J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....guarantees, even in the presence of data skew. Recursive functions Declarative query languages are limited in expressive power. For instance the relational algebra cannot express transitive closure. This limitation remains true for extensions of the relational algebra to nested relations [55, 78]. To increase computing power, applications use a query language like SQL embedded in a general purpose programming language. There are limitations on the data structures that can be exchanged between the two languages (the impedance mismatch problem) as well as limitations in optimization ....

Jan Paredaens and Dirk Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....of conjunctive queries [Ull89] and also corresponds to natural fragments of other query languages for complex objects. In particular, our equivalence result partially answers an open problem for testing equivalence of sequences of nest and unnest operations in [GPG90] Paredaens and Van Gucht [PG92] and Wong [Won93] prove that the nested relational algebra for complex objects is a conservative extension of the relational algebra. This however does not reduce the containment problem for complex objects to that of flat relations: we show here that the former is more difficult. A particular ....

Jan Paredaens and Dirk Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....of set brackets. The height of an expression is defined as the maximum height of all types that appear in its typing derivation. More specifically, if f : s 1 s 2 takes flat relations to flat relations and is expressible in NRC, then it is also expressible in the standard flat relational algebra [19, 5]. It is a common misconception that the relational algebra is the same as SQL. The truth is that all versions of SQL come with three features that have no equivalence in relational algebra: SQL extends the relational calculus by having arithmetic operations, a group by operation, and various ....

....application of sri . In the rest of this report, we concentrate on IES(SQL) because it precisely models real relational databases. However, for convenience and clarity, we give proofs using IES(NRC aggr ) It is worth pointing out that previous results on the conservative extension of NRC[19, 23] and NRC aggr [22, 15] do not imply the collapse of IES(NRC aggr ) to IES(SQL) The conservative extension property[23] implies that if the input and output of a (update) function are flat, then the function can be implemented using only flat intermediate data. In an IES(NRC aggr ) having a ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

.... N, where f : t N, with semantics P [f ] fx 1 ; x n g) f(x 1 ) f(x n ) The language NRL obtained from NRL nat by removing the arithmetic operations is equivalent to the nested relational algebra, which is a generalization of 6 relational algebra to complex objects (cf. [13, 16]) According to [11] for any boolean query q of type fb Theta bg bool in NRL nat , there exists a number k such that for any l 1 , l 2 k and any two cycles C 1 and C 2 of length l 1 and l 2 respectively, q(C 1 ) q(C 2 ) Thus, NRL nat cannot define a query that is equivalent to ....

....to the first order case. So far, all inexpressibility results for the nested relational languages were proved in the following way. First, a conservativity result is established that shows that expressive power of the language is independent of the depth of set nesting in intermediate results (see [10, 13, 16] for examples of such results) Then the desired results are proved by reduction to the first order case, when no nested relations are allowed. For example, the flat fragment of the nested relational algebra is equivalent to the relational algebra [13, 16] Hence, recursive queries such as ....

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J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17 (1992), 65--93.

....3.3, we conclude the following. Theorem 3.4 frNRC has the conservative extension property. 2 Paredaens and Van Gucht gave a translation for mapping nested relational algebra expressions having flat relations as input to an equivalent expression in first order logic with bounded quantification [11]. This translation can be easily adapted to provide a translation for mapping frNRC expressions of height 1 to first order logic with polynomial constraints. Next result follows from this and Theorem 3.4. Corollary 3.5 If f : s 1 s 2 is a function expressible in frNRC and s 1 and s 2 have ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....of tuples of base types) have height 1. Given an expression e, the height of e is defined as the maximal height of all types that appear in the typing derivation of e. For example, S f S ff(x; y)g j x 2 Rg j y 2 Sg is an expression of height 1 if both R and S are flat relations. It is known [26, 28] that when restricted to expressions of height 1, NRC( is equivalent to the usual relational algebra. We also write NRC( b ) when the equality test is restricted to base types b, B , and Q. We sometimes list the free variables in an expression in brackets like: e(R; x) As was mentioned, the ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM TODS, 17(1):65--93, March 1992.

....explicitly typed calculus in this section. This calculus is based on the formalism of [2, 12, 26] which is recently being used as a query language for complex heterogenous genomic data sources [9] The fragment dealing with sets is known [26] to be equivalent to several nested relational algebras [20, 17, 5, 15]. The types used in the explicitly typed calculus are either eq types t or noneq types T . The eq types are those types where equality tests can be performed on their objects; the noneq types are those types where equality tests are not available. This distinction is necessary because we require ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....frNRC has the conservative extension property. 2 3 The symbol = denotes semantic equivalence. Paredaens and Van Gucht gave a translation for mapping nested relational algebra expressions having flat relations as input to an equivalent expression in first order logic with bounded quantification [17]. This translation can be easily adapted to provide a translation for mapping frNRC expressions of height 1 to first order logic with polynomial constraints. Next result follows from this and Theorem 3.4. Corollary 3.5 If f : s 1 s 2 is a function expressible in frNRC and s 1 and s 2 have ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....is very little hope for finding nice tools for analyzing the expressiveness of such languages. Fortunately, queries of the nested relational algebra were shown to be independent of the height of set nesting in intermediate data. The first result of this kind was proved by Paredaens and Van Gucht [46] for queries over flat relations. It was later generalized by Wong [59] to arbitrary queries. Recently, we showed that it continues to hold in the presence of aggregate functions [36] and that it holds even in the presence of a large variety of polymorphic functions [37] This property provides ....

....[54] Colby [13] and Schek and Scholl [49] NRL(eq) is a better language for the purpose of this paper than these older languages because it has simpler semantics and it is easily extensible with operations such as aggregate functions. A very important property of NRL(eq) is its conservativity [46, 59] the class of NRL(eq) queries from flat relations to flat relations is precisely the class of relational algebra, or first order definable queries. Another advantage of NRL is its extensibility. In particular, it is easy to see what constructs should be used if bags are used instead of sets. ....

[Article contains additional citation context not shown here]

J. Paredaens, D. Van Gucht, Converting nested relational algebra expressions into flat algebra expressions, ACM Transaction on Database Systems 17, No. 1 (1992), 65--93.

....of tuples of base types) have height 1. Given an expression e, the height of e is defined as the maximal height of all types that appear in the typing derivation of e. For example, S f S ff(x; y)g j x 2 Rg j y 2 Sg is an expression of height 1 if both R and S are flat relations. It is known [41, 44] that when restricted to expressions of height 1, NRC( is equivalent to the usual relational algebra. We also write NRC( b ) when the equality test is restricted to base types b, B , and Q. We sometimes list the free variables in an expression in brackets like: e(R; x) As was mentioned, the ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....and prove that its range restricted fragment is equivalent to NRA with bounded fixpoints. Keywords: nested relational algebra, conservative extension, complex objects, fixpoints, indexes. 1 Introduction Several query languages for databases with complex objects have been studied in recent years [AGVW89, AK89, AB88, BNTW95, dB92, GV91, GF88, GG91, GG92, HS89, HS91, PG92]. A natural way of designing such a language is to extend first order logic to a logic for hereditary finite sets, and consider only domainindependent queries, like in the case of first order logic. Abiteboul and Beeri follow this path in [AB88] define safe queries, and show that the resulting ....

....one of which has the same expressive power as the algebra without powerset of [AB88] Following established tradition, they call this language the nested relational algebra (NRA) and show that, like in first order logic, all queries expressible in NRA are in PTIME. Paredaens and Van Gucht in [PG92] and Wong in [Won93] show that NRAis a conservative extension of first order logic. That means that a query mapping flat relations to flat relations is expressible in NRA if and only if it is expressible in first order logic. This result is surprising, because such queries in NRA may have ....

[Article contains additional citation context not shown here]

Jan Paredaens and Dirk Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....languages studied in this paper. Queries expressible in the augmented language are proved, in section 6, to be independent of the height of set nesting of intermediate results. This is a significant generalization of the conservative extension result of Wong [38] and Paredaens and Van Gucht [29]. In particular, it implies that nested relational queries whose input and output are flat relations can be expressed in a language like SQL, even if aggregate operators such as average and count are used. The conservativeness of transitive closure, bounded fixpoints, and powerset operators is ....

.... sort(dom(R) where ffl dom(R) fx j (x; y) 2 Rg [ fy j (x; y) 2 Rg ffl encode(R; C) f(a; b) j (r; s) 2 R; r; a) 2 C; s; b) 2 Cg ffl decode(R; C) f(r; s) j (a; b) 2 R; r; a) 2 C; s; b) 2 Cg 2 Conservative extension property was first studied by Paredaens and Van Gucht [29] and later by Van den Bussche [10] They proved that NRC(eq) has it when input and output are restricted to flat relations. It was then extended by Wong [38] to any input and output. More recently, Suciu [32] managed to prove the remarkable theorem that NRC(eq; bfix) note the absence of natural ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, 1992.

....of tuples of base types) have height 1. Given an expression e, the height of e is defined as the maximal height of all types that appear in the typing derivation of e. For example, S f S ff(x; y)g j x 2 Rg j y 2 Sg is an expression of height 1 if both R and S are flat relations. It is known [27, 29] that when restricted to expressions of height 1, NRC( is equivalent to the usual relational algebra. We also write NRC( b ) when the equality test is restricted to base types b, B , and Q. We sometimes list the free variables in an expression in brackets like: e(R; x) As was mentioned, the ....

....is bigger than all the roots of the non zero polynomials in e(G) As a result, e(G) is equivalent to an expression in NRC( b ) whenever card(G) k. Then we apply the conservative extension property of NRC( and conclude that e(G) is equivalent to an expression e 00 (G) in relational algebra [29, 27], whenever card(G) k. Thus e 00 (G) implements e(G) for each G in G r;k . Now we deal with those G in S k Gamma1 j=1 G r;j . Let e i (G) denote the expression obtained from e(G) replacing card(G) with i; it is definable in relational algebra. Since for every constant i, the test card(G) i ....

J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....These languages have polynomial time complexity. A similar but weaker result was obtained in [21] by assuming the presence of a powerset operation. We then show that R augmented with equality testing is equivalent to the well known nested relational algebra of Thomas and Fischer [57] By [49] it follows that our nested relational language is conservative with respect to flat relational algebra. That is, the queries with flat relations as input and flat relations as output are expressible in (flat) relational algebra. Because, both flat and nested relational algebra are now seen as ....

....of Thomas and Fischer is via their selection operation. 2 It is an immediate corollary of this theorem that Corollary 4.6 Every function from flat relations to flat relations expressible in R( is also expressible in flat relational algebra. Proof. It is known from Paredaens and Van Gucht [49] that every function from flat relations to flat relations expressible in Thomas F ischer is also expressible in flat relational algebra. The corollary thus follows from Theorem 4.5. 2 In fact, elsewhere we are able to strengthen the theorem of Paredaens and Van Gucht to a general theorem on the ....

J. Paredaens, D. Van Gucht, Converting nested relational algebra expressions into flat algebra expressions, ACM Transaction on Database Systems 17, No. 1 (1992), 65--93.

....[BBW92] offers a rational reconstruction of the nested relational algebras (NRA) starting from the monad constructs as well as connections between s.r. and the algebra with powerset of [AB88] That NRA at relational types gives us a language equivalent to the relational algebra was shown in [PG92, Won93] even if nesting is used in intermediate results. This fails in the presence of powerset [HS91] Another result of this kind [Suc94] shows that adding a bounded fixed point construct to NRA gives us, again at relational types, inflationary datalog, and in [LW94a, LW94b] it is shown that ....

Jan Paredaens and Dirk Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....is a conservative extension of NRL i;o;k as a language. Consequently, the class NRL i;o;k is independent of k. Thus the ability to use intermediate expressions of great height does not increase expressive power. This complements work by other researchers. To begin with, Paredaens and Van Gucht [15, 16] showed that the nested relational algebra of Thomas and Fischer [18] is conservative with respect to flat relational algebra in the sense we have described. This result implies that NRL i;o;k 1 is conservative with respect to NRL i;o;k when i = o = 1. Our result generalizes this to ....

....to height of input output. Secondly, the proof we give for relative set abstraction relies on a set based semantics. This is in line with the work of many researchers as reported in Abiteboul et al. 1] Abiteboul and Beeri [2] Hull and Su [11] Grumbach and Vianu [7] Paredaens and Van Gucht [15, 16], Gyssens and Van Gucht [9] etc. But our languages can also be given interpretations based on bags and lists. It is desirable to know whether our main result holds when the languages are used to manipulate nested lists and bags. We prove that it does. Moreover, the proof is uniform across these ....

[Article contains additional citation context not shown here]

J. Paredaens and D. Van Gucht. Converting Nested Relational Algebra Expressions into Flat Algebra Expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

....as the algebra without powerset in [AB88] while NRA(powerset) has the same expressive power as the algebra. NRA has essentially the same expressive power as Schek and Scholl s NF 2 relational algebra [SS86] as Thomas and Fischer s algebra [TF86] and as Paredaens and Van Gucht s nested algebra [PG88, PG92]. In defining NRA, we follow the formalism in [BBW92] The nested relational algebra NRA is a typed language. Its types are build from the following base types: B (the booleans) unit (the single valued type, unit = f( g) and N (the natural numbers) and are given then by the grammar: t : ....

....are in PTIME, NRA(powerset) can obviously express exponential queries. More interestingly, NRA(powerset) can express PTIME queries, which are not expressible in NRA: Abiteboul and Beeri prove in [AB88] that transitive closure can be expressed in NRA(powerset) In contrast, we know from Paredaens [PG92] and Wong [Won93] that transitive closure is not expressible in NRA. But the obvious way of expressing transitive closure in NRA(powerset) is through an exponential space query. We prove in the following that exponential space is indeed needed. 3 The Complexity of Evaluation in NRA(powerset) ....

Jan Paredaens and Dirk Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

.... In defining NRA we have followed closely [BBW92] but the language has essentially the same expressive power as other formalisms for complex objects, such as Schek and Scholl s NF 2 relational algebra [SS86] as Thomas and Fischer s algebra [TF86] and as Paredaens and Van Gucht s nested algebra [PG88, PG92]. To substantiate this claim, we follow [BBW92] and show in the following examples how to express some of the primitives present in the latter languages. Example 2.2 The database projection Pi t 1 ;t 2 1 : ft 1 Theta t 2 g ft 1 g and Pi t 1 ;t 2 2 : ft 1 Theta t 2 g ft 2 g are defined ....

....parity : fZg B , with the meaning: tc(r) def = fhu; vi j 9n 1; 9u 1 ; un :u = u 1 ; un = v; 8i:1 i n; hu i ; u i 1 i 2 rg parity(x) def = T iff card(x) is even Throughout the paper, card(x) denotes the cardinality of the set x (i.e. the number of its elements) Proposition 2. 9 [PG92, Won92] The queries tc and parity cannot be expressed in NRA. powerset t : ftg fftgg Figure 2: The definition of powerset. 2.2.2 NRA(powerset) Now we consider for every type t a new primitive operation, powerset t , see Figure 2. Its meaning is: powerset t (x) def = fy j y xg We denote ....

Jan Paredaens and Dirk Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

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J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

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J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM TODS, 17(1):65--93, March 1992.

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J. Paredaens, D. Van Gucht, Converting nested relational algebra expressions into flat algebra expressions, ACM Transaction on Database Systems 17, No. 1 (1992), 65--93. 23

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J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

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J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17(1):65--93, March 1992.

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J. Paredaens, D. Van Gucht, Converting nested relational algebra expressions into flat algebra expressions, ACM Transaction on Database Systems 17, No. 1 (1992), 65--93.

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J. Paredaens and D. Van Gucht. Converting nested relational algebra expressions into flat algebra expressions. ACM Transaction on Database Systems, 17 (1992), 65--93.

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