L. Libkin and L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 155--166, May 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has been studied extensively, and was shown to be significant for the expressive power of query languages. In particular it was shown that fixpoint logic expresses all PTIME functions on ordered inputs [Imm86, Var82] The impact of allowing duplicates in relations was studied in [GM93, GMK93, LW94] It was also shown to have a strong effect on the expressive power of query languages, and in particular to result in an increased expressive power. In this paper, we study the interaction between order and duplicates, and the effect that the combined features have on the expressive power of ....

....the tuples, are unordered, and the relations contain no duplicates, and cannot be nested. Relaxing these assumptions leads to numerous distinct data types, such as the complex objects (nested sets) Jac82, AB87, KV84, KRS85, AG91] the bags (sets with duplicates) BK90, Mum90, Alb91, BS91, GM93, LW94] the lists (internal order) the ordered sets, and the pomsets (partially ordered multisets) Pra84] 2.1 Partially Ordered Multisets Pomsets The pomset type generalizes sets, bags, lists, trees, and other ordered types, and therefore provides a uniform representation for all these types. ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proc. 13th ACM Symp. on Principles of Database Systems, 155-166, 1994.

.... has led to increasing interest in the problem of e#ciently elaborating aggregate queries, which are widely used in such systems [24,39] Thus, due to its practical importance, the study of logics and declarative languages with aggregates has received significant attention in the literature [13,15,16,19,38]. Most of the research has concentrated on the study of declarative programs containing recursive predicates with monotonic aggregates [10,33,28] These works pursue the general objective of ensuring the A preliminary version of this paper appeared in [14] This work has been partially supported ....

Libkin, L., and Wong, L., New Techniques for Studying Set Languages, Bag Languages and Aggregate Functions. Proc. Int. Symp. on Principles of Database Systems, 155--166 (1994).

....has been studied extensively, and was shown to be significant for the expressive power of query languages. In particular it was shown that fixpoint logic expresses all PTIME functions on ordered inputs [Imm86, Var82] The impact of allowing duplicates in relations was studied in [GM93, GMK93, LW94] It was also shown to have a strong effect on the expressive power of query languages, and in particular to result in an increased expressive power. In this paper, we study the interaction between order and duplicates, and the effect that the combined features have on the expressive power of ....

....the tuples, are unordered, and the relations contain no duplicates, and cannot be nested. Relaxing these assumptions leads to numerous distinct data types, such as the complex objects (nested sets) Jac82, AB87, KV84, KRS85, AG91] the bags (sets with duplicates) BK90, Mum90, Alb91, BS91, GM93, LW94] the lists (internal order) the ordered sets, and the pomsets (partially ordered multisets) Pra84] 2.1 Partially Ordered Multisets Pomsets The pomset type generalizes sets, bags, lists, trees, and other ordered types, and therefore provides a uniform representation for all these types. ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proc. 13th ACM Symp. on Principles of Database Systems, 155-166, 1994.

....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, New Techniques for Studying Set Languages, Bags Languages, and Aggregate Functions, Proc. ACM PODS, 155--166, 1994

....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 ....

....parity of the cardinality of a relation is not first order definable even in the presence of an order relation. This is easily proved using EhrenfeuchtFra iss e [Ehr61, Fra54] games. This shows the difference between BALG 1 and RALG in presence of an order on the domain. It has been shown in [LW94], that the parity of the cardinality of a relation is not definable in BALG 1 in general, that is without assuming an order relation over the domain. The proof of [LW94] is done in a more general setting than BALG 1 . It was also proved in the same paper, that the transitive closure of a ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proc. 13th ACM Symp. on Principles of Database Systems, 1994. 33

....quantifiers to first order logics. The construction in the next section will give an example of the use of indiscernibles to reduce every formula in the language L(Ct) to a first order formula. Since several previous expressibility bounds on aggregates make use of some sort of Count Elimination [19], we hope to investigate this phenomenon in more generality in forthcoming work. 4 Nonstandard Models and Unary Counters Consider the logics L(Ct) L(Tm) and L(Ct; N) defined in Section 2. We are interested in investigating the relative expressive powers of these languages, using the techniques ....

.... We are interested in developing a usable set of rewrite rules for simplifying languages such as L(Ct) and L(Tm) and getting semantic characterizations of the definable transactions that are available (along the lines of Gaifman s locality theorem [11] or the bounded degree property of [19]) We are interested in studying the relationship of the language L(Tm) to various other counting languages (those discussed, for example, in [16] In particular, it would be helpful to know whether languages with binary counters can express all sentences of L(Tm) and similarly for n ary ....

L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proceedings of the 13th Conference on Principles of Database Systems, Minneapolis MN, May 1994, pages 155--166.

....the user desires multiset semantics. Our framework integrates the treatment of relations as lists, multisets, and sets. Grumbach and Milo [GM93] study the expressive power of algebras for manipulating bags. In particular, they study how bag nesting affects expressive power. Libkin and Wong [LW94] provide new techniques for studying the expressive powers of set languages and bag languages that have aggregate functions. We do not focus on studying the expressive power of our proposed algebra other than showing that it extends the conventional relational algebra. More than a dozen temporal ....

L. Libkin and L. Wong. New Techniques for Studying Set Languages, Bag Languages and Aggregate Functions. In Proceedings of ACM PODS, Minneapolis, Minnesota, pp. 155--166 (1994). 35

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 155--166, May 1994.

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L. Libkin and L. Wong, New techniques for studying set languages, bag languages, and aggregate functions. in "Proceedings of 13th ACM Symposium on Principles of Database Systems," Minneapolis, Minnesota, 1994.

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 155--166, Minneapolis, Minnesota, May 1994. Full version to appear in JCSS.

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In PODS'94, pages 155-166.

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proc. of the 13th Symposium on Principles of Database Systems, Minneapolis MN, May 1994, pages 155--166.

.... and record formation, conditional and the equality test gives us precisely the nested relational algebra [3] but the presentation is nicer than the standard ones, such as in [32] This approach to the language design has proved extremely fruitful and allowed to solve some open problems (e.g. [24]) and develop languages for other collections (e.g. 22, 23] In order to apply it to the approximation constructs, we first need formal models of those, and then the universality properties for those models. Remark. There is no mysticism in the diagrams above these are constructions well ....

L. Libkin and L. Wong, New techniques for studying set languages, bag languages and aggregate functions, In PODS-94, pages 155--166.

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Leonid Libkin and Limsoon Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 155--166, Minneapolis, Minnesota, May 1994. See also UPenn Technical Report MS-CIS-9395.

.... from an adjunction to the Kleisli category of its monad, and the fact that ext and map and are interchangeable follows from the general properties of the categorical notion of a monad, see [BW90] This approach to the language design was shown to be extremely useful in the past few years, see [LW94a, LW94b, Suc94] Here we apply it to partial information; the reader has probably already noticed the similarity between diagrams in figure 6 and proposition 13, which will give us the operations of the language. 3.2 Language for sets and its sublanguages 22 General operators and pairs g : u ....

....ffi for any x = x ffi , such f is definable in NRL a (p) We do not know if NRL( b ) is conservative over NRL a . However, we can show that it is conservative when augmented with aggregate functions. Instead of choosing a restricted set of aggregates, we use a general template suggested by [LW94a, LW94b] This is the higher order function P (f) that takes a function f : t N and returns P (f) ftg N given by P (f) fx 1 ; xng) f(x 1 ) f(xn ) Other operations on the type of naturals include multiplication and modified subtraction (monus) The key idea in ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 155--166, Minneapolis, Minnesota, May 1994.

....We present two main results. First, we extend the order theoretic approach from sets to bags. This results in two orderings that must be used for bags under the open and the closed world assumptions. Second, we prove that these orderings are not definable in basic bag algebras such as those in [2, 11]. This may impact query language design for bag based databases. While in the set case, orderings are definable in any language that extends relational calculus, in the bag case one may have to enrich basic languages with primitives capable of expressing these orderings. In addition, as a somewhat ....

....has enough power to lift to H and P . The situation is very different in the bag case. In order to demonstrate this result, we need a standard language for bags that has a role similar to that of relational calculus for sets. Such a language has recently been proposed and studied [2, 11]. The object types are given by the grammar t : b j t Theta t j fjtjg where b ranges over a collection of base types (among them bool , the type of Booleans) t 1 Theta t 2 is the type of pairs (we use pairs rather than records to keep notation simple) and values of type fjtjg are 4 finite ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proc. of the 13th Symposium on Principles of Database Systems, Minneapolis MN, May 1994, pages 155--166.

....[15] modifying a result by Hanf [19] for the finite case, proved that if a certain criterion relating the numbers of small neighborhoods in two structures holds, then these structures agree on sentences whose quantifier rank is determined by the size of those neighborhoods. The author and Wong [25] showed that if first order query operates on graphs, then the number of different in and out degrees in the output is below a bound given by the query and the maximal degree in the input graph. That is, if locally the input looks simple, then so does the output of a first order query. We called ....

....cleaner inexpressibility proofs (see [10] However, no extension of first order logic is known to satisfy an analog of Gaifman s theorem. Finally, we have the bounded degree property, whose proof is based on Gaifman s theorem, and which leads to particularly simple inexpressibility proofs, cf. [10, 25]. Very recently, with considerable amount of effort, it was shown that the bounded degree property holds for certain queries in a first order relational language extended with aggregate functions [10] this language has substantial counting power) The goal of this paper is to study the ....

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L. Libkin, L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In PODS'94, pages 155--166. Full version to appear in JCSS.

.... It was also shown that the basic bag algebra essentially adds the correct evaluation of aggregate functions to the relational algebra, and this continues to hold when nested relations are present [20] There are a number of deep results on the complexity and expressive power of 1 bag languages [11, 2, 19, 20, 21, 22, 31]. The main goal of this paper is to lay the foundation for incremental maintenance of views defined in the bag algebra. We advocate an approach based on equational reasoning. That is, for each primitive in the bag algebra we derive an equation that shows how the result of applying this primitive ....

....Section 7. Finally, we conclude in Section 8 with some remarks concerning future work. All proofs can be found in [10] 2 Notation and Problem Statement 2. 1 The Bag Algebra, BA As we mentioned in the introduction, several equivalent approaches to bag based database languages have been proposed [11, 19, 22]. As our basic language in which we formulate the change propagation algorithm we take 3 a restriction of those languages to flat bags (that is, bag valued attributes are not allowed) In what follows, base relation names are denoted by the symbols R, R 1 , R 2 , Let p range over ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In PODS-94, pages 155--166.

....Only sets have been considered in [INV91a, INV91b, LW93a, Rou91] but many practical languages are based on bags (multisets) In the past few years several approaches to design of bag languages have been proposed. Moreover, most approaches agree on what constitutes the basic set of bag operations [Alb91, GM93, LW93b, LW94]. Thus, we believe the normalization mechanism must be extended to bags. ffl Normalization may cause exponential blowup in the size of objects. For objects of size n, the size of their normal forms is bounded (roughly) by n Delta 1:45 n [LW93a] Therefore, we need better normalization tools. ....

....expressive power of query languages. Recall that the AbiteboulBeeri algebra A B [AB88] is the nested relational algebra (general and set operators in figure 2) plus the powerset operator. While the nested relational algebra cannot express recursive queries such as transitive closure (tc) [LW94], A B can express tc by first producing all possible relations on a given set of nodes and then selecting those that contain a given one and are transitive. Of course this way of computing tc uses exponential space. A remarkable result of [SP94] says that no matter how we write an A B expression ....

L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In PODS-94, pages 155-- 166.

....it here as an ambient language. Note that we use the version based on bags (multisets) rather than sets. This is necessary because keeping duplicates is very important for the normalization process [11] Our ambient language contains standard languages for nested bags, such as BALG [5, 6] and BQL [13, 14], as its sublanguages. To obtain the corresponding results for sets, one can use the techniques of [11] in a straightforward way, so here we only present results for bags. Organization. We define normal forms, partial normal forms, the ambient language, and prove the generalized normalization ....

....more than one element of a bag have nondominated F values, selects one nondeterministically. The semantics of gen is given by gen(n) fj1; njg (this function plays an important role in establishing equivalences between set and bag languages with structural recursion and power operators [13, 14]) Now opt pnorm rand is defined in two steps. First, we define one iteration step iter opt pnorm rand s (P) o; T ) opt apnorm time s (P) random s (o) T ) and then opt pnorm rand s (P) o; T; n) if n 1 then iter opt pnorm rand s (P) o; T ) else select best( 2 ; F ) b ....

L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Symp. Principles of Database Systems 1994, ACM Press, pages 155--166.

....(node with out degree zero) the root has out degree 1, the leaf has in degree one, and all other nodes have both in and out degrees 1. All these conditions are firstorder definable. That chains and cycles are not firstorder definable can be proved straightforwardly using standard techniques (cf. [15, 23]) 2 For the rest of the paper Psi c c denotes a first order sentence in the language of E( Delta; Delta) that defines C Cgraphs. 3.1 Languages not verifiable over FO In this subsection we prove Theorem B from the introduction. That is, Theorem 2 Assume that TL can express one of the ....

....nor deterministic transitive closure is in WPC(L) Proof sketch. The proof for FO eq card and FO c( Omega Gamma is essentially the same as the proof of theorem 2, as we have to show that connectivity and testing for chain are not definable in those logics. This follows from the results of [23], 7] and [19] In the case of monadic Sigma 1 1 , the proof for tc is the same as before, since connectivity is not expressible in monadic Sigma 1 1 [5] For dtc, we have to show that testing whether a graph is a chain is not monadic Sigma 1 1 definable. This can be done using the ....

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Proceedings of the 13th Symposium on Principles of Database Systems, Minneapolis MN, May 1994, pages 155--166.

....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 2 presents the notations that are used throughout the paper. We also explain the active and natural semantics of relational calculus and state two previous results relating them. Section 3 ....

L. Libkin and L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 155--166, May 1994.

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages, and aggregate functions. In Proceedings 13th ACM Symposium on Principles of Database Systems, pages 155--166. ACM Press, 1994.

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L. Libkin and L. Wong. New Techniques for Studying Set Languages, Bag Languages and Aggregate Functions. In Proceedings of ACM PODS, Minneapolis, Minnesota, pp. 155--166 (1994). 35

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L. Libkin and L. Wong. New techniques for studying set languages, bag languages and aggregate functions. In Thirteenth ACM SIGACT SIGMOD SIGART Symp. on Principles of Database Systems, pages 155#166, 1994.

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