L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, to appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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 ....

Hella, L., Libkin, L., Nurmonen, J., and Wong, L., Logics with Aggregate Operators. Proc. Int. Conf. on Logic in Computer Science. 35--44 (1999).

....that certain properties of finite structures are not expressible in first order logic, and it seems that this was Gaifman s main motivation. More recently, Libkin and others considered this technique of proving inexpressibility results using locality in a complexity theoretic context (see, e.g. [5, 14, 13, 15]) A completely different application of Gaifman s theorem has been proposed in [11] It can be used to evaluate first order sentences in certain finite structures quite efficiently. In general, it takes time n (l) to decide whether a structure of size n satisfies a firstorder sentence of size ....

L. Hella, L. Libkin, J. Nurmonen, and L. Wong. Logics with aggregate operators. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science, 1999.

....that certain properties of finite structures are not expressible in first order logic, and it seems that this was Gaifman s main motivation. More recently, Libkin and others considered this technique of proving inexpressibility results using locality in a complexity theoretic context (see, e.g. [5, 15, 14, 16]) A completely different application of Gaifman s theorem has been proposed in [11] It can be used to evaluate first order sentences in certain finite structures quite efficiently. In general, it takes time n (l) to decide whether a structure of size n satisfies a first order sentence of size ....

L. Hella, L. Libkin, J. Nurmonen, and L. Wong. Logics with aggregate operators. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science, 1999.

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, to appear.

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, pages 35--44.

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. J. ACM, 48 (2001), 880--907.

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. Journal of the ACM 48(4), pages 880--907, 2001.

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. JACM 48(4): 880-907 (2001).

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. Journal of the ACM 48(4), pages 880--907, 2001.

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. JACM 48(4): 880--907 (2001).

....property. Section 5 de nes an aggregate logic L aggr and shows a simple translation of the algebra with aggregates Alg aggr into this logic. Then, in Section 6, we present a self contained proof of locality of L aggr (and thus of Alg aggr ) In previous papers on the expressive power of SQL [24,25,22,18], we used languages of a rather di erent avor, based on structural recursion [4] and comprehensions [30] In Section 7, we show that those languages are at most as expressive as Alg aggr . In Section 8, we consider an extension Alg aggr of Alg aggr in which nonnumerical order comparisons are ....

....the expressiveness of rst order logic (FO) over nite structures, since relational algebra has the same power as FO. So perhaps if we could put aggregates and arithmetic directly into logic, we would be able to prove expressivity bounds in a nice and simple way That program was carried out in [18], and I shall survey the results below. One problem with [18] is that it inherited too much unnecessary machinery from its predecessors [24,8,25,22,23] one had to deal with languages for complex objects and apply conservativity results to get down to SQL; logics were in nitary to start with, ....

[Article contains additional citation context not shown here]

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. Journal of the ACM, 48 (2001), 880-907.

....de nable in relational calculus is local. It is rather pleasant that locality can be established for the language that is essentially plain SQL. SQL, the dominant language of commercial databases, adds two main features to the relational calculus: grouping and aggregation. In a number of papers [23, 25, 18] we studied a theoretical reconstruction of plain SQL and its expressive power. Our approach was as follows. To model the grouping feature, we considered a nested relational language, as in [3] If one deals with the usual queries from at relational databases to at relational databases, then ....

....the standard aggregate functions such as AVG, TOTAL, COUNT. Then [25] established locality of relational queries in such a language (that is, queries that do not have values of the numerical type in their input and output, but can use them for intermediate steps of the computation) Furthermore, [18] showed (a stronger form of) locality under the assumption that every arithmetic function and every aggregate operator is present in the language. 5 Another very useful result is that queries de nable in relational calculus in the presence of a built in order relation are local, provided they ....

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L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. Journal of the ACM, to appear. Extended abstract in LICS'99, pages 35-44.

....that starts with the position ( a; b) i.e. each f i sends a to b. This condition implies that for a FO (or FO(Q u ) formula ( x) of quantifier rank n, A j= a) iff B j= b) 12] We extend this to L 1 (C) Note that the lemma below follows from a slightly more general result of [14]. Lemma 2 Let (x 1 ; xm ) be a L 1 (C) formula in the language of oe, with all free variables of the first sort. Let (A; a) j bij rk( B; b) where a 2 A m ; b 2 B m . Then A j= a) iff B j= b) QED The following is the key lemma, which is proved by a technique ....

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, pages 35--44. Full version to appear in J. ACM.

....of these tuples are isomorphic; again d is determined by k. It was shown that Hanf s theorem is strictly stronger than Gaifman s, and that both apply to a variety of logics that extend FO with counting mechanisms and limited in nitary connectives [Grohe and Schwentick 2000; Hella et al. 1999a; Hella et al. 1999b; Libkin 2000; Nurmonen 1996] These results found applications in descriptive complexity and database theory. Since the complexity class TC 0 (with the appropriate notion of uniformity) can be captured by FO with counting quanti ers [Barrington et al. 1990] locality can be used to prove lower bounds for ....

....of this logic, which is one sorted, and uses arbitrary unary generalized quanti ers [Hella 1996; Hella et al. 1999a] however, expressing counting properties with unary quanti ers is often quite awkward, and thus we chose to use a two sorted version with counting terms here. Fact 2.7. See [Hella et al. 1999b; Libkin 2000]. Queries expressed by L 1 (C) formulae without free variables of the second sort are Hanf local and Gaifman local. Gaifman locality of L 1 (C) was proved by a simple direct argument in [Libkin 2000] Hanf locality was shown in [Hella et al. 1999b] using bijective EhrenfeuctFra ss e ....

Hella, L., Libkin, L., Nurmonen, J. and Wong, L. 1999b. Logics with aggregate operators. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science (LICS'99, Trento, Italy, July), IEEE Press, Piscataway, NJ, 35-44. Full version to appear in J. ACM.

....the expressiveness of first order logic (FO) over finite structures, since relational algebra has the same power as FO. So perhaps if we could put aggregates and arithmetic directly into logic, we would be able to prove expressivity bounds in a nice and simple way That program was carried out in [18], and I ll survey the results below. One problem with [18] is that it inherited too much unnecessary machinery from its predecessors [23, 8, 24, 21, 22] one had to deal with languages for complex objects and apply conservativity results to get down to SQL; logics were infinitary to start with, ....

....since relational algebra has the same power as FO. So perhaps if we could put aggregates and arithmetic directly into logic, we would be able to prove expressivity bounds in a nice and simple way That program was carried out in [18] and I ll survey the results below. One problem with [18] is that it inherited too much unnecessary machinery from its predecessors [23, 8, 24, 21, 22] one had to deal with languages for complex objects and apply conservativity results to get down to SQL; logics were infinitary to start with, although infinitary connectives were not necessary to ....

[Article contains additional citation context not shown here]

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, pages 35--44.

....schema oe) are shown in Figure 1. We adopt the convention of omitting the explicit type superscripts in these expressions whenever they can be inferred from the con6 text. The complete definitions of the concept of a free variable of an expression and the semantics of the language can be found in [15]; here we explain the main features. The set of free variables of an expression e is defined by induction on the structure of e and we often write e(x 1 ; xn ) to explicitly indicate that x 1 , xn are free variables of e. Expressions S fe 1 j x 2 e 2 g, P fe 1 j x 2 e 2 g, and Aggr ....

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. Bell Labs, Technical Memo, 1998.

.... considered a sequence of more and more powerful logics, each of which was proved to be local, starting with FO with counting quantifiers, and ending with a logic that permits arbitrary predicates on natural numbers, a limited form of infinitary connectives [Li00] and even aggregate functions [HLNW99]. However, it was not clear how much one can add to these logics and still preserve its locality. Our goal, therefore, is to give a precise characterization of local logics. Note that the abstract notions of locality were previously characterized on finite structures of bounded valence (e.g. for ....

....TBD, TBD TBD. 6 Delta Leonid Libkin Fact 2.7. see [HLNW99; Li00] Queries expressed by L 1 (C) formulae without free variables of the second sort are Hanf local and Gaifman local. 2 Gaifman locality of L 1 (C) was proved by a simple direct argument in [Li00] Hanf locality was shown in [HLNW99] using bijective Ehrenfeuct Fraiss e games of [He96] The game is played by two players, called the spoiler and the duplicator, on two structures A; B 2 STRUCT[oe] For the n round game, in each round i = 1; n, the duplicator selects a bijection f i : A B, and the spoiler selects a point ....

[Article contains additional citation context not shown here]

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, pages 35--44.

....logics, and described outputs of local queries. We now briefly discuss applications and new directions. First, one can add counting over the numerical domain to L 1 (C) This has to be done with care, as the value of a counting term may then be infinite. Such an extension was given recently in [18]. Moreover, 18] showed that L 1 (C) can be extended by means of aggregate functions. An example would be a formula (x; j) j P (y;i) x; y; i) such that (x; j) holds if j = P (y;i) x;y;i) holds i. This is a typical operation in database query languages [1] It was shown in [18] that ....

....outputs of local queries. We now briefly discuss applications and new directions. First, one can add counting over the numerical domain to L 1 (C) This has to be done with care, as the value of a counting term may then be infinite. Such an extension was given recently in [18] Moreover, [18] showed that L 1 (C) can be extended by means of aggregate functions. An example would be a formula (x; j) j P (y;i) x; y; i) such that (x; j) holds if j = P (y;i) x;y;i) holds i. This is a typical operation in database query languages [1] It was shown in [18] that L 1 (C) has ....

[Article contains additional citation context not shown here]

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In Proc. 14th IEEE Symp. on Logic in Computer Science (LICS'99), Trento, Italy, July 1999, pages 35--44.

.... in the n move bijective game that starts with the position ( a; b) This condition implies that for a FO (or FO(Qu ) formula ( x) of quantifier rank n, A j= a) iff B j= b) 16] We extend this to L 1 (C) Note that the lemma below follows from a slightly more general result of [18], but it also has a simple direct proof, see [25] Lemma 1 Let (x 1 ; xm ) be a L 1 (C) formula in the language of oe, with all free variables of the first sort. Let (A; a) j bij rk( B; b) where a 2 A m ; b 2 B m . Then A j= a) iff B j= b) 2 The following is the ....

L. Hella, L. Libkin , J. Nurmonen and L. Wong. Logics with aggregate operators. This volume.

....schema ) are shown in Figure 1. We adopt the convention of omitting the explicit type superscripts in these expressions whenever they can be inferred from the con6 text. The complete de nitions of the concept of a free variable of an expression and the semantics of the language can be found in [15]; here we explain the main features. The set of free variables of an expression e is de ned by induction on the structure of e and we often write e(x 1 ; x n ) to explicitly indicate that x 1 , x n are free variables of e. Expressions S fe 1 j x 2 e 2 g, P fe 1 j x 2 e 2 g, and ....

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. Bell Labs, Technical Memo, 1998.

.... in the n move bijective game that starts with the position ( a; b) This condition implies that for a FO (or FO(Q u ) formula ( x) of quantifier rank n, A j= a) iff B j= b) 13] We extend this to L 1 (C) Note that the lemma below follows from a slightly more general result of [15]. 19 Lemma 2 Let (x 1 ; xm ) be a L 1 (C) formula in the language of oe, with all free variables of the first sort. Let (A; a) j bij rk( B; b) where a 2 A m ; b 2 B m . Then A j= a) iff B j= b) 2 The following is the key lemma, which is proved by a ....

L. Hella, L. Libkin , J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, to appear.

....show how to evaluate presumably intractable queries) the incremental techniques can well be used in practice. However, proving that certain queries cannot be incrementally evaluated in SQL within some complexity bounds appears beyond reach, as doing so would separate some complexity classes, cf. [15]. Organization In the next section, we give preliminary material, such as a theoretical language SQL capturing the grouping and aggregation features of SQL, the definition of incremental evaluation system IES, a nested relational language, and the relationship between the incremental evaluation ....

....at the first level, and explain the relationship between the classes in both IES(SQL) k and IES(SQL ) k hierarchies. 2 Preliminaries Languages SQL and NRC A functional style language that captures the essential features of SQL (grouping and aggregation) has been studied in a number of papers [18, 5, 15]. While the syntax slightly varies, choosing any particular one will not affect our results, as the expressive power is the same. Here we work with the version presented in [15] The language is defined as a suitable restriction of a nested language. The type system is given by Base : b j Q rt ....

[Article contains additional citation context not shown here]

L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS'99, to appear.

No context found.

L. Hella, L. Libkin, J. Nurmonen, and L. Wong. Logics with aggregate operators. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science, pages 35-44, 1999.

No context found.

L. Hella, L. Libkin, J. Nurmonen, and L. Wong. Logics with aggregate operators. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science, pages 35-44, 1999.

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