S, I, δ, F ) ove the alhabet Σ, the coesonding 2way automaton A E is: (Σ A = Σ {$}, S A = S {s f } {s s S}, I, δ A, {s f }) whee δ A is defined as follows: (s 2, 1) δ A (s 1, ), fo each tansition s 2 δ(s 1, ) of E ente backwad mode: (s, 1) δ A (s, l), fo each s S and l Σ A exit backwad mode: (s 2, 0) δ A (s1, ), fo each s 2 δ(s 1, ) of E (s f, 1) δ A (s, $), fo each s F. = w satisfies E iff w$ L(A E ). Mauizio Lenzeini 59

61 2way automata and 2RPQs. a b c q d. Automaton fo Q Q = ( q) q q s 1 δ(s 0, ), s 2 δ(s 1, ), s 2 δ(s 1, q), s 3 δ(s 2, ), s 4 δ(s 3, ), s 5 δ(s 4, q), s 5 δ(s 5, q) 2way automaton (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 1, q), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) Mauizio Lenzeini 60

62 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 0 Tansition: (s 1, 1) δ A (s 0, ) Mauizio Lenzeini 61

63 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 1 Tansition: (s 2, 1) δ A (s 1, ) Mauizio Lenzeini 62

64 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 2 Tansition: (s 2, 1) δ A (s 2, q) Mauizio Lenzeini 63

65 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 2 Tansition: (s 3, 0) δ A (s 2, ) Mauizio Lenzeini 64

66 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 3 Tansition: (s 4, 1) δ A (s 3, ) Mauizio Lenzeini 65

67 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 4 Tansition: (s 5, 1) δ A (s 4, q) Mauizio Lenzeini 66

68 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q Automaton fo Q (s 1, 1) δ A (s 0, ), (s 2, 1) δ A (s 1, ), (s 2, 1) δ A (s 2, q), (s 3, 0) δ A (s 2, ), (s 4, 1) δ A (s 3, ), (s 5, 1) δ A (s 4, q), (s f, 1) δ A (s 5, $) State: s 5 Tansition: (s f, 1) δ A (s 5, $) Mauizio Lenzeini 67

69 2NFA and 2RPQs. a b c q d. Wod: q $ Quey: Q = ( q) q q State: s f (a, d) satisfies quey Q, and the ath fom a to d is acceted by the 2NFA coesonding to Q: the comutation fo 2RPQs is catued by 2way automata. Mauizio Lenzeini 68

70 2NFA and view extensions Global schema G: ( q q ) Souces: q ( ) ( q) (q ) (d 1,d 2 ) (d 4,d 5 ) (d 4,d 2 ) (d 3,d 3 ) (d 2,d 3 ) Database fo G: q d 1 d 2 d 3 d 4 d 5 Mauizio Lenzeini 69

71 2NFA and view extensions q d 1 d 2 d 3 d 4 d 5 Database B as a wod: $d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q To veify that (d 1, d 3 ) satisfies Q in the above database B, we build A (Q,d1,d 3 ), by exloiting not only the ability of 2way automata to move on the wod both fowad and backwad, but also the ability to jum fom one osition in the wod eesenting a node to any othe osition (eithe eceding o succeeding) eesenting the same node. Mauizio Lenzeini 70

72 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 0, 1) δ A (s 0, l), fo each l Σ A Mauizio Lenzeini 71

73 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 0, 1) δ A (s 0, l), fo each l Σ A Mauizio Lenzeini 72

74 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 0, 1) δ A (s 0, l), fo each l Σ A Mauizio Lenzeini 73

75 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 0, 1) δ A (s 0, l), fo each l Σ A Mauizio Lenzeini 74

76 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 0, 1) δ A (s 0, l), fo each l Σ A Mauizio Lenzeini 75

77 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 0, 1) δ A (s 0, l), fo each l Σ A Mauizio Lenzeini 76

78 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 0 Tansition: (s 1, 0) δ A (s 0, d 1 ), s 1 initial state fo Q Mauizio Lenzeini 77

79 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 1 Tansition: (s 1, 1) δ A (s 1, d 1 ) Mauizio Lenzeini 78

80 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 1 Tansition: (s 2, 1) δ A (s 1, ), tansition coming fom Q Mauizio Lenzeini 79

81 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 2 Tansition: ((s 2, d 2 ), 1) δ A (s 2, d 2 ), seach fo d 2 Mauizio Lenzeini 80

82 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 2, d 2 ) Tansition: ((s 2, d 2 ), 1) δ A ((s 2, d 2 ), $), seach fo d 2 Mauizio Lenzeini 81

83 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 2, d 2 ) Tansition: ((s 2, d 2 ), 1) δ A ((s 2, d 2 ), d 4 ), seach fo d 2 Mauizio Lenzeini 82

84 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 2, d 2 ) Tansition: ((s 2, d 2 ), 1) δ A ((s 2, d 2 ), ), seach fo d 2 Mauizio Lenzeini 83

85 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 2, d 2 ) Tansition: (s 2, 0) δ A ((s 2, d 2 ), d 2 ), exit seach mode Mauizio Lenzeini 84

86 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 2 Tansition: (s 2, 1) δ A (s 2, d 2 ), backwad mode Mauizio Lenzeini 85

87 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 2 Tansition: (s 3, 0) δ A (s 2, ), tansition coming fom Q Mauizio Lenzeini 86

88 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 3 Tansition: (s 4, 1) δ A (s 3, ), tansition coming fom Q Mauizio Lenzeini 87

89 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: s 4 Tansition: ((s 4, d 2 ), 1) δ A (s 4, d 2 ), seach fo d 2 Mauizio Lenzeini 88

90 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 4, d 2 ) Tansition: ((s 4, d 2 ), 1) δ A ((s 4, d 2 ), $), seach fo d 2 Mauizio Lenzeini 89

91 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 4, d 2 ) Tansition: ((s 4, d 2 ), 1) δ A ((s 4, d 2 ), d 3 ), seach fo d 2 Mauizio Lenzeini 90

92 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 4, d 2 ) Tansition: ((s 4, d 2 ), 1) δ A ((s 4, d 2 ), ), seach fo d 2 Mauizio Lenzeini 91

93 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 4, d 2 ) Tansition: ((s 4, d 2 ), 1) δ A ((s 4, d 2 ), ), seach fo d 2 Mauizio Lenzeini 92

94 A un of A (Q,d1,d 3 ) q d 1 d 2 d 3 d 4 d 5 Wod: $ d 4 d 5 $ d 1 d 2 $ d 4 d 2 $ d 3 d 3 $ d 2 q d 3 $ Q = ( q) ( ) q q State: (s 4, d 2 ) Tansition: ((s 4, d 2 ), 1) δ A ((s 4, d 2 ), d 3 ), seach fo d 2 Mauizio Lenzeini 93

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