W. P. Hanf. Model-theoretic methods in the study of elementary logic. In J. W. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models. NorthHolland, Amsterdam, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....z: 9y:R 1 (x; y) R 2 (y; z) z) c 10 6) 4. AGGREGATE LOGIC: EXPRESSIVE POWER In this section we 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 [19] and Gaifman [16] and recently, following [14] they were a subject of renewed attention (see, e.g. 11; 30; 31; 33; 37] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

....of SQL queries What kind of general statement can one provide that would give us strong evidence that SQL cannot express recursive queries For that purpose, we shall use the locality of queries. Locality was the basis of a number of tools for proving expressivity bounds of rst order logic [15,13,11], and it was recently studied on its own and applied to more expressive logics [17,23] The general idea of this notion is that a query can only look at a small portion of its input. If the input is a graph, small means a neighborhood of a xed radius. For example, Fig. 1 shows that ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132-145.

....[LW98] Despite its importance as an ingredient for more expressive logics, it is well known that the expressive power of FO logic is rather limited. It can only express properties that depend on the local appearance of a structure. This intuition has been formalized in different ways by Hanf [Han65] and Gaifman [Gai82] Hanf showed that, for every first order formula , there is an r such that whether holds in a structure A ( A j= only depends on the multiset of isomorphism types of all r spheres in A. Here an r sphere is a 1365 8050 c 1999 Maison de l Informatique et des Mathematiques ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132--145. North Holland, 1965.

....duplicator wins. Several such conditions have been identified. Among these are (a) a formulation of secondorder Ehrenfeucht Fraiss e games by Ajtai and Fagin [AF90] for which it seems easier to prove that the duplicator has a winning strategy; b) a sufficient condition (due essentially to Hanf [Han65]) for the duplicator to have a winning strategy; and (c) the idea of playing EhrenfeuchtFra iss e games over random structures. Techniques (a) and (c) were used by Ajtai and Fagin [AF90] and all three techniques were used by Fagin, Stockmeyer, and Vardi [FSV95] Thus a library of tools seems ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132--145. North Holland, 1965.

....structures G 0 and G 1 are before the spoiler s coloring step occurs. 6 Hanf s technique In this section, we state a result from [FSV95] which provides a simple but very useful sufficient condition for guaranteeing that A r B for two structures A; B. The proof is based on a technique of Hanf [Han65] Let A be an L structure, where L = fP 1 ; P s g, and where R i is the interpretation in A of the relation symbol P i , for 1 i s. Let a and b be two points in (the universe of) A. We 13 say that a and b are adjacent (in A) if either a = b or there is some R i and some tuple t such ....

...., or else both have at least m points with d type . Intuitively, A and B are (d; m) equivalent if for every d type , they have the same number of points with d type , where we can count only as high as m. The following result is proved in [FSV95] based on the proof of a similar result of Hanf [Han65] This version of Hanf s result from [FSV95] is useful in the context of finite structures, whereas Hanf s original result is not. Theorem 6.1: Let r; f be positive integers. There are positive integers d; m, where d depends only on r, such that whenever A and B are (d; m) equivalent structures ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132--145. North Holland, 1965.

....z: 9y:R 1 (x; y) R 2 (y; z) z) c 10 6) 4. AGGREGATE LOGIC: EXPRESSIVE POWER In this section we 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 [19] and Gaifman [16] and recently, following [14] they were a subject of renewed attention (see, e.g. 11; 30; 31; 33; 37] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

....elements a; b 2 A if there is a relation R 2 and a tuple c 2 R A such that both a and b occur in c. The valence of a structure A, denoted by val(A) is de ned to be the valence of its Gaifman graph, that is, val(A) max jfb j E G(A) a; b)gj a 2 Ag. Using Hanf s Sphere Theorem [Han65], Seese proved the following: Theorem 5 ( See96] For every vocabulary , the following parameterized problem is in FPL: Input: structure A, FO sentence Parameter: val(A) j j) Problem: Decide if A j= A more general approach is based on Gaifman s locality theorem [Gai82] The ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132-145. North Holland, 1965.

....of SQL queries What kind of general statement can one provide that would give us strong evidence that SQL cannot express recursive queries For that purpose, we shall use the locality of queries. Locality was the basis of a number of tools for proving expressivity bounds of first order logic [15, 13, 11], and it was recently studied on its own and applied to more expressive logics [17, 22] b : oe : r r : a oe Upsilon Sigma Pi Xi Upsilon Sigma Pi Xi Fig. 1. A local formula cannot distinguish (a; b) from (b; a) ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

....As the numerical domain, we choose the set of rational numbers Q, although other domains (e.g. Z; R) can be chosen. The resulting logic L aggr defines every arithmetic operation and every aggregate function. We then show that it has very nice behavior: its formulae satisfy analogs of Hanf s [8, 12] and Gaifman s [10] theorems, meaning that it can only express local properties. In particular, properties such as connectivity of graphs cannot be expressed. We then consider a theoretical language RL aggr , similar to those defined in [3, 24] and explain how it models all the features of ....

.... Sigma z: 9y:R 1 (x; y) R 2 (y; z) z) c 10 6 ) 4 Aggregate logic: Expressive power In this section 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 ....

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

....Several proposals have been made to formalize the notion of locality. Gaifman [16] proved that every firstorder formula is equivalent to a local one, in the sense that only a small part of the input is relevant for evaluating a query. Fagin, Stockmeyer and Vardi [15] modifying a result by Hanf [19] for the finite case, proved that if a certain criterion relating the numbers of small neighborhoods in two structures holds, then these structures agree on sentences whose quantifier rank is determined by the size of those neighborhoods. The author and Wong [25] showed that if first order query ....

....and describe the basic notions of locality. We start by reviewing Gaifman s theorem, and note that it leads to two notions, called Gaifman s locality and strong Gaifman s locality. The result of [16] then says that first order logic has both. We review the modification of Hanf s technique [19] for the finite case [15] and define the notion of Hanf s locality property. We review the bounded degree property of [10, 25] which is implied by Gaifman s locality [10] In Section 3, we show that Hanf s locality implies Gaifman s locality and the bounded degree property. We use these results ....

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

....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 nontrivial counting properties. One way in which locality theorems are applied is the following. First, a form of locality based on Hanf s condition (see [10, 15]) is shown for a logic; this form is closely tied to a gamecharacterization of the logic. Then results of [17, 22] show that the logic also satisfies Gaifman s locality condition [11] and the bounded degree property [24] which are much easier to apply to prove expressivity bounds. However, no ....

....that no power is lost due to this restriction. Proving Locality. How does one prove that formulae in a counting logic (e.g. FO(C) FO(Qu ) L 1 (C) only express local properties, as shown in Figure 1 Currently, with the exception of FO, such results are established via Hanf s criterion [10, 15] that relates the number of isomorphism types of small neighborhoods in two structures. This criterion is closely tied to a game characterization of a logic, and may not work if such a characterization does not exist. Also, one needs to adjust the implication results for two sorted logics. Here, ....

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132-- 145.

....answer would imply that the lower bounds of [27] apply to TC 0 . However, 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 ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

....elements for each natural number m. It can be seen that the game of Immerman and Lander is equivalent to the bijective game BP k 1 . Hence for every k, L k 1 (Q 1 ) has the same expressive power as the logic L k 1 (C) see also [KV95] 5 2. 2 Hanf s technique and unary quanti ers Hanf [Han65] introduced a technique based on the number of local isomorphism types to guarantee elementary equivalence of two structures ( nite or in nite) with respect to rst order logic. Fagin, Stockmeyer and Vardi [FSV95] formulated this technique in a form which is better suited for nite model theory. ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132-145. North Holland, 1965.

....quantifier. Our approach works for arbitrary (relational) vocabulary. In Section 3 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 ....

....respect to FO(D n ) Often rather complicated EhrenfeuchtFra iss e type game theoretical methods can be replaced by the combinatorial argument. The method was based on counting the number of isomorphism types of a fixed radius of points. In that sense, this paper can be seen as a continuation of [FSV95, Han65, Nur96]. Inexpressibility results with built in linear order were also considered. We showed that sufficiently large linear orders of modulo n r 1 equal length cannot be distinguished by a sentence of FO(D n ) with quantifier rank at most r. With this observation we showed that the majority language as ....

W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132--145. North Holland, 1965.

....Mathematical Logic General Terms: Languages, Theory Additional Key Words and Phrases: Locality, logic, counting 1. INTRODUCTION It is well known that rst order logic (FO) only expresses local properties. Two best known formal results stating locality of FO are Hanf s and Gaifman s theorems [Hanf 1965; Gaifman 1982] They both found numerous applications in computer science, due to the fact that they are among relatively few results in rst order model theory that apply to both nite and in nite structures. Gaifman s theorem itself works for both nite and in nite structures, while for Hanf s ....

....in A and B. That is, A d B if there exists a bijection f : A B such that N A d (a) N B d (f(a) for every a 2 A. We also write (A; a) d (B; b) if there is a bijection f : A B such that N A d ( ac) N B d ( bf(c) for every c 2 A. De nition 2. 3 (Hanf Locality) See [Hanf 1965; Fagin et al. 1995; Hella et al. 1999a] An m ary query Q, m 0, is called Hanf local if there exist a number d 0 such that for any two structures A; B and any a 2 A m ; b 2 B m , A; a) d (B; b) implies a 2 Q(A) i b 2 Q(B) The minimum d for which this holds is called Hanf ....

[Article contains additional citation context not shown here]

Hanf, W. 1965. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland Publishing Co., Amsterdam, 132-145.

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W. P. Hanf. Model-theoretic methods in the study of elementary logic. In J. W. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models. NorthHolland, Amsterdam, 1965.

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, pages 132--145. North Holland, 1965. 81

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, pages 132--145, 1965.

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al., eds., The Theory of Models, North Holland, 1965, pages 132-145.

No context found.

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, pages 132--145, 1965.

No context found.

W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al., eds., The Theory of Models, North Holland, 1965, pages 132--145.

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

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W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132--145.

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