L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, June 1996. 11  Home/Search   Document Details and Download   Summary   Related Articles   Check

....types they had to support. It turned out that a set language capturing essentially the expressive power of a language for bags, could also model all the essential features of SQL [24] The problem was that the language dealt with nested relations, or complex objects. But then [24] extending [28,31], proved a conservativity result, stating that nested relations are not really needed if the input and output do not have them. That made it possible to use a non nested fragment of languages inspired by structural recursion [4] and comprehensions [30] as a theoretical reconstruction of SQL. ....

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52 (1996), 495-505. 31

....a query that is not expressible without the powerset. To conclude this introduction we should mention that an analogue of the flat flat theorem in a complex object formalism different from, but equivalent to, the nested relational model was proved by Wong using a remarkably elegant argument [18]. 2 Preliminaries on nested relations Basically we assume the existence of a countably infinite supply of atomic attributes. The set of atomic attributes is denoted by U . The set HF(U) of hereditarily finite sets with atoms in U is the smallest set containing U , such that if X 1 ; X n ....

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, 1996. 19

....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 ....

Limsoon Wong. Normal forms and conservative extension properties for query languages over collection types. JCSS, 52(3):495--505, 1996. 54 Syntax Operational Semantics Complexities

....Also the other operators of MA are shown to be primitive. 1 Generalizations of Codd s theorem to extensions of the relation model have always been a popular research topic (e.g. 13, 15, 2, 11, 5] 2 Analogous conservative extension properties are known for complex object databases [17, 26, 23] and spatial databases [16] The present paper is a follow up on an earlier paper by three of us [25] There, we studied the expressive power of evaluating stored relational algebra programs in a completely untyped setting. Relational algebra programs were encoded in data relations, and the ....

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505,

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L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, June 1996. 11

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L. Wong. Normal forms and conservative extension properties for query languages over collection types. JCSS, 52:495--505, 1996.

....like to emphasize the the choice of a particular language is not central to our problems. In particular, our results extend to any language with the same power as the language NRC aggr presented below. The language NRC aggr is obtained by extending the nested relational calculus NRC( of [5, 23] by arithmetics and aggregate functions. The motivation for considering NRC aggr is that it is a much more realistic query language than relational algebra. Indeed, as explained later, one can consider NRC aggr to be a theoretical reconstruction of SQL, the de facto relational query language ....

....to e 2 if e 1 is true and evaluates to e 3 if e 1 is false . We provided equality test on every type s. However, this is equivalent to having equality test restricted to base types together with emptiness test for set of base types [22] NRC possesses the so called conservative extension property [23]: if a function f : s 1 s 2 is expressible in NRC, then it can be expressed using an expression of height no more than that of s 1 and s 2 . The height of a type is defined as its depth of nesting of set brackets. The height of an expression is defined as the maximum height of all types that ....

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L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, June 1996. This article was processed using the L A T E X macro package with LLNCS style

....be customized by different rules and strategies. When a query is submitted to Kleisli, it is first processed by the CPL Module which translates it into an equivalent expression in NRC, an abstract nested relational calculus NRC developed in [6] Various suitable fragments of NRC have been shown [6, 14, 24] to directly correspond to relational calculus, nested relational calculus, and SQL. It is chosen as the internal query representation because it is easy to manipulate and amenable to machine analysis. The NRC expression is then analyzed by the Type Module to infer the most general valid type for ....

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, 1996. 10

.... BioKleisli system [7, 15] as our query engine for retrieving scientific abstracts and literature from (possibly multiple) online bibliographic resources on the Internet [17] BioKleisli is a well tested query system that has been built on the formal foundations of modern query languages [4, 3, 16] and specially designed for integrating data from disparate sources across many geographies and systems; 2. BioNLP . We have developed the BioNLP module to process the free texts in the scientific abstracts retrieved by BioKleisli. BioNLP identifies protein names mentioned in the free texts and ....

Wong, L., Normal forms and conservative extension properties for query languages over collection types, Journal of Computer and System Sciences, 52:495--505, 1995.

....and the analysis of relevant language properties, has been left to future work. The aim of this paper is the definition of a model and a query language for finitely representable nested relations, overcoming some limitations of the previous proposals. Our language is obtained by extending NRC [14] to deal with possibly infinite relations, finitely representable by using the real polynomial constraint theory, and it is called frNRC. NRC is similar to the well known comprehension mechanism in functional programming and its formulation is based on structural recursion [5] and on monads [13] ....

....in the same framework. One of the main result about this language is that frNRC as NRC, has the conservative extension property. This means that, when input and output are restricted to deal with a specific degree of nesting, any higher degree of nesting generated by the computation is useless [14]. Thus, when input and output relations represent constraint flat relations, frNRC expressions can be mapped in first order logic extended 1 with polynomial constraints. Note that, even if this result is a consequence of properties of NRC and constraint query languages, nobody has proved this ....

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L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, 1996. 8 A Some Relevant Proofs Proposition 5.2 For every function f : s 1 ! s 2 in frNRC, there is a function (f)

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L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52, 1996. To appear.

....and the analysis of relevant language properties, has been left to future work. The aim of this paper is the definition of a model and a query language for finitely representable nested relations, overcoming some limitations of the previous proposals. Our language is obtained by extending NRC [20] to deal with possibly infinite relations, finitely representable by using a decidable logical theory admitting variable elimination (this is a necessary and sufficient condition to the definition of constraint query languages [10] and is called frNRC. NRC is similar to the well known ....

....the same framework. One of the main results about this language is that frNRC, as NRC, has the conservative extension property. This means that, when input and output are restricted to deal with a specific degree of nesting, any higher degree of nesting generated by the computation is useless [20]. Thus, when input and output relations represent constraint flat relations, frNRC expressions can be mapped into first order logic extended with the considered logical theory. Conservative extension is a very important property for optimization purposes, since it guarantees that complex ....

[Article contains additional citation context not shown here]

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, 1996.

....of invariant code out of a loop. ffl Reducing the size of intermediate data by reducing the number of fields, which corresponds to moving projections down to intermediate results and base relations in the relational algebra. Details of these optimizations are beyond the scope of this paper (see [51, 52]) however, the important thing to notice is that the monadic optimizations generalize many known optimizations for the relational algebra to complex objects. Moreover, they allow us to move the largest possible SQL subquery of a CPL expression to the relational server, minimizing the amount of ....

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52, 1996. To appear.

....our nested relational calculus with arrays. NRCA is an extension of the nested relational calculus NRC [7] We chose this presentation of nested relations because it leads to an appealing comprehension syntax [6, 9] and because it comes with an equational theory that gives us useful optimizations [7, 33, 34]. We will touch on these aspects of the calculus in sections 3 and 5. The constructs of NRC and their typing rules are given in the top third of figure 1. Here, we briefly review their meanings. The constructs for functions are standard. For products, e 1 ; e k ) and i;k (e) are just ....

.... 2 ] for subscripting into an array, dim k (e) for extracting the dimensions of an array, and index k (e) for converting an 3 Note that from an expressivity standpoint we need only include = over the base types, because their liftings to all other complex object types will be definable in NRCA [20, 34]. NRC constructs Functions and k ary Products (k 2) Gamma; x : s x : s Gamma; x : s e : t Gamma x:e : s t Gamma e 1 : s t Gamma e 2 : s Gamma e 1 (e 2 ) t Gamma e 1 : t 1 : Gamma e k : t k Gamma (e 1 ; e k ) t 1 Theta Delta Delta Delta ....

[Article contains additional citation context not shown here]

L. Wong. Normal forms and conservative extension properties for query languages over collection types. JCSS, to appear. Extended abstract in PODS'93.

....of invariant code out of a loop. Reducing the size of intermediate data by reducing the number of fields, which corresponds to moving projections down to intermediate results and base relations in the relational algebra. Details of these optimizations are beyond the scope of this paper (see [60, 61]) however, the important thing to notice is that the monadic optimizations generalize many known optimizations for the relational algebra to complex objects. Other non monadic optimizations include the ability to capture general properties of primitives (such as commutativity) and characteristics ....

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52, 1996. To appear.

....like to emphasize the the choice of a particular language is not central to our problems. In particular, our results extend to any language with the same power as the language NRC aggr presented below. The language NRC aggr is obtained by extending the nested relational calculus NRC( of [5, 22] by arithmetics and aggregate functions. The motivation for considering NRC aggr is that it is a much more realistic query language than relational algebra. Indeed, as explained later, one can consider NRC aggr to be a theoretical reconstruction of SQL, the de facto relational query language ....

....to e 2 if e 1 is true and evaluates to e 3 if e 1 is false. We provided equality test on every type s. However, this is equivalent to having equality test restricted to base types together with emptiness test for set of base types [21] NRC possesses the so called conservative extension property [22]: if a function f : s 1 s 2 is expressible in NRC, then it can be expressed using an expression of height no more than that of s 1 and s 2 . The height of a type is defined as its depth of nesting of set brackets. The height of an expression is defined as the maximum height of all types that ....

[Article contains additional citation context not shown here]

L. Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, June 1996.

....that practical relational databases, as well as more advanced systems like Kleisli [6] possess remarkable power in a way that was little suspected before. 2 Nested Relational Calculus with Aggregates The language NRC aggr is obtained by extending the nested relational calculus NRC( of [4, 23] by arithmetics and aggregate functions. The motivation for considering NRC aggr is that it is a much more realistic query language than relational algebra. Indeed, as explained later, one can consider NRC aggr to be a theoretical reconstruction of SQL, the de facto relational query language ....

....type s. However, this is equivalent to having equality test restricted to base types together with emptiness test for set of base types [22] Also, set member, union, intersection, etc. can obviously be defined in terms of equality test. NRC possesses the so called conservative extension property [23]: if a function f : s 1 s 2 is expressible in NRC, then it can be expressed using an expression of height no more than that of s 1 and s 2 . The height of a type is defined as its depth of nesting of set brackets. The height of an expression is defined as the maximum height of all types that ....

[Article contains additional citation context not shown here]

Limsoon Wong. Normal forms and conservative extension properties for query languages over collection types. Journal of Computer and System Sciences, 52(3):495--505, June 1996.

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