L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Proceedings of 4th International Workshop on Database Programming Languages, New York, pages 282-294, August 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A j= a) and 1 otherwise. Theorem 6.4. Let Theta be monoidal. Then L aggr (All; Theta) captures RL (All; Theta) Moreover, L g) captures RL contains ( Gamma; Xi) and L g) captures RL contains ( Gamma; Xi; root) Proof. The proof of Proposition 5. 2 in [32] can be adapted to show that for Theta monoidal, every expression e of RL (All; Theta) is equivalent to an expression e in which for every subexpression of the form fe 1 j x 2 e 2 g, and Aggr F fe 1 j x 2 e 2 g, the expression e 2 is one of R 2 oe. Moreover, if is the only ....

L. Libkin and L. Wong. Aggregate functions, conservative extensions, and linear orders. In Proc. Database Programming Languages

....equality and membership in our languages and these are interpreted, since they have fixed meanings. 29 Note, however, that aggregates may require the use of bags rather than sets, for producing correct results. The problem can be addressed either by including bag as a type constructor, see e.g. [46, 47, 48], or using, e.g. Klug s approach [39] We do not consider the issue further in the paper. 56 ( for each function f in Delta fn , f Gamma1 is also in Delta fn . Note that when a function is not 1 1, its inverse is set valued; since we have sets in the model, that is not a problem. Our ....

Leonid Libkin and Limsoon Wong, Aggregate Functions, Conservative Extension, and Linear Orders, Proc. ACM PODS, 282--294, 1993.

....The algebra considered here was introduced in [GM93] Various calculi for bags based on some of the operations of [GM93] were presented in [GMK93] and links with various weak arithmetics were established. The expressive power of languages for bags has been investigated in another setting in [Won93, LW93b, LW93a, LW94]. The paper is organized as follows. In the next section, we briefly present the main definitions. Section 3 is devoted to the algebra. In the following sections, we study the expressive power and the complexity of the algebra, when restricted to bags with one level of nesting (Section 4) two ....

....over bags over a single constant, is satisfied by a finite or co finite number of bags, as noted in [LW93b] On the other hand, the parity of the cardinality of a relation (bag with no duplicates) becomes definable in BALG 1 in the presence of an order on the domain. This was first shown in [LW93a] for a language equivalent to BALG 1 . The following boolean expression states that the parity of the cardinality of relation R is even: oe x (MAP [a] oe y (yx) R) MAP [a] oe y (x y) R) R) 6= The previous expressions states the existence of an x such that the number of elements smaller ....

L. Libkin and L. Wong. Aggregate functions, conservative extensions, and linear orders. In Proc. 4th Int. Workshop on database programming languages, 1993.

....The use of grouping is modeled by nesting of sets. At first glance, it seems that sets of sets must be produced to answer this query. That is, the restriction that ts and ss in Figure 1 be record types is eliminated. However, such a nested language has the conservative extension property [24], which says that every query can be written using the height of set nesting not exceeding that of its input and output types. In particular, it means that every query whose input and output types are flat relations, can be written in RA aggr (N) that is, without using nesting) even if it ....

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Proc. 4th Internat. Workshop on Database Programming Languages, New York, 1993, Springer Verlag, pages 282--294.

....to the constraint model, other extensions of Codd s relational model have received much attention in the past few years. Among them are extensions to complex objects and bags. It is interesting to compare the addition of constraints with those extensions. Several conservativity results proved in [33, 20, 21] show that the addition of nested relations (as in complex objects) does not add relational expressive power. On the other hand, the addition of bags does add relational expressive power [9, 19, 22] In particular, parity test is definable using bags on ordered domains [19] Organization. Section ....

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Proceedings of 4th International Workshop on Database Programming Languages, New York, pages 282-294, August 1993.

....many statistical computations on numeric series are essentially based on the evaluation of the expected value of some scalar functions, defined as the mean of the function applied to the elements of the series. A similar result has been obtained in the context of a nested relational language [18]. Actually, we can show that conjunctive md cal(sum; if; expresses other aggregate functions, including the generalized quantifiers (set predicates such as some, all, and at least) introduced in [19] and special operators (like rank, ratio to report, and n tile) introduced in some ....

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Workshop on Database Programming Languages, pages 282--294, 1993.

....than relational algebra. In particular, the de facto practical query language SQL has a GROUP BY operator, arithmetic operators, and aggregate functions. These extra primitives give these languages extra power; for example, they can test whether a set of numbers has even or odd cardinality [12]. In this section, we consider decremental recomputation of transitive closure in these more practical languages. First, we define a theoretical SQL , which is a formally presented language that has the main features that distinguish SQL from relational calculus. Then we prove analogs of our ....

....3.1 A Theoretical Reconstruction of SQL As we mentioned, there are two main features that make SQL more expressive that first order logic. First, SQL allows nesting by using the GROUP BY operator. Second, SQL has arithmetic operations and a number of aggregate functions. It was proposed in [12] to use the language NRL aggr as a theoretical language to study the expressive power of SQL. It is obtained from the nested relational algebra (which is a standard language for manipulating complex objects, see [3, 4, 15] by adding simple rational arithmetic operators , Gamma, Delta, and ....

[Article contains additional citation context not shown here]

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Proceedings of 4th International Workshop on Database Programming Languages, New York, pages 282-294, August 1993.

.... sets, tuples, and conditionals come from the equational theory of NRC, described in [7, 34] They include rules for vertical and horizontal fusion of set loops, filter promotion, and column reduction [5] The rules for summation and arithmetic come from an extension of NRC to arithmetic given in [18]. Here, we describe the new rules for arrays. Since the syntax for arrays was inspired by viewing them as functions, it is not surprising that the rules for arrays are also based on this view of arrays as (partial) functions. There are three rules for arrays: fi p ) e 1 j i e 2 ] e 3 ] ....

L. Libkin and L. Wong. Aggregate functions, conservative extensions, and linear orders. In DBPL'93, pages 282--294.

....and Beeri [1] without the powerset operator. Then Wong [30] proved that the language has the conservative extension property at all input output heights. That is, the expressive power of the language is independent of the height of set nesting in the intermediate data. Then Libkin and Wong [19] showed that in the presence of very simple arithmetic operators conservativity can be extended uniformly to all input output heights for languages augmented with bounded fixpoint operator, transitive closure, powerset and many other operators. In [17] Libkin and Wong extended the use of the ....

....U or N Gamma U is finite. Proof sketch: Assume there is an infinite and co infinite property U of natural numbers that is expressible in BQL. Then by theorem 4. 2 a function f : N N such that f(n) 1 for n 2 U and f(n) 0 for n 62 U is expressible in NRL(N; Sigma; Delta; eq) In [19] we proved that expressions of NRL(N; Sigma; Delta; eq) are independent of the height of the intermediate data. Careful analysis of functions of type N N that do not involve set constructs shows that they coincide with polynomials almost everywhere and hence can not have infinitely many ....

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. This volume.

....us a language equivalent to the relational algebra was shown in [25, 35] even if nesting is used in intermediate results. Another result of this kind [27] shows that by adding a bounded fixed point construct to comprehension syntax gives us, again at relational types, inflationary datalog, and in [19, 21] it is shown that nesting at intermediate types does not add expressiveness in presence of aggregate functions and certain generic queries. Other results on expressive power are to be found in [19, 20, 21] Our approach can be used for different collections: languages for or sets were studied in ....

....to comprehension syntax gives us, again at relational types, inflationary datalog, and in [19, 21] it is shown that nesting at intermediate types does not add expressiveness in presence of aggregate functions and certain generic queries. Other results on expressive power are to be found in [19, 20, 21]. Our approach can be used for different collections: languages for or sets were studied in [17, 13] and bag languages in [18] 29] shows that transitive closure, which is efficiently expressible using structural recursion, has a necessarily exponential implementation in complex object algebra ....

L. Libkin and L. Wong. Aggregate Functions, Conservative Extension, and Linear Orders. In Proceedings of 4th International Workshop on Database Programming Languages, Manhattan, New York, Springer Verlag, 1993.

....without using any intermediate data whose depth of set nesting is greater than k. Hence the expressive power of the nested relational calculus is independent of the depth of set nesting allowed in the intermediate data. This property is called the conservative extension property. Libkin and Wong [5] showed that when the nested relational calculus is endowed with aggregate functions, it retains the conservative extension property. The aim of this paper is to demonstrate that the nested relational calculus continues to retain the conservative extension property even in the presence of any ....

....relational calculus continues to retain the conservative extension property even in the presence of any family of functions that are generic and do not invent new values. The strategy is to combine the rewriting technique of Wong [10] and the effective lifting of linear orders of Libkin and Wong [5] to encode any such function into one whose height is minimum. To appear in Information Processing Letters. Lambda Calculus and Products x s : s e : t x s :e : s t e 1 : s t e 2 : s e 1 e 2 : t ( unit e : s Theta t 1 e : s 2 e : t e 1 : s e 2 : t (e 1 ; e 2 ) s Theta t ....

[Article contains additional citation context not shown here]

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Proceedings of 4th International Workshop on Database Programming Languages, Springer Verlag, 1993, pages 279--292.

No context found.

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Proceedings of 4th International Workshop on Database Programming Languages, New York, pages 282-294, August 1993.

No context found.

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In C. Beeri, A. Ohori, and D. Shasha, editors, Proceedings of 4th International Workshop on Database Programming Languages, New York, August 1993.

No context found.

L. Libkin and L. Wong. Aggregate functions, conservative extensions, and linear orders. In DBPL'93, pages 282--294.

No context found.

L. Libkin and L. Wong. Aggregate functions, conservative extension, and linear orders. In Workshop on Database Programming Languages, pages 282#294, 1993.

No context found.

L. Libkin and L. Wong. Aggregate functions, conservative extensions, and linear orders. In Proc. DBPL'93, Springer, 1994, pages 282--294.

No context found.

L. Libkin and L. Wong. Aggregate functions, conservative extensions, and linear orders. In Proc. DBPL'93, Springer, 1994, pages 282-294.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC