Jan Van den Bussche. Departement WNI, Limburgs Universitair Centrum (LUC), Universitaire Campus, B-3590 Diepenbeek, Belgium

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1 Theoretial Computer Siene 254 (2001) Simulation of the nested relational algera y the at relational algera, with an appliation to the omplexity of evaluating powerset algera expressions Jan Van den Busshe Departement WNI, Limurgs Universitair Centrum (LUC), Universitaire Campus, B-3590 Diepeneek, Belgium Reeived July 1998; revised April 1999 Communiated y J.D. Ullman Astrat Paredaens and Van Guht proved that the at relational algera has the same expressive power as the nested relational algera, as far as queries over at relations and with at results are onerned. We provide a new, very diret proof of this fat using a simulation tehnique. Our tehnique is also applied to partially answer a question posed y Suiu and Paredaens regarding the omplexity of evaluating powerset algera expressions. Speially, we show that when only unary at relations are into play, any powerset algera expression is either equivalent to a nested algera expression, or its evaluation will produe intermediate results of exponential size Elsevier Siene B.V. All rights reserved. Keywords: Nested relational dataases; Query languages; Expressive power 1. Introdution The formal asis for relational dataase systems is provided y the relational data model [5, 4]. A dataase is modeled as a olletion of relations among asi data values. These relations an e manipulated using ve asi operators whih together form the relational algera. The nested relational model, designed in order to e ale to represent omplex data strutures in a more natural and diret way [10, 3], is a typed higher-order extension of the lassial at relational model. In a nested relation, a tuple may onsist not only of asi values ut also of relations in turn. By WWW: vduss/. address: vduss@lu.a.e (J. Van den Busshe) /01/$ - see front matter 2001 Elsevier Siene B.V. All rights reserved. PII: S (99)

2 364 J. Van den Busshe / Theoretial Computer Siene 254 (2001) anonially generalizing the operators of the relational algera to work on nested relations, and adding the two operators of nesting and unnesting [9], one otains the nested relational algera [15]. These days, nested relations are known as omplex ojets [4]. The expressive power of the nested relational algera as a query language is well understood, as well as its extensions with iteration, reursion, or the powerset operator, and extensions in the ontext of more general omplex ojet data models involving not only sets ut also ags, lists, arrays, and the like [4]. Two partiular results we will e interested in the present paper are those y Paredaens and Van Guht [12], and y Suiu and Paredaens [14]. Paredaens and Van Guht proved the Flat Flat Theorem: the at relational algera has the same expressive power as the nested relational algera, as far as queries over at relations and with a at result are onerned. Their proof was rather iruitous however, and they posed the prolem of nding a diret proof. In this paper we will provide suh a proof, ased on a very diret simulation of the nested algera y the at algera. Under this simulation, a nested relation is represented y a numer of at relations, the numer depending on the sheme of the nested relation. Related simulations have een known in several variants sine the 1980s [2, 13, 7]. Moreover, several researhers in the eld ([17, 1], see also [4, Theorem 20:7:2]) have suggested the possiility of a proof along the lines we will present. Consequently, we have written this paper not to lay any laim, ut eause we elieve it is worthwhile to make the omplete argument generally known (and atually, the detailed write-up turned out to e a rather intriate task). Another goal of the present paper, however, is to demonstrate that the simulation tehnique y whih we prove the at at theorem an also nd other appliations. Speially, we partially answer a question raised y Suiu and Paredaens onerning the omplexity of evaluating powerset algera expressions. (The powerset algera is the extension of the nested algera with the powerset operator.) Suiu and Paredaens onjetured that for any expression in the powerset algera that is not equivalent to a nested algera expression, its evaluation will produe intermediate results of exponential size. They onrmed their onjeture for expressions dening the transitive losure of a inary relation, and more generally, for expressions dening queries on single inary relations having the form of a hain. We will onrm the onjeture for the ase of multiple unary relations. The general ase remains open; atually, we would not e surprised if it turned out to e false, eause it is not inoneivale that there are lasses of strutures that are reognizale without using the powerset operator, and that have very weird ominatorial properties, suh as having identiale suparts of logarithmi size, to whih we ould apply the powerset operator without lowing up, and use the result to express a query that is not expressile without the powerset. To onlude this introdution we should mention that an analogue of the at at theorem in a omplex ojet formalism dierent from, ut equivalent to, the nested relational model was proved y Wong using a remarkaly elegant argument [18].

3 J. Van den Busshe / Theoretial Computer Siene 254 (2001) Preliminaries on nested relations Basially, we assume the existene of a ountaly innite supply of atomi attriutes. The set of atomi attriutes is denoted y U. The set HF(U) of hereditarily nite sets with atoms in U is the smallest set ontaining U, suh that if X 1 ;:::;X n HF(U) then also {X 1 ;:::;X n } HF(U). An element of HF(U) U is alled a sheme if no atomi attriute ours more than one in it. 1 Shemes are also alled omplex attriutes. Assume further given a domain V of data values. Let e a sheme. A relation over is a nite set of tuples over. Here, a tuple over is a mapping t on, suh that for A U; t(a) V, and for X U; t(x ) is a relation over X.We will also refer to t(x )asaomplex value of type X. A dataase sheme is a nite set S of shemes. A dataase over S is a mapping on S that assigns to eah sheme S a relation over. Fix a dataase sheme S. The set NA of nested relational algera expressions over S is indutively dened as follows. For eah expression e we also dene its result sheme, denoted y e. Eah sheme is in NA; its result sheme equals itself. If e 1 and e 2 are in NA, with e1 = e2 =, then (e 1 e 2 ) and (e 1 e 2 ) are in NA, also with result sheme. If e 1 and e 2 are in NA, suh that no atomi attriute ours oth in e1 and e2, then (e 1 e 2 )isinna, with result sheme e1 e2. Let e e in NA. Projetion: ifz e, then Z (e) isinna, with result sheme Z. Seletion: ifx; Y e and is a permutation of U suh that (X )=Y, then X = Y (e) isinna, with result sheme e. Renaming: if is a permutation of U, then (e) isinna, with result sheme ( e ). Nesting: ifz e, then Z (e) isinna, with sheme ( e Z) {Z}. Unnesting: ifx e U, then X (e) isinna, with sheme ( e {X }) X. Let e a dataase over S, and let e e a nested relational algera expression over S. The result of evaluating e on, denoted y e(), is indutively dened as follows: If e is a sheme, then e():=(). (e 1 e 2 )():=e 1 () e 2 (); (e 1 e 2 )():=e 1 () e 2 (). (e 1 e 2 )():={t 1 t 2 t 1 e 1 () and t 1 e 2 ()}. Z (e)():={t Z t e()}. X = Y (e)():={t e() (t(x )) = t(y )}. 2 (e)():={ (t) t e()}. 1 We say that X ours in if X, orx ours in some Y. 2 Permutations of U are applied to tuples and relations in the ovious way. If t is a tuple over, then (t) is the tuple over () dened y (t)( (X )) = t(x ).

4 366 J. Van den Busshe / Theoretial Computer Siene 254 (2001) Z (e)():={t tuple over ( Z) {Z} t e() : t Z = t Z and t(z)= {t Z t e() and t Z = t Z }}. X (e)():={t tuple over ( {X }) X t e() : t {X } = t {X } and t X t (X )}. So, a nested relational algera expression with result sheme denes a mapping from dataases to relations over. Suh mappings are alled queries. The in a seletion operation X = Y is important when X and Y are omplex, eause it speies how the two omplex values t(x ) and t(y ), for some tuple t, are to e ompared. For example, if X = {A; B} and Y = {C; D}, then X = 1 Y, where 1 (A)=C and 1 (B)=D, has a dierent semantis than X = 2 Y, where 2 (A)=D and 2 (B)=C. When X and Y are atomi, the is irrelevant (there is only one way to ompare two atomi values) and we will omit it. 3. Representing nested relations y at dataases A sheme is alled at if all its elements are atomi attriutes. A dataase sheme is alled at if all its shemes are at. In order to formally dene a representation of nested relations y at dataases, we must rst make some tehnial assumptions aout the set U of atomi attriutes. We partition the atomi attriutes in ordinary attriutes and identier attriutes. Unless expliitly speied otherwise, an atomi attriute is always assumed to e ordinary, and a sheme is always assumed to e uilt up from ordinary atomi attriutes only. For eah sheme X, we assume we have the innitely many identier attriutes id(x ) 1 ; id(x ) 2 ; id(x ) 3 ; ::: suh that if X Y or i j then id(x ) i id(y ) j. Our representation of a nested relation y a at dataases uses identiers for omplex values. These identiers are tuples of atomi values ourring in the nested relation. The width of these tuples an vary depending on the type of omplex value represented. Apart from the at relation representing the nested relation itself, the at dataase will have auxiliary relations, one for eah omplex attriute X, holding the identiers representing a omplex value of type X, and speifying whih omplex values are represented y these identiers. Why do we need identiers that are tuples? Why annot we just use identiers that are atomi values? The point is that later we want to simulate the nested relational algera y the at relational algera. Operations that introdue new omplex values, notaly nesting, will have to e simulated y introduing identiers for these new omplex values. The relational algera annot invent new atomi values. Hene, it has to onstrut the identiers as tuples of the atomi values existing in the dataase. If we ould only use identiers of length 1, and there are n distint atomi values in the dataase, we would only have n dierent identiers at our disposition, whih is muh

5 J. Van den Busshe / Theoretial Computer Siene 254 (2001) too little. For example, the following nested relational algera expression, starting from a at relation over {A}, introdues n 2 new omplex values of type {C; D}: one for eah pair of atomi values. 3 {A;B} B = D A = C ({A} (AB) ({A}) (AC) ({A}) (AD) ({A})) The lengths of the identiers, depending on the type of omplex value they represent, are given y an identier-width assignment, dened next: Denition 1. Let e a sheme. An identier-width assignment (i:w:a:) over is a mapping from the set of omplex attriutes ourring in to the natural numers. For a omplex attriute X ourring in, we will denote the set {id(x ) 1 ;:::;id(x ) (X ) } y ID (X ). This set ontains the atomi attriutes of the identier tuples for omplex values of type X. Given a sheme and an i.w.a. over, we an now dene the at dataase sheme listing the at shemes of the at relations that together make up a at dataase representation of a nested relation over. This at dataase sheme is denoted y at (). Denition 2. The at dataase sheme at () onsists the sheme rep ():=( U) {ID (X ) X U}; together with, for all omplex attriutes X ourring in, the shemes rep (X ):=(X U) {ID (Y ) Y X U} ID (X ): Flat dataases over at () represent nested relations over. To dene this representation formally in Denition 4, we need the following auxiliary tehnial denition: Denition 3. Let e a sheme, let X e a omplex attriute ourring in, and let e an i.w.a. over. Let e a dataase over at () and let t e a tuple over ID (X ). Let X e the restrition of to the omplex attriutes ourring in X. Then t is the dataase over at X (X ) dened y t (rep X (X )) := {t repx (X ) t (rep (X )) and t ID(X ) = t}; and, for eah omplex attriute Y ourring in X, t (rep X (Y )) := (rep (Y )): 3 We use the standard notation (A B) for the permutation that exhanges A and B.

6 368 J. Van den Busshe / Theoretial Computer Siene 254 (2001) To larify the notation used in the aove denition, it should e pointed out that rep X (X )isnot the same as rep (X ); the latter inludes ID (X ) as a suset, while the former does not, eause for X ; X is the top-level sheme. We see learly in Denition 2 that rep (Z) only ontains ID (Z) as a suset if Z is a lower-level sheme (a omplex attriute ourring in the top-level sheme). Of ourse, for the Y s in the aove denition, whih are lower-level even when viewing X as the top-level, rep X (Y ) is the same as rep (Y ), whih is why at these Y the denition of t is simpler. We an now dene: Denition 4. Let e a sheme, let e an i.w.a. over, and let e a dataase over at (). The nested relation represented y, denoted y nested(), equals {t tuple over t (rep ()): t U = t U and X U: t(x ) = nested( t ID(X ) )}: Note how the tuple t ID(X ) serves as an identier for the omplex value nested( t ID(X ) ). As a (perhaps too trivial) example, let = {{A}} with A U, and let ({A})=1. Then at () onsists of rep ()={id({a}) 1 } and rep ({A})={id({A}) 1 ;A}. Let e the following dataase over at (): { id({a}) 1 } a d { id({a}) 1 } A a a a a d d Then nested() equals {{A} } a d

7 J. Van den Busshe / Theoretial Computer Siene 254 (2001) As a nal remark we note that ourrenes of the empty omplex value of some type X would also e represented y an identier, ut this identier would not show up in the orresponding rep (X ) relation sine it represents the empty set. 4. Simulation of nested algera y at algera A nested relational algera expression is alled at if it is dened over a at dataase sheme and does not use the nesting () and the unnesting () operators. In this setion we will prove: Theorem 1. Let e e a nested relational algera expression over a at dataase sheme S. Then there exists an i.w.a. over e and at relational algera expressions e X over S for X = e or X a omplex attriute ourring in e ; suh that for eah dataase over S; nested( e )=e(); where e is the dataase over at ( e ) dened y e (rep (X )) = e X () for eah X. As a orollary, we get: Theorem 2 (Paredaens-Van Guht). Let e e a nested relational algera expression over a at dataase sheme S; suh that e is at. Then there exists a at relational algera expression e over S suh that for eah dataase over S; e ()=e(): Proof. Sine e is at, at ( e ) onsists simply of e itself and for any dataase over at ( e ); nested( )= ( e ). Theorem 1 thus tells us there exists a at relational algera expresion e = e e suh that for eah ; e ()=e(), as desired. Before we prove Theorem 1, we illustrate the most intriate part of the proof, the simulation of nesting, with a simple example. Let S onsist of the single relation sheme = {A; B}. Consider the dataase over S dened y () = { A B } a a d :

8 370 J. Van den Busshe / Theoretial Computer Siene 254 (2001) First, onsider the expression {A} (). Evaluating it on yields the following relation: {{A} B } a a d We an represent this relation y the following at dataase: { id{a} 1 B } d d { id{a} 1 A } a a d Next, onsider the expression {B} {A} (). Its evaluation on yields the following relation: {{A} {B} } a d This relation an e represented y the following at dataase: { id{a} 1 id{b} 1 } d d We now present: { id{a} 1 A } a a d { id{b} 1 B } d d Proof of Theorem 1. In the proof, we will rely on the well-known fat that the at relational algera is as powerful as the tuple relational alulus [16, 11], a variant

9 J. Van den Busshe / Theoretial Computer Siene 254 (2001) of rst-order logi. 4 So instead of writing algera expressions we will often write alulus formulas whenever this is more onvenient. We egin y stating the following: Lemma 1. Let e a sheme; let X e a omplex attriute ourring in ; and let e an i.w.a. over. Then there exists a tuple relational alulus formula equal (t 1 ;t 2 ) suh that for eah dataase over at () and tuples t 1 ;t 2 over ID (X ); equal (t 1 ;t 2 ) is true in i nested( t1 ) = nested( t2 ). In the formulation of the aove lemma, note that t1 and t2 are at dataases over at X (X ), and thus nested( t1 ) and nested( t2 ) are omplex values of type X. Proof of Lemma 1. We indutively onstrut a formula suset (t 1 ;t 2 ) and dene equal (t 1 ;t 2 )assuset (t 1 ;t 2 ) suset (t 2 ;t 1 ). The formula suset (t 1 ;t 2 ) is dened as t rep (X ):t ID(X ) = t 1 t rep (X ):t ID(X ) = t 2 t X U = t X U equal (t ID(Y );t ID(Y )): Y X U For any sheme Z, we will use the areviation equiv (Z)(t 1 ;t 2 ) for the formula t 1 Z U = t 2 Z U X Z U equal (t 1 ID(X );t 2 ID(X )): This formula learly expresses that the at tuples t 1 and t 2 represent the same nested tuple of type Z. The atual proof of Theorem 1 now proeeds y indution on the struture of the expression e. e is a (at) sheme. In this asi ase we dene e := : (There is no need to dene as there are no omplex attriutes ourring in.) e = Z (e ). By indution, we have an i.w.a. over e satisfying the theorem. Let us denote W := (( e Z) U) {ID (X ) X ( e Z) U}: 4 This is with the understanding that we use the ative-domain semantis of the alulus [4].

10 372 J. Van den Busshe / Theoretial Computer Siene 254 (2001) Dene (Z):= W and (Y ):= (Y ) for all other omplex attriutes ourring in e. Let e a ijetion from W to ID (Z). Then we dene and e e := {t (t) t W (e e )}; e Z := {t tuple over rep (Z) t ;t W (e e ): t = 1 (t ID(Z)) equiv ( e Z)(t ;t ) (t rep (Z) ID (Z) t ) e }; e with the understanding that in the formula equiv ( e Z), every ourrene of a sheme rep (Y ) (for some Y ) is replaed y the orresponding expression e Y. Finally, for any other omplex attriute X Z ourring in e e, dene e X := e X. e = X (e ). Dene as the restrition of to the omplex attriutes ourring in e. Let e a permutation of U suh that (ID (X )) is disjoint from rep ( e ) rep (X ). Dene e e := rep ( e) idx1 = (idx 1) ::: idx (X ) = (idx (X ) )(e e (e X )) and dene e Y := e Y for omplex attriutes Y ourring in e. e =(e 1 e 2 ). By indution we have i.w.a. s 1 for e 1 and 2 for e 2 satisfying the theorem. For a omplex attriute X ourring in e, dene (X ):= 1 (X ) and e X := (e 1 ) X if X ours in e1, and (X ):= 2 (X ) and e X := (e 2 ) X if X ours in e2 (y the syntax of the nested relational algera exatly one of the two ases holds). Dene e e := (e 1 ) e1 (e 2 ) e2. e =(e 1 e 2 ). Dene and (X ) := max{ 1 (X ); 2 (X )} +2 e X := (e 1 ) X (e 2 ) X ; where (e 1 ) X and (e 2) X are dened as follows. (e 1 ) X is otained y taking the Cartesian produt ( ) of(e 1) X with the following fators for all Y (X U), as well as, if X e, for Y = X itself: {t tuple over {idy 1(Y )+1; idy 1(Y )+2;:::;idY (Y ) } t(idy 1(Y )+1) adom t(idy 1(Y )+1)=t(idY 1(Y )+2)= = t(idy (Y ) )}: (e 2 ) X is otained y taking the Cartesian produt ( ) of(e 2) X with the following fators for all Y (X U), as well as, if X e, for Y = X itself: {t tuple over {idy 2(Y )+1; idy 2(Y )+2;:::;idY (Y ) } t(idy 2(Y )+1) t(idy 2(Y )+2) adom (idy 2(Y )+2)=t(idY 2(Y )+3)= = t(idy (Y ) )}:

11 J. Van den Busshe / Theoretial Computer Siene 254 (2001) These Cartesian produts make sure that identier tuples appearing in the result of evaluating (e 1 ) X will e dierent from identier tuples appearing in the result of evaluating (e 2 ) 5 X, so that no mix-up ours when taking the union. e =(e 1 e 2 ). This ase is exatly the same as the ase e =(e 1 e 2 ), exept that we dene e e as {t (e 1 ) e t (e 2 ) e : equiv ( e )(t; t )}; with the understanding that in the formula equiv ( e ), every ourrene of a sheme rep (Y ) (for some Y ) is replaed y the orresponding expression e Y. e = Z (e ). Then equals the restrition of to the omplex attriutes ourring in Z, e e := rep (Z)(e e ); and e X := e X for eah omplex attriute X ourring in Z. e = X = Y (e ). Take := and dene e Z := e Z for eah omplex attriute Z ourring in e.ifx and Y are atomi, e e is simply X = Y (e e ). If X and Y are omplex, we need the following lemma similar to Lemma 3 (the proof is analogous). Lemma 2. Let e a sheme; let X; Y U; let e a permutation of U suh that (X )=Y; and let e an i.w.a. over. Then there exists a tuple relational alulus formula equal (t 1 ;t 2 ) suh that for eah dataase over at (); tuple t 1 over ID (X ); and tuple t 2 over ID (Y ); equal (t 1 ;t 2 ) is true in i (nested( t1 )) = nested ( t2 ). Proof. We then dene e e := {t e e equal (t ID (X );t ID (Y ))}: e = (e ). Extend to identier attriutes in the following anonial manner: (idx i ):=id (X ) i. Then ( (X )) := (X ) and e X := (e X ) for X = e or X a omplex attriute ourring in e. 5. Complexity of evaluating powerset algera expressions over unary relations The powerset algera is the extension of the nested relational algera with the powerset operator (). Syntatially, if e is an expression, then (e) is also an expression, with output sheme { e }. Semantially, on any dataase, (e)() equals {t tuple over { e } t( e ) e()}. 5 This trik does not work if adom would ontain only one element. But this is harmless, eause we an treat this ase (as well as the ase where adom is ompletely empty) entirely separately, eause in this ase there are, up to isomorphism, only a nite numer of possile dataases: we an test for these possiilities and return, for eah possiility, diretly the right result, ypassing the simulation.

12 374 J. Van den Busshe / Theoretial Computer Siene 254 (2001) Hull and Su [8] showed that in the powerset algera preisely all queries omputale in elementary time are expressile. So the powerset algera is a very powerful query language. It does not seem to e a very pratial language, however, in the sense that no example is known of a query not expressile in the nested relational algera, that is expressile in the powerset algera y an expression whose evaluation never generates intermediate results of exponential size (in spite of appliations of the powerset operator). In this setion, we prove that no suh query exists if it is over a unary dataase sheme. A dataase sheme is alled unary if all its shemes are singletons of the form {A}, with A atomi. The proof will e easy one we have estalished the following: Lemma 3. Let S e a unary dataase sheme and let e e a nested relational algera expression over S. Let e : N N e the mapping on the natural numers dened as follows: e (n) is the maximal ardinality of e(); where is a dataase over S with ative domain of ardinality n. Then either e = O(1) or e = (n). The ative domain of a dataase, denoted y adom(), is the set of all data values appearing in the relations of the dataase. Proof of Lemma 3. Apply Theorem 1 to otain an i.w.a. over e and at relational algera expressions e X whih simulate e in the sense desried y the theorem. Let e a dataase over S. Let e e the dataase over at ( e ) desried y Theorem 1. Consider the following equivalene relation on tuples in e (rep ( e )): t 1 t 2 if equiv ( e )(t 1 ;t 2 ) holds in e. 6 So, t 1 t 2 i t 1 and t 2 represent the same nested tuple in e(). Hene, the ardinality of e() equals, the index (numer of dierent equivalene lasses) of. 7 Sine the relations of e an e omputed y the same expressions e X for any given, and sine equiv ( e ) is a tuple relational alulus formula, there is one tuple relational alulus formula (t 1 ;t 2 ) suh that for every dataase, t 1 t 2 i (t 1 ;t 2 ) holds in. Sine the tuple relational alulus is equivalent to the domain relational alulus (essentially rst-order logi) [16], we an equivalently express as a domain, rather than a tuple, relational alulus formula (whih we also denoted y y ause of notation). Variales now range over the ative domain of the input dataase. A dataase over S onsists of a set of relations over unary shemes. We naturally view a relation over a unary sheme {A} as a set of data values (formally it is a set of mappings from {A} to V ). The unary relations of indue a partition on adom(). Eah partition lass is determined y some non-empty suset X S 6 The denition of equiv (Z) was given after Lemma 3. 7 We denote the index of an equivalene relation R y R.

13 J. Van den Busshe / Theoretial Computer Siene 254 (2001) and equals [X ]:= (Y ) Y X Y S X (Y ): The automorphisms of are preisely the permutations of adom() that leave every [X ] invariant. Let k e the numer of dierent variales used in the formula. Let us all any funtion from the non-empty susets of S to {0;:::;k} a lassier. We lassify the dataases over S using these lassiers as follows: for a lassier, S denotes the family of all dataases over S for whih = (X ) if (X ) k; and ardinality of [X ] k if (X )=k for eah X: The families S with a suh that (X ) k for every X are nite (up to isomorphism) and an e disarded. If we an show for all other that restrited to S is either O(1) or (n), we are ready. Indeed, if all of them are O(1), then also gloally is O(1); if at least one of them is (n), then also gloally is (n) sine the families are innite (we just disarded the nite ones). So x a family S ; we only onsider dataases in this family. It is a routine exerise in logi (ompare Exerise 1:3:11 in [6]) to show that any formula over S that uses at most k dierent variales is equivalent, on S, to a quantier-free formula. This holds in partiular for formula. We may assume without loss of generality that is in disjuntive normal form, and that eah onjuntion in this disjuntion is maximally onsistent. Note that a maximally onsistent onjuntion of literals over S, with m variales, serves as an automorphism type (also alled m-type): it desries an m-tuple of data values entirely up to appliation of an automorphism, speifying the partition lass of every omponent, as well as all equalities and non-equalites that hold among the omponents. In the ase of, m equals 2, where is the ardinality of rep ( e ). Note also that a 2 -type is nothing ut the onjuntion of two - types and the speiation of the equalities and non-equalities holding aross these two types. We distinguish etween the following possiilities: is the empty disjuntion, i.e., equivalent to false. Then is the empty equivalene relation on eah. But sine is dened on rep e ( e ), i.e., on e e, this means that e e (), and thus also e(), is empty on eah. In this ase e is everywhere zero and thus trivially O(1). is not false, and there is an -type suh that the 2 -type desriing those pairs of tuples (t 1 ;t 2 ) suh that t 1 and t 2 oth satisfy ; t 1 and t 2 are equal outside Z ; and t 1 and t 2 are disjoint on Z,

14 376 J. Van den Busshe / Theoretial Computer Siene 254 (2001) is not in. Here, Z is the set of those attriutes for whih the orresponding variale in is speied y to take a value in the partition lass determined y an X S with (X )=k. So for any dataase and any pair (t 1 ;t 2 ) of tuples in as aove, t 1 t 2.By augmenting with a xed numer of new data values, plaed in the appropriate partition lasses, we get a third tuple t 3 not equivalent to t 1 or t 2. We an keep on doing this, so that is (n). is not false, and there is no suh -type as in the previous item. But then, for large enough, any two tuples t 1 ;t 2 of the same -type that are equal outside Z are equivalent. Indeed, we an always nd a third tuple t 3 of the same type disjoint from oth t 1 and t 2 on Z ut equal outside; y assumption we then have t 1 t 3 t 2 and thus t 1 t 2. Hene, in this ase is O(1), eing ounded y the numer of dierent -types, whih is a xed numer, and the numer of dierent values a tuple of some type an have outside Z, whih is also xed y denition of Z (omponents outside Z elong to a partition lass determined y an X S with (X ) k and thus of xed size). As a orollary, we get: Theorem 3. Let e e a powerset algera expression over a unary dataase sheme S. Then either e is equivalent to a nested relational algera expression, or for some suexpression e of e, e is (2 n ). Proof. Let e e a minimal suexpression of e of the form (e ). By Lemma 3, e is either O(1) or (n). If e is (n), then learly e is (2 n ). If e is O(1), we an simulate the appliation of the powerset operator in the nested relational algera. Indeed, let K e the maximal ardinality of e (). Then e is equivalent to the following expression: (let := e ) {} ( = K () = 1()(e 1 (e ) K (e )). = K ()(e 1 (e ) K (e )) ) ; where 1 ;:::; K are permutations of U suh that every atomi attriute ours in at most one of, 1 (), :::, K (), and = () is an areviation for the sequene of all X = (X ) with X. Referenes [1] S. Aiteoul, Personal ommuniation, [2] S. Aiteoul, N. Bidoit, Non rst normal form relation: An algera allowing data restruting, J. Comput. System Si. 33 (3) (1986)

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