L. Libkin. On the forms of locality over finite models. In Proceedings of 12th IEEE Symposium on Logic in Computer Science, 204--215, Warsaw, Poland, 1997.  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. ....

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997.

....AC 0 or even TC 0 queries are local in the sense that if two points of a structure of size n have isomorphic neighborhoods of some radius r(n) As long as r(n) n 2 1) they are indistinguishable.This would still be sufficient to separate LOGSPACE from these classes. Libkin also considered [11] stronger notions of locality, the most important being Hanf locality. It remains an open question whether order invariant first order logic is also Hanf local. ....

L. Libkin. On forms of locality over finite models. In Proceedings of the 12th IEEE Symposium on Logic in Computer Science, pages 204--215, 1997.

....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. ....

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997.

....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 di#erent 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. ....

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997.

....addition and multiplication cannot express the transitive closure of S, and therefore fails to capture LOGSPACE. The proof in the journal version of the paper relies on the fact that first order logic with counting quantifiers can only express local properties, which follows from [22] Libkin [18, 19, 20] also considered local properties of a variety of logics involving counting quantifiers. Unfortunately, in the presence of a total ordering, these proof techniques do not apply anymore, since all elements of the structure are directly connected by the ordering. The result of this paper may ....

L. Libkin. On the forms of locality over finite models. Proceedings LICS'97, pages 204--215, 1997.

....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. ....

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997.

....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 Libkin et al. [DLL97, Lib97]. It is easy to see that the straightforward attempt to replace the isomorphism type of a sphere 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 the sphere) does not work. A counterexample is given by the simple ....

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997.

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L. Libkin. On the forms of locality over finite models. In Proceedings of 12th IEEE Symposium on Logic in Computer Science, 204--215, Warsaw, Poland, 1997.

....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 considered a sublanguage that only permits aggregation over columns of natural numbers, returning natural numbers as well (for example, AVG is not allowed) Then [30] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity ....

....syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [30] which considered a sublanguage that only permits aggregation over columns of natural numbers, returning natural numbers as well (for example, AVG is not allowed) Then [30] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [30; 37] The encoding of [30] was extended to aggregation over rational numbers [34] it did allow more aggregates (e.g. AVG) and more arithmetic, at the ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

....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 considered a sublanguage that only permits aggregation over columns of natural numbers, returning natural numbers as well (for example, AVG is not allowed) Then [30] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity ....

....syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [30] which considered a sublanguage that only permits aggregation over columns of natural numbers, returning natural numbers as well (for example, AVG is not allowed) Then [30] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [30; 37] The encoding of [30] was extended to aggregation over rational numbers [34] it did allow more aggregates (e.g. AVG) and more arithmetic, at the ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

....TC 0 from classes such as DLOGSPACE;NLOGSPACE, P, etc. is thus reduced to proving that their complete problems are inexpressible in FO(C) However, it appears that the presence of an order relation is a major obstacle to proving such expressivity bounds for FO(C) Several partial results [8,21] show that there are problems complete for DLOGSPACE that 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. For example, in the graph of a successor ....

....from 0 to n Gamma 1 are realized. Thus, in order to move closer to proving expressivity bounds in the presence of an order relation, one has to at least be able to lift the results from constant degrees to those that depend on the size of the input. A result in this direction was proved in [21] using a definition of moderate degree by Fagin, Stockmeyer and Vardi [9] We say that a class C of graphs (more generally, relational structures) is of moderate degree, if degmax C (n) the maximal in or out degree of an n element graph from C, is at most log o(1) n. That is, for some function ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

....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 first order logic with counting, FO(C) it will be formally defined later) The restriction was to natural numbers, thus excluding aggregates such as AVG. The encoding is bound to be indirect, since SQL is capable of expressing queries that ....

....was to natural numbers, thus excluding aggregates such as AVG. The encoding is bound to be indirect, since SQL is capable of expressing queries 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 ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215.

....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 permits aggregation over columns of natural numbers (for example, AVG is not allowed) Then [21] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [21, 28] The ....

....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 permits aggregation over columns of natural numbers (for example, AVG is not allowed) Then [21] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [21, 28] The encoding of [21] was extended to aggregation over rational numbers [25] it did allow more aggregates (e.g. AVG) and more arithmetic, at the ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215.

....a problem complete for DLOG under first order reductions that cannot be defined by FO(C) in the presence of a successor relation. The result of [10] also shows that dtc, deterministic transitive closure, is not in FO(C) succ, while FO dtc succ captures the class DLOG. This was extended in [16] as follows. Fact 2 ( 16] Deterministic 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 ....

....under first order reductions that cannot be defined by FO(C) in the presence of a successor relation. The result of [10] also shows that dtc, deterministic transitive closure, is not in FO(C) succ, while FO dtc succ captures the class DLOG. This was extended in [16] as follows. Fact 2 ([16]) Deterministic 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 ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215. Full paper "Notions of locality and their logical characterization over finite models" by L. Hella, L. Libkin and J. Nurmonen is available as Bell Labs Technical Memo.

....more importantly, cannot be extended to show that such queries are local. Here we give a much simplified proof that implies Gaifman s locality, not just the bounded degree property. It is based on simulating relational queries in logic with counting. Complete proofs are given in the full version [23]. 2 Notions of locality Notations Unless explicitly stated otherwise, all structures are assumed to be finite. A relational signature oe is a set of relation symbols fR 1 , R l g, with an associated arity function. In what follows, p i ( 0) denotes the arity of R i . We write oe n for oe ....

....of Q. By choosing p large enough so that all numbers produced during evaluating Q and all encodings of tuples are under p(n) we guarantee that the simulation can be done in FO COUNT . It turns out that only coding of tuples of fixed length is needed. For more details, see the full version [23]. Now by Fact 2.7 and Theorem 3.1, Q p is Gaifman local, and thus Q is. This proves the theorem. 2 Corollary 5.4 (see [10] Every relational query in RA aggr (N) has the bounded degree property. Consequently, deterministic) transitive closure is not definable in RA aggr (N) RA aggr (N) ae ....

L. Libkin. On the forms of locality over finite models. Bell Labs, Technical Memo, 1996.

....very recently. For example, 9] used the games of [21] to prove that there is an L complete problem that is not definable in FO(C) this implies that connectivity of finite graphs is not definable in FO(C) In [19] nondefinability of connectivity is shown for FO(Qu ) More bounds were obtained in [24], which used the results of [27] to prove an analog of Gaifman s locality theorem [11] for those logics. Currently, most bounds for extensions of FO with various counting quantifiers can be derived from its local properties, as shown in [17; 24; 27] exceptions include the bound of [5] a result ....

....write a A;B d b. If A = B, we write a A d b. Given tuples a = a 1 ; an ) and b = b 1 ; b m ) and an element c, we write a b for the tuple (a 1 ; an ; b 1 ; b m ) and ac for (a 1 ; an ; c) Definition 2. 2 (Gaifman locality) cf. [24]) A formula ( x; in a twosorted logic is called Gaifman local if there exists a number r 0 such that, for any structure A and any a; b over A, a A r b implies A j= a; iff A j= b; for all ae N. The minimum such r is called the locality rank of , and is ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

....were proved very recently. For example, 9] used the games of [19] to prove that an L complete problem is not definable in FO(C) this implies that connectivity of finite graphs is not definable in FO(C) In [18] nondefinability of connectivity is shown for FO(Qu ) More bounds were obtained in [22], which used the results of [26] to prove an analog of Gaifman s locality theorem [11] for those logics. Currently, most bounds for extensions of FO with various counting quantifiers can be derived from its local properties, as shown in [17, 22, 26] exceptions include the bound of [5] a result ....

....is shown for FO(Qu ) More bounds were obtained in [22] which used the results of [26] to prove an analog of Gaifman s locality theorem [11] for those logics. Currently, most bounds for extensions of FO with various counting quantifiers can be derived from its local properties, as shown in [17, 22, 26]; exceptions include the bound of [5] a result in [4] on counting the sizes of equivalence classes, and the hierarchy result in [14] Locality of a logic gives us a general statement that it lacks a recursion mechanism, much in the same way as 0 1 laws tell us that a logic cannot express ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215.

....e.g. FO and FO(C) 3, 30] Our main goal is to study the impact of auxiliary re1 lations, such as orderings, on the expressive power of counting. The primary motivation comes from complexity theory: while good expressivity bounds exists for counting logics, e.g. FO(C) over unordered structures [8, 23, 24], no nontrivial bounds are known for the ordered case. As we mentioned, FO(C) over ordered structures, captures TC 0 , the class of problems solvable by polynomial size, constant depth threshold circuits, under DLOGTIME uniformity, see [2] This is an important complexity class: problems such ....

....in G are below k, the conclusion is that the number of different degrees in Q(G) is below fQ (k) It is known that over unordered structures FO definable graph queries have the BNDP. This was proved in [26] using Gaifman s locality theorem. More recently, this property was shown to hold in FO(C) [23] and L 1 (C) 24] again, over unordered structures) and very recently it was proved for FO in the ordered case [13] assuming that queries are order invariant. Informally, our main result can be then stated as follows: In the presence of relations which are almosteverywhere linear orders, ....

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L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215.

....extended by all unary generalized quantifiers [38] for the case of finite structures. Proofs of applicability of Hanf s technique typically are not very difficult [17, 15, 38, 40] We will see some examples in Section 4. The above results have motivated a study of general notions of locality [32, 25]. We review this line of work in Section 5. We show that Gaifman s theorem gives rise to two general notions, 2 one for sentences and one for open formulas. We formulate an abstract notion of locality that captures Hanf s condition, and study the relationship between the notions of locality. We ....

....particularly simple to use in proving expressivity bounds. 5.1 Gaifman s locality We start by analyzing Gaifman s theorem. This theorem says that only local neighborhoods are important for elementary equivalence in first order logic. This is captured by the following definition. Definition 5. 1 ([32, 25]) A formula (x 1 ; xm ) is Gaifman local if there exists r 0 such that for every A 2 STRUCT[oe] and for every two m ary vectors a; b 2 A m , tp r ( a) tp r ( b) implies A j= a) if and only if A j= b) The minimum r for which this holds is called the locality ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

....of FO COUNT could also be extended to the case of order independent queries. Indeed, this would imply that deterministic transitive closure is not in TC 0 , which in turn would imply the separation of TC 0 and DLOGSPACE. However, the following counterexample shows that this conjecture, made in [21], is false. Proposition 7.1 There is an order independent query in FO COUNT which does not have the bounded degree property, and hence is not Gaifman local. Proof: Consider structures of the type A = hA; P; Ei, where P A and hA; Ei is a directed graph such that E P 2 is the graph of a ....

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, Warsaw, Poland, June--July 1996, pages 204--215.

....in [10] One problem with the proofs of [32, 10] 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 [29] which considered a sublanguage that only permits aggregation over columns of natural numbers, returning natural numbers as well (for example, AVG is not allowed) Then [29] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity ....

....syntactic rewritings of queries, rather than on the semantic properties of those. An attempt to remedy this was made in [29] which considered a sublanguage that only permits aggregation over columns of natural numbers, returning natural numbers as well (for example, AVG is not allowed) Then [29] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [29, 36] The encoding of [29] was extended to aggregation over rational numbers [33] it did allow more aggregates (e.g. AVG) and more arithmetic, at the ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

.... TC 0 from classes such as DL;NL, P, etc, is thus reduced to proving that their complete problems are inexpressible in FO(C) However, it appears that the presence of an order relation is a major obstacle to proving such expressivity bounds for FO(C) Several partial results (for example, [8, 22]) show that there are problems complete for DL that 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. For example, in the graph of a successor ....

....from 0 to n Gamma 1 are realized. Thus, in order to move closer to proving expressivity bounds in the presence of an order relation, one has to at least be able to lift the results from constant degrees to those that depend on the size of the input. A result in this direction was proved in [22], using a definition on moderate degree by Fagin, Stockemeyer and Vardi [9] We say that a class C of graphs (more generally, relational structures) is of moderate degree, if degmax C (n) the maximal in or out degree of an n element graph from C, is at most log o(1) n. That is, for some ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215.

....output. It turns out that an analog of Theorem 3.1 can be proved for queries of arbitrary arity, with d depending on both locality rank and the arity. For details, see [31] Another problem mentioned in [9] was to develop techniques for proving languages local. One such technique was proposed in [30] which showed that queries in any reasonable logic that satisfies an analog of Hanf s theorem [24, 16] are local. Using this, and results of [25, 38] the paper [30] showed that first order logic extended with unary generalized quantifiers is local. In [31] a technique was presented that allows ....

....see [31] Another problem mentioned in [9] was to develop techniques for proving languages local. One such technique was proposed in [30] which showed that queries in any reasonable logic that satisfies an analog of Hanf s theorem [24, 16] are local. Using this, and results of [25, 38] the paper [30] showed that first order logic extended with unary generalized quantifiers is local. In [31] a technique was presented that allows one to prove locality without a recourse to Hanf s theorem. The same paper showed a version of infinitary logic that can define every numerical property, but ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proceedings of 12th IEEE Symposium on Logic in Computer Science, 204--215, Warsaw, Poland, 1997.

....a problem complete for DLOG under first order reductions that cannot be defined by FO(C) in the presence of a successor relation. 2 Etessami s result also shows that dtc, deterministic transitive closure, is not in FO(C) succ, while FO dtc succ captures the class DLOG. This was extended in [19] as follows. Fact 2 ( 19] Deterministic transitive closure cannot be defined by FO(C) in the presence of auxiliary relations, whose degrees are bounded by a fixed constant k. 2 If we talk about directed graphs, by degrees we mean in and out degrees of nodes. A more general definition can be ....

....DLOG under first order reductions that cannot be defined by FO(C) in the presence of a successor relation. 2 Etessami s result also shows that dtc, deterministic transitive closure, is not in FO(C) succ, while FO dtc succ captures the class DLOG. This was extended in [19] as follows. Fact 2 ([19]) Deterministic transitive closure cannot be defined by FO(C) in the presence of auxiliary relations, whose degrees are bounded by a fixed constant k. 2 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 ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In Proc. 12th IEEE Symp. on Logic in Computer Science, 1997, pages 204--215. Full paper "Notions of locality and their logical characterization over finite models" by L. Hella, L. Libkin and J. Nurmonen is availabe as Bell Labs Technical Memo.

....appears to be no commonality between Gaifman s proof of locality for first order [16] and our proof of (restricted) locality of NRC aggr . We also believe that this restriction can be eliminated. Conjecture 8.1 Every relational query in NRC aggr is local. A step in this direction was made in [24] which proved that a sublanguage of NRC aggr obtained by replacing rational arithmetic with natural arithmetic does have the property that every relational query is local. It was also shown in [24] how to use the results in [14, 30, 31] similar to the proof of Hanf s lemma [15] for extensions of ....

....Conjecture 8.1 Every relational query in NRC aggr is local. A step in this direction was made in [24] which proved that a sublanguage of NRC aggr obtained by replacing rational arithmetic with natural arithmetic does have the property that every relational query is local. It was also shown in [24] how to use the results in [14, 30, 31] similar to the proof of Hanf s lemma [15] for extensions of first order logic to show that they satisfy an analog of Gaifman s theorem. These extensions include first order logic with counting [22] and first order logic with unary quantifiers [21, 30] The ....

L. Libkin. On the forms of locality over finite models. In LICS'97, to appear.

....in [6] One problem with the proofs of [24, 6] 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 permits aggregation over columns of natural numbers (for example, AVG is not allowed) Then [21] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [21, 28] The ....

....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 permits aggregation over columns of natural numbers (for example, AVG is not allowed) Then [21] gave a somewhat complicated encoding of the language in first order logic with counting quantifiers, for which expressivity bounds are known [21, 28] The encoding of [21] was extended to aggregation over rational numbers [25] it did allow more aggregates (e.g. AVG) and more arithmetic, at the ....

[Article contains additional citation context not shown here]

L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204--215.

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L. Libkin. On forms of locality over finite models. In Proceedings of the 12th IEEE Symposium on Logic in Computer Science, pages 204--215, 1997.

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