G. Dong, L. Libkin, L. Wong. Local properties of query languages, Tech. Memo, Bell Labs, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the isomorphism types in Hanf s condition by something weaker and to get rid of the disjoint r neighbourhoods constraint in Gaifman s condition. For very interesting recent results concerning Hanf s and Gaifman s theorems from a different point of view see the papers of Libkin and Dong et al. [Lib97, DLW97]. It is easy to see that the straightforward attempt to replace the isomorphism type of a sphere S in Hanf s condition by its Hintikka type for some d (i.e. by the set of formulas of quantifier depth at most d that hold in S) does not work. A counterexample is given by the set of clique graphs. ....

G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proc. Int. Conf. on Database Theory, LNCS, pages 140--154. Springer-Verlag, 1997.

....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 ....

G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proceedings of the 5th International Conference on Database Theory, volume 1186 of Lecture Notes in Computer Science, pages 140--154. Springer-Verlag, 1997.

....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 ....

G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proceedings of the 5th International Conference on Database Theory, volume 1186 of Lecture Notes in Computer Science, pages 140--154. Springer-Verlag, 1997.

....taking the big union of e 1 [o=x] over each o in the set e 2 . NRC (suitably extended) is implemented by the NRC Module of Kleisli and is the abstract counterpart of CPL, a la Wadler s equations relating monads and comprehensions[30] The expressive power of NRC and its extensions are studied in [28, 15, 19, 9, 29]. The impact of these and other theoretical results on the design of CPL and Kleisli is that CPL adopts NRC(Q , Delta, Gamma, Xi, P , Q ; as its core, while allowing for full fledged recursion and other operators to be imported easily as needed into the system. NRC(Q , ....

G. Dong et al. Local properties of query languages. In Proc. 6th International Conference on Database Theory, pages 140--154, 1997. 23

....expressed by a first order formula. For example, to decide whether there is a path between two vertices of a graph it clearly does not suffice to look at small neighborhoods of these vertices. Hence by locality, s t connectivity is not expressible in first order logic. Recently, Libkin and others [2, 9 13] systematically started to explore locality as a tool for proving inexpressibility results. The ultimate goal of this line of research would be to separate complexity classes, in particular, to separate the class TC 0 from LOGSPACE. 1 A result of Hella and the first author (unpublished) showing ....

G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proceedings of the 5th International Conference on Database Theory, volume 1186 of Lecture Notes in Computer Science, pages 140--154. Springer-Verlag, 1997.

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G. Dong, L. Libkin, L. Wong. Local properties of query languages, Tech. Memo, Bell Labs, 1995.

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G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proceedings of 6th International Conference on Database Theory, pages 140--154, Delphi, Greece, January 1997.

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G. Dong, L. Libkin, L. Wong. Local properties of query languages. In Proceedings of International Conference on Database Theory (ICDT'97), Springer LNCS vol. 1186, pages 141--154.

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G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Theoretical Computer Science, to appear. Extended abstract in ICDT'97.

....adding arithmetic operations. In [9] it is shown that the transitive closure of a graph is not expressible in an aggregate extension of first order logic if DLOGSPACE 6= NLOGSPACE. In [33] this is proved without any complexity assumptions; a generalization of [33] to many other queries is given in [11]. One problem with the proofs of [33; 11] is that they are very syntactic they work for a particular presentation of the language, and rely heavily on complicated syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [30] which ....

....the transitive closure query cannot be expressible in any logic that has the BNDP. The relationship between the notions of locality we introduced is the following, when one deals with one sorted finite structures: Proposition 4.8. a) see [21] Every Hanf local formula is Gaifman local. b) see [11]) Every query defined by a Gaifman local formula has the BNDP. 2 These results are not affected by the transfer to pure two sorted structures. 4.3 Locality of In [37] it was proved that the extension of first order logic by all unary generalized quantifiers is Hanf local. The proof was based ....

G. Dong, L. Libkin and L. Wong. Local properties of query languages. Theoretical Computer Science, 239 (2000), 277--308.

....of transitive closure in a way that was very unlikely to extend to other queries. It relied on a complicated syntactic rewriting that would not work even for a slightly di erent language. And the proof would not work if one added more aggregate functions. The rst limitation was addressed in [8] where a certain general property of queries expressible in SQL was established. However, the other two problems not only remained, but were exacerbated: the rewriting of queries became particularly unpleasant. In an attempt to remedy this, 22] gave an indirect encoding of a fragment of SQL into ....

....that FO(C) cannot express. The encoding showed that for any query Q in SQL, there exists an FO(C) query Q that shares some nice properties with Q. Then [22] established some properties of FO(C) queries and transferred them to that fragment of SQL. The proof was much cleaner than the proofs of [24,8], at the expense of a less expressive language. After that, 25] showed that the coding technique can be extended to SQL with rational numbers and the usual arithmetic operations. The price to pay was the readability of the proof the encoding part became very unpleasant. 7 That was a good time ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin and L. Wong. Local properties of query languages. TCS 239 (2000), 277-308.

....adding arithmetic operations. In [9] it is shown that the transitive closure of a graph is not expressible in an aggregate extension of first order logic if DLOGSPACE 6= NLOGSPACE. In [33] this is proved without any complexity assumptions; a generalization of [33] to many other queries is given in [11]. One problem with the proofs of [33; 11] is that they are very syntactic they work for a particular presentation of the language, and rely heavily on complicated syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [30] which ....

....in any logic that has the BNDP. The relationship between the notions of locality we introduced is the following, when one deals Logics with Aggregate Operators Delta 15 with one sorted finite structures: Proposition 4.8. a) see [21] Every Hanf local formula is Gaifman local. b) see [11]) Every query defined by a Gaifman local formula has the BNDP. 2 These results are not affected by the transfer to pure two sorted structures. 4.3 Locality of LC In [37] it was proved that the extension of first order logic by all unary generalized quantifiers is Hanf local. The proof was based ....

G. Dong, L. Libkin and L. Wong. Local properties of query languages. Theoretical Computer Science, 239 (2000), 277--308.

....gives us the usual notions of in and out degree. By deg set (A) we mean the set of all degree j (R A i ; a) realized in A, and deg count(A) stands for the cardinality of deg set (A) We use the notation STRUCT k [oe] for fA 2 STRUCT[oe] j deg set(A) f0; 1; kgg. Definition 2 (see [24,5,21]) An m ary query Q, m 1, is said to have the bounded number of degrees property 2 , or BNDP, if there exists a function fQ : N N such that deg count(Q(A) fQ (k) for every A 2 STRUCT k [oe] QED The BNDP is very easy to use for proving expressivity bounds [24] For example, it is very ....

....from Gaifman s theorem [10] that every FO definable query is local. Moreover, if Q is definable by a formula ( x) then lr(Q) 7 qr( Gamma 1) 2. It was shown in [21,22] that every FO(Q u ) FO(C) and L 1 (C) definable query is local. Furthermore, lr(Q) 2 rk( 22] Fact 1 (see [5]) Every local query has the bounded number of degrees property. QED Thus, without auxiliary relations, queries such as transitive closure cannot be expressed in FO(C) and even in L 1 (C) Proviso: When we deal with queries in L C and (L C)w , which are defined on structures (A; A 0 ) A 0 ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin and L. Wong. Local properties of query languages. TCS, 239 (2000), 277--308.

....logics coming very close to capturing TC 0 [Etessami 1995; Libkin and Wong 1998] In database theory, logics with counting mechanisms model aggregate functions commonly found in commercial query languages. Thus, locality was used to prove expressivity bounds for query languages with aggregation [Dong et al. 2000; Hella et al. 1999b] For applications to automata, see [Schwentick and Barthelmann 1998] The above mentioned papers considered a sequence of more and more powerful logics, each of which was proved to be local, starting with FO with counting quanti ers, and ending with a logic that permits ....

....Q(A) A query Q is de nable in a logic L if there exists an L formula (x 1 ; xm ) such that Q(A) hA; f a j A j= a)gi. If m = 0, then Q is naturally associated with a subclass of STRUCT[ and de nability means de nability by a sentence of L. De nition 2. 1 (Gaifman Locality) See [Dong et al. 2000; Hella et al. 1999a] An m ary query Q, m 1, is called Gaifman local if there exists a number r 0 ACM Transactions on Computational Logic, Vol. 2, No. 1, January 2001. 4 Leonid Libkin such that, for any structure A and any a; b 2 A m a A r b implies a 2 Q(A) i b 2 ....

Dong, G., Libkin, L., and Wong, L 2000. Local properties of query languages. Theoretical Computer Science 239, 1, 277-308.

....of transitive closure in a way that was very unlikely to extend to other queries. It relied on a complicated syntactic rewriting that wouldn t work even for a slightly different language. And the proof wouldn t work if one added more aggregate functions. The first limitation was addressed in [8] where a certain general property of queries expressible in SQL was established. However, the other two problems not only remained, but were exacerbated: the rewriting of queries became particularly unpleasant. In an attempt to remedy this, 21] gave an indirect encoding of a fragment of SQL into ....

....that FO(C) cannot express. The encoding showed that for any query Q in SQL, there exists a FO(C) query Q 0 that shares some nice properties with Q. Then [21] established some properties of FO(C) queries and transferred them to that fragment of SQL. The proof was much cleaner than the proofs of [23, 8], at the expense of a less expressive language. After that, 24] showed that the coding technique can be extended to SQL with rational numbers and the usual arithmetic operations. The price to pay was the readability of the proof the encoding part became very unpleasant. That was a good time ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin and L. Wong. Local properties of query languages. TCS 239 (2000), 277--308. Extended abstract in ICDT'97.

....arithmetic operations. In [4] it is shown that the transitive closure of a graph is not expressible in the aggregate extension of first order logic if DLOGSPACE 6= NLOGSPACE. In [24] this is proved without any complexity assumptions; a generalization of [24] to many other queries is given in [5]. One problem with the proofs of [24, 5] is that they are very syntactic they work for a particular presentation of the language, and rely heavily on complicated syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [21] which ....

....it is shown that the transitive closure of a graph is not expressible in the aggregate extension of first order logic if DLOGSPACE 6= NLOGSPACE. In [24] this is proved without any complexity assumptions; a generalization of [24] to many other queries is given in [5] One problem with the proofs of [24, 5] is that they are very syntactic they work for a particular presentation of the language, and rely heavily on complicated syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [21] which considered a sublanguage that only ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin and L. Wong. Local properties of query languages. TCS, to appear. Extended abstract in ICDT'97, pages 140--154.

No context found.

G. Dong, L. Libkin, L. Wong. Local properties of query languages. Proc. Int. Conf. on Database Theory, Springer LNCS 1186, 1997, pages 140--154.

....transitive closure cannot be defined by FO(C) in the presence of auxiliary relations, whose degrees are bounded by a fixed constant k. If we talk about directed graphs, by degrees we mean in and out degrees of nodes. A more general definition can be given for arbitrary relational structures, cf. [7]. In the successor relation, every node has in and out degree either 0 or 1. In contrast to these two results, in a linear order on an n element set, all n different (in and out ) degrees from 0 to n Gamma 1 are realized. Thus, in order to move closer to proving expressivity bounds in the ....

....linear orders. We also write L instead of L O 1 for the class of queries definable in L in the presence of built in order relation. 3 Local queries over finite models A number of notions of locality have been introduced in finite model theory in order to prove inexpressibility results, cf. [9, 12, 11, 7, 16]. Here we describe one of these notions, which will serve as a main technical tool. Given a structure A, its Gaifman graph [9, 12] G(A) is defined as hA; Ei where (a; b) is in E iff there is a tuple t 2 R A i for some i such that both a and b are in t. For example, if A is a graph itself, ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin, L. Wong. Local properties of query languages. Proc. Int. Conf. on Database Theory, Springer LNCS 1186, 1997, pages 140--154.

....start by describing our ambient query language. We want this language to be more powerful than the relational calculus in two ways: it will deal with nested relations, and will use aggregate functions. There are many choices for such a language. We use the language similar to those considered in [5, 15, 17, 8]. These languages have been extensively studied and they are easier to work with than most other nested formalisms. However, we would 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 ....

....so that A n = g(I o ; u; R o ) f(f(x; y) j (x; u) 2 I o ; y; v) 2 I o g; I o ; u; R o ) Notice that A o does not appear in the input to g. Now it can be shown that this function g is not definable in NRC aggr this follows from the bounded degree property of NRC aggr [8] which says that on inputs of small degree, any NRC aggr query can only produce outputs that realize a small (not depending on the input) number of distinct degrees, provided those outputs do not contain numbers. Consequently, f cannot be defined in NRC aggr , and thus IES(SQL) 1 cannot ....

G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proceedings of 6th International Conference on Database Theory, pages 140--154, Delphi, Greece, January 1997.

....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 this the bounded degree property. It was generalized to queries on arbitrary finite structures by Dong, Wong and the author [10]. At a more intuitive level, the weakness of first order logic is often attributed to its inability to count (e.g, parity of cardinality is not definable) and lacking a mechanism for doing recursion (e.g. transitive closure is not definable) Usually, the proofs of inexpressibility of ....

.... completely trivial [7] Proofs of applicability of Hanf s technique are usually not very hard, see [15, 13, 26, 27] Further down the road one has Gaifman s locality theorem, whose proof is harder than that of Hanf s technique, but which leads to simpler and 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 ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin, L. Wong. Local properties of query languages. Proc. Int. Conf. on Database Theory, Springer LNCS 1186, 1997, pages 140--154.

No context found.

G. Dong, L. Libkin and L. Wong. Local properties of query languages. Theoretical Computer Science, 239 (1) (2000), ???--???.

....forms of locality, based on Hanf s and Gaifman s conditions, and we show that in both cases the maximum radii are the same for all the counting logics listed above. In Section 6, we consider open local formulae as queries that map finite structures to finite structures. Extending a result from [7], we prove a bound on the number of different degrees realized in the output of a local query, and apply it to counting logics, thereby connecting this measure of complexity of the output with the syntactic parameters of a query. In Section 7, we prove analogs of Gaifman s theorem [11] for FO ....

....gives us the usual notions of in and out degree. By deg set(A) we mean the set of all degrees realized in A, and deg(A) stands for the cardinality of deg set(A) We use the notation STRUCT k [oe] for fA 2 STRUCT[oe] j deg set(A) f0; 1; kgg. Definition 2. 6 (Bounded Degree Property) see [24, 7, 22]) A query q, that is, a function that maps A 2 STRUCT[oe] to an m ary relation on A, m 1, is said to have the bounded degree property, or BDP, if there exists a function f q : N N such that deg(q(A) f q (k) for every A 2 STRUCT k [oe] 2 The intuition is that if A locally looks simple, then ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin and L. Wong. Local properties of query languages. ICDT'97, pages 140--154.

....we shall show (as a corollary of the main result) that the answer to the above question is negative. To prove the main result, we exploit the locality techniques in finite model theory. Originated in the work by Hanf [15] and Gaifman [10] they were recently a subject of renewed attention [5, 9, 13, 26, 23, 24, 28, 34]. The BNDP is typically proved by showing that a logic satisfies an analog of either Hanf s or Gaifman s theorem [23] However, those fail for L 1 (C) in the presence of several classes of preorders. Nevertheless, we prove a statement, weaker than Gaifman s theorem, for counting logics in the ....

....directed graphs, this gives us the usual notions of in and out degree. By deg set(A) we mean the set of all degrees realized in A, and deg count(A) stands for the cardinality of deg set(A) We use the notation STRUCT k [oe] for fA 2 STRUCT[oe] j deg set(A) f0; 1; kgg. Definition 2 (see [26, 5, 23]) An m ary query Q, m 1, is said to have the bounded number of degrees property 1 , or BNDP, if there exists a function fQ : N N such that deg count(Q(A) fQ (k) for every A 2 STRUCT k [oe] 2 The BNDP is very easy to use for proving expressivity bounds [26] For example, it is very easy to ....

[Article contains additional citation context not shown here]

G. Dong, L. Libkin and L. Wong. Local properties of query languages. TCS, to appear. Extended abstract in ICDT'97, pages 140--154.

No context found.

G. Dong, L. Libkin, and L. Wong. Local properties of query languages. In Proceedings of the 5th International Conference on Database Theory, volume 1186 of Lecture Notes in Computer Science, pages 140-154. Springer-Verlag, 1997.

No context found.

G. Dong, L. Libkin and L. Wong. Local Properties of Query Languages. Proceedings of ICDT'97. Delphi, Greece 1997.

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