J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....extensions of FO can be embedded into it. We also relied on bijective games to prove the main result. However, bijective games characterize expressiveness of a logic which defines all queries on ordered finite structures. Thus, in the ordered case one cannot use the generic techniques from [12,21,22,26] that apply to a variety of counting logics. It was shown in [8] that if there is a proof of inexpressibility of some property in FO(C) then there must be a proof of that based on the counting games of [17] The counting game is weaker than the bijective game; on the other hand, it does not ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....Such a sentence fl clearly distinguishes C 1 n from C 2 n . This finishes the proof for the case of (deterministic) transitive closure. Now let q 2 SG tree ; we show that q is not verifiable over L. When L = FO count , we can apply the proof of claim 3 in theorem 2, since, by the result of [30] (see also [15] for each k it is possible to find a number r such that any two structures that realize the same number of all r neighborhoods cannot be distinguished by a FO count sentence of quantifier rank k. For FO c( Omega Gamma0 first define an order relation OE on U that is isomorphic to ....

....to the reviewers for a number of valuable suggestions; we especially thank the reviewer who found an error in an earlier version of the proof of Theorem 3. Thanks to Ron Fagin for clarifying the use of the Ajtai Fagin games and pointing out [16] and to Moshe Vardi for bringing the results of [30] to our attention. ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 755--778.

....Y ) 2 Q, which means, by definition, that the sentence Qx( x : is true in B. Thus Qx( x : cannot, after all, define the property P . Theorem 4 has been successfully used to show that various graph properties (e.g. planarity) are not expressible in terms of unary quantifiers [23]. Such results can be seen as formalizations of the vague intuition that some properties of binary relations cannot be reduced to properties of cardinalities of definable sets. What is really remarkable about Theorem 4 is that it generalizes both to non unary quantifiers and to extensions of ....

J. Nurmonen, On winning strategies with unary quantifiers, Journal of Logic and Computation, 6(6): 779-798, 1996.

.... if there exists r 0 such that, for every A 2 STRUCT[oe] and for every two m ary vectors a, b of elements of A, N r ( a) N r ( b) implies A j= a) iff A j= b) The minimum r for which this holds is called the locality rank of , and is denoted by lr( Based on results of [13, 17], the following was shown in [16] Fact 4 Every FO(C) formula without free second sort variables is local, and every FO(Qu ) formula is local. 2 4 Expressivity bounds for FO(C) and FO(Qu ) in the presence of relations of large degree We start by giving a general technique for proving ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

.... seems to make life easier: for example, it simplifies the proof that connectivity is not monadic Sigma 1 1 [15] quite a lot, compared to [3] but sometimes the combinatorial argument is not 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 ....

....1 ) k (x k ; a k ) iff the oe unary k structure whose ith relation is fa 2 A j A j= i (a; a i )g is in K. Examples of unary quantifiers include the usual 9 and 8, as well as Rescher and Hartig quantifiers. We use FO(Qu ) for FO extended with all unary quantifiers. Fact 2. 6 (see [26, 27]) Every FO(Qu ) sentence is Hanf local. Moreover, hlr( Psi) 3 qr( Psi) 2 Etessami [13] studied first order logic with counting FO COUNT , which is defined as a two sorted logic, with second sort being the sort of natural numbers. On natural numbers one has 1; max; and the BIT predicate ....

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J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

.... 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 in [4] on counting the sizes of ....

....as x, A; a) d (B; b) implies A j= a) iff B j= b) It can be seen [10] that A d B implies A r B for r d; in particular, if A d B, then j A j=jB j. Fact 2.5. a) see [10] If A 3 nB, then A jn B. In particular, A and B agree on all FO sentences of quantifier rank up to n. b) see [27]; bound from [17] Let n 0. Then A (3 n Gamma1 Gamma1) 2 B implies A j bij n B. c) see [17; 24] Every Hanf local formula (without free second sort variables, if one deals with a two sorted logic) is Gaifman local. 2 Next, we review results on outputs of local queries. With each formula (x ....

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 6 (1996), 779--798.

.... 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 in [4] on counting the sizes of ....

....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 ....

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J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....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 ....

....extensions of FO can be embedded into it. We also relied on bijective games to prove the main result. However, bijective games characterize expressiveness of a logic which defines all queries on ordered finite structures. Thus, in the ordered case one cannot use the generic techniques from [16, 23, 24, 28] that apply to a variety of counting logics. It was shown in [8] that if there is a proof of inexpressibility of some property in FO(C) then there must be a proof of that based on the counting games of [20] The counting game is weaker than the bijective game; on the other hand, it does not ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....extensions of FO can be embedded into it. We also relied on bijective games to prove the main result. However, bijective games characterize expressiveness of a logic which defines all queries on ordered finite structures. Thus, in the ordered case one cannot use the generic techniques from [13, 22, 23, 27] that apply to a variety of counting logics. It was shown in [8] that if there is a proof of inexpressibility of some property in FO(C) then there must be a proof of that based on the counting games of [18] The counting game is weaker than the bijective game; on the other hand, it does not ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....with counting, 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 ....

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 6(6):779--798, 1996.

....in the proof of our main result) In the case of Monadic NP there are a lot of nonexpressibility results for more natural graph problems. Fagin [Fag75] showed that connectivity of undirected graphs a Monadic coNP property is not in Monadic NP. This result was extended by a variety of papers [dR87, FSV93, Sch94, Sch95, Nur95] to cases where some kinds of built in relations or generalized quantifiers are allowed. Ajtai and Fagin [AF90] proved that directed reachability, in contrast to undirected reachability, is not in Monadic NP. By means of reductions some of these results can be transferred to other problems ....

J. Nurmonen. On winning strategies with unary quantifiers. Preprint 77, Department of mathematics, University of Helsinki, 1995.

....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 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 ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....total ordering, can not express ORD. Our original proof of this in the conference version of this paper ( Ete95] employed an Ehrenfeucht Fraiss e Game for first order logic with counting ( IL90] Here we show how the result follows from a more general one obtained independently by Nurmonen, [Nur96]. We then show that an appropriately modified version of the EF game is complete for the logic with counting in the sense that it provides a necessary and sufficient condition for expressibility in the logic. We observe that the L complete problem ORD is essentially sparse if we ignore ....

.... Our original lower bound proof (in the conference version of this paper, Ete95] uses a version of the Ehrenfeucht Fraiss e Game (hereafter called EF game) for first order logic with counting quantifiers (see [IL90, CFI92, EI95] A recent more general result obtained independently by Nurmonen [Nur96] implies this lower bound. Here, instead of providing the full EF game proof as before, we will describe Nurmonen s result and show how it implies the lower bound. We also show that an appropriately modified version of the counting game is complete for first order logic with counting in that ....

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J. Nurmonen. On winning strategies with unary quantifiers, 1996. To appear in Journal of Logic and Computation.

.... used as the basis for a systematic analysis of optimization problems from the point of view of descriptive complexity [38, 49, 50] Normal forms for Sigma 1 2 have been studied in [18] The definability of problems in fragments of monadic second order logic has been studied to some depth, e.g. [3, 10, 16, 21, 47, 52, 69], and there are strong links between definabliity in monadic SO on strings and finite automata and formal language theory (see [72] 3 Beyond first order logic: fixed points Whilst SO (resp. Sigma 1 1 ) captures PH (resp. NP) from a complexity theoretic viewpoint we have jumped over ....

J. Nurmonen, On winning strategies with unary quantifiers, J. Logic Computat. 6 (1996) 779--798.

....Qy holds . the duplicator responds by selecting ff(x) j x 2 Sg as his chosen subset of the universe of G 1 . What is not so clear (but is true) is that the converse also holds. 5 Thus, the duplicator has a winning strategy in the r round counting game over G 0 ; G 1 iff G 0 r G 1 . Nurmonen [Nur96] showed the following strengthening of Theorem 3.3. Theorem 3.6: Nur96] Let r be a positive integer. There is a positive integer d such that whenever G 0 and G 1 are d equivalent structures, then G 0 r G 1 . In fact, as shown in [Nur96] we can take d = 3 r in Theorem 3.6. Etessami [Ete95] ....

....chosen subset of the universe of G 1 . What is not so clear (but is true) is that the converse also holds. 5 Thus, the duplicator has a winning strategy in the r round counting game over G 0 ; G 1 iff G 0 r G 1 . Nurmonen [Nur96] showed the following strengthening of Theorem 3.3. Theorem 3. 6: [Nur96] Let r be a positive integer. There is a positive integer d such that whenever G 0 and G 1 are d equivalent structures, then G 0 r G 1 . In fact, as shown in [Nur96] we can take d = 3 r in Theorem 3.6. Etessami [Ete95] considered the following problem of defining an order: in a structure with ....

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J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 1996. To appear.

....cf. 14] These languages adequately model features present in commercial languages (like SQL) but often ignored in database theory. It was shown recently [19, 21] that (a theoretical reconstruction of) such languages can be simulated by FO(C) This, and known expressivity bounds for FO(C) [11, 22, 23, 19] answer some of the open problems [7, 14] about expressivity of such languages without order relation on elements of base types. However, typically base types, such as int and string, come equipped with an ordering, or at least a preorder (for example, elements of type real can be only preordered ....

.... for every A 2 STRUCT[oe] and for every two m ary vectors a, b of elements of A, N r ( a) r N r ( b) implies A j= a) iff A j= b) The minimum r for which this holds is called the locality rank of , and is denoted by lr( Based on earlier results by Hella [15] and Nurmonen [22], the following was shown in [19] Fact 4 Every FO(C) formula without free second sort variables is local, and every FO(Q u ) formula is local. 2 Note that a direct application of locality is of no help if a built in (pre)order relation is avilable, because then d(a; b) 1 for any two elements, ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....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 previous results do not seem to apply to ....

....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 previous results do not seem to apply to ordered structures: indeed, by taking any input and returning the graph of the underlying linear order, we violate the bounded degree property. Thus, it does not hold in NRC aggr ( b ) which is NRC aggr augmented with a linear order on type b. ....

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....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 on bijective Ehrenfeucht Fraiss e games [20] which characterize equivalence of structures with respect to all unary quantifiers. We now use these games to prove the ....

....Theorem 4.11. Over pure two sorted structures, every formula of without free second sort variables is Hanf local. Proof. Let ( x) be a formula of LC , where x are first sort variables and rk( r. Let A and B be finite one sorted oe structures, and let a 2 A . It was proved in [37] that if (A; a) d (B; b) for d = 3 , then the duplicator has a winning strategy in the bijective game b) and hence by Lemma 4.10, A b) Thus is Hanf local. 2 By Theorem 4.2, we get as a consequence the Hanf locality of the full aggregate logic L aggr (All; All) Corollary 4.12. Over ....

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 6 (1996), 779--798.

....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 on bijective Ehrenfeucht Fraiss e games [20] which characterize equivalence of structures with respect to all unary quantifiers. We now use these games to prove the ....

....of LC without free second sort variables is Hanf local. Logics with Aggregate Operators Delta 17 Proof. Let ( x) be a formula of LC , where x are first sort variables and rk( r. Let A and B be finite one sorted oe structures, and let a 2 A n and b 2 B n . It was proved in [37] that if (A; a) d (B; b) for d = 3 r , then the duplicator has a winning strategy in the bijective game BEF r (A; a; B; b) and hence by Lemma 4.10, A j= a) if and only if B j= b) Thus is Hanf local. 2 By Theorem 4.2, we get as a consequence the Hanf locality of the full ....

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 6 (1996), 779--798.

.... 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 expense of a very unpleasant and complicated encoding procedure. This shows that first order logic with counting quantifiers is inadequate as a logic ....

....we deal with expressiveness of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [12] and Gaifman [10] and recently, following [8] they were a subject of renewed attention (see, e.g. [5, 21, 22, 24, 28] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for queries definable by logical formulae. For example, if we deal with queries on graphs, then the number of ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

....we give a general criterion that guarantees elementary equivalence of two finite structures in FO with counting modulo n quantifier D n , where n is a positive integer. The method is based on the work of Hanf [Han65] Especially in the context of finite model theory, this method was considered in [FSV95, Nur96]. Our criterion has been tailored for the logic FO(D n ) It gives an easy combinatorial way to prove undefinability results for FO(D n ) We show that it is enough to count the number of isomorphism types of neighborhoods of a fixed radius of points in our structures. If the result of this ....

....elementary equivalence of these structures up to a certain quantifier rank. 3.1. Theorem. FSV95] Let r and f be positive integers. There are positive integers d and m, such that whenever A and B are (d; m) equivalent structures where every point has degree at most f , then A j r FO B . In [Nur96] we proved that d equivalence is actually enough to guarantee elementary equivalence in FO(Q u ) where Q u is the set of all unary generalized quantifiers. 3.2. Theorem. Nur96] Let r be a positive integer. There is a positive integer d, such that whenever A and B are d equivalent structures, ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 1996. To appear.

....certain criteria, then the structures considered are guaranteed to be elementary equivalent in a certain logic. This technique has been modified for first order logic [17] first order logic with counting modulo quantifiers [39] and first order logic 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. ....

....technique has been modified for first order logic [17] first order logic with counting modulo quantifiers [39] and first order logic 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 ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

.... connectivity is not monadic Sigma 1 1 in [2] and [11] may still involve somewhat nontrivial combinatorial argument (see, e.g. 5] On the other hand, Hanf s theorem being close to game characterization of logics, its extensions have been proved for several extensions of first order logics [9, 24, 26]. Thus, it would be desirable to understand the relationship between various locality notions for first order logic and its extensions. This constitutes the main goal of the paper. We isolate the locality notions underlying Gaifman s and Hanf s theorems, and prove a chain of implications among ....

....shown by Luosto [23] using Ramsey theory) On the other hand, each vectorization of a unary quantifier can be defined in L1 (Q u ) 20] As mentioned in the previous section, several extensions of Theorem 2.4 are known. One such extension can be given for L1 (Q u ) This is because in [24] it was shown that d equivalence, for large enough d, guarantees a winning strategy for the duplicator in the n round bijective Ehrenfeucht Fraiss e game. Theorem 3.3 (see [24, 26] Every L1 (Q u ) sentence Psi is Hanf local. Moreover, hlr( Psi) 3 qr( Psi) 2 In the next section we give ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

.... 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 expense of a very unpleasant and complicated encoding procedure. Thus, first order logic with counting quantifiers is inadequate as a logic for ....

....deal with the expressive power of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [18] and Gaifman [15] and recently, following [13] they were a subject of renewed attention (see, e.g. [10, 29, 30, 32, 36] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for queries definable by logical formulae. See Subsection 4.2. As there are several ways to define locality, ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 6 (1996), 779--798.

....[Han65] introduced a technique based on the number of local isomorphism types to guarantee elementary equivalence of two structures (finite or infinite) with respect to FO. Fagin, Stockmeyer and Vardi [FSV95] formulated this technique in a form which is better suitable for finite model theory. In [Nur96] it was shown that with this technique one gets 1 bijective equivalent finite structures. In the following we explain shortly this technique. Let A be a oe structure and a; b 2 A. Elements a and b are adjacent, if there is some R i 2 oe and a tuple t 2 R i (A ) such that a and b are components ....

....a) Two structures A and B are e equivalent if for every e type , they have exactly the same number of elements with e type . The concept of e equivalence gives an easy way to prove inexpressibility results for FO(Q 1 ) The following theorem is used later in this paper. 2.5 Theorem. [Nur96]) For every positive integer r there is a positive integer e such that whenever A and B are e equivalent structures, then A j B (FO r (Q 1 ) 2.3 Extended vectorization of quantifiers Let oe be a relational vocabulary, where a relation symbol R in oe has arity r. We denote by oe k the ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation, 1996. To appear.

.... 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 expense of a very unpleasant and complicated encoding procedure. This shows that first order logic with counting quantifiers is inadequate as a logic ....

....we deal with expressiveness of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [13] and Gaifman [11] and recently, following [9] they were a subject of renewed attention (see, e.g. [6, 21, 22, 24, 28] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for queries definable by logical formulae. For example, if we deal with queries on graphs, then the number of ....

[Article contains additional citation context not shown here]

J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

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J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

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J. Nurmonen. On winning strategies with unary quantifiers. Journal of Logic and Computation 6(6): 779--798 (1996).

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J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779--798.

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