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 ....

[Article contains additional citation context not shown here]

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 ....

[Article contains additional citation context not shown here]

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, ....

[Article contains additional citation context not shown here]

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.

....of this kind. We consider Gaifman s theorem [18] for first order logic, which shows that every first order formula is equivalent to a local one, in the sense that only a small part of a structure is relevant for evaluating the query given by a formula. We also study modifications of Hanf s result [21]. In this approach one counts the number of isomorphism types of fixed radius neighborhoods of points. If the result of this counting satisfies certain criteria, then the structures considered are guaranteed to be elementary equivalent in a certain logic. This technique has been modified for ....

....cannot define the transitive closure, since (a; b) 2 TRCL(A) but (b; a) 62 TRCL(A) 10 4. 2 Hanf s theorem and its modifications While Gaifman s theorem helps prove expressivity bounds for FO directly, without resorting to establishing a winning strategy for the duplicator 1 , Hanf s theorem [21] and its numerous modifications [17, 25, 38, 39] provide criteria for the existence of a strategy for the duplicator that is based on counting of small neighborhoods in two structures. Hanf s theorem was originally proved for infinite structures. It was observed by Fagin, Stockmeyer and Vardi ....

[Article contains additional citation context not shown here]

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 de nes 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 de ned in [3, 24] and explain how it models all the features of SQL. ....

....(Aggr 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 satis es 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 ....

[Article contains additional citation context not shown here]

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.

....portions of the input. Several proposals have been made to formalize the notion of locality. Gaifman [12] proved that the outcome of a first order definable query depends only on the isomorphism types of neighborhoods of a fixed radius. Fagin, Stockmeyer and Vardi [11] modifying a result by Hanf [16] 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. Libkin and Wong [22] showed that if a first order query ....

....notions of locality. We start by reviewing Gaifman s theorem, and note that it leads to two properties, called the Gaifman locality and the strong Gaifman locality. The result of [12] then says that first order logic has both of these properties. We review the modification of Hanf s technique [16] for the finite case [11] and define the notion of Hanf locality. We review the bounded degree property of [6, 22] which is implied by the Gaifman locality [6] In Section 3 we review the extensions of first order logic we consider in this paper. These are fragments of infinitary logic, logics ....

[Article contains additional citation context not shown here]

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.

....that connectivity of finite graphs is not definable in monadic Sigma 1 1 , while it is easy to see that it is in monadic Pi 1 1 . Many proofs for the same result have been given afterwards. One of them was presented in [6] where a certain kind of Ehrenfeucht Fraiss e game was used. In [8] Hanf gave a condition that guarantees a winning strategy for the duplicator, and in [6] Fagin, Stockmeyer and Vardi gave it in a form, which is more suitable for finite model theory. Furthermore, with this method and with some probabilistic arguments, Fagin, Stockmeyer and Vardi proved that ....

....wins, since the duplicator can not make the first or the third move. Using this game Ajtai and Fagin prove the next result. 3.2. Theorem. A class C of oe structures is Mon Sigma 1 1 if and only if there are c and r such that the spoiler has a winning strategy in AF c;r (C) Hanf proved in [8], that a winning strategy for the duplicator in the game EF r (A ; B ) for finite and infinite structures) is guaranteed by counting d types. In [6] this condition was presented in a form, which is more suitable for finite model theory. 3.3. Lemma. Let r be a positive integer. There is a ....

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

....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 [13, 18] and Gaifman s [15] 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 [5, 32] and explain how it models all the features of ....

....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 [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 ....

[Article contains additional citation context not shown here]

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.

....limited. It can only express properties that depend on the local appearance of a structure. This intuition has been formalized in di#erent ways by 1365 8050 c # 1999 Maison de l Informatique et des Mathematiques Discretes (MIMD) Paris, France 110 Thomas Schwentick and Klaus Barthelmann 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 = # ) only depends on the multiset of isomorphism types of all r spheres in A. Here an r sphere is a substructure of A which is induced by all elements of A that ....

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. 124 Thomas Schwentick and Klaus Barthelmann

....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 say that a and b are adjacent (in A) if either a = b or there is some R i and some tuple t such that ....

...., 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. 14 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 ....

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. 52

....We will refer to this as the multiplicity argument later on. Of course, this does not already give a winning strategy for Duplicator, as A and A 0 might differ in some other way that is useful for Spoiler. The condition of Fagin, Stockmeyer and Vardi is based on a similar method of Hanf [Han65] on infinite structures. It essentially says: If two structures have bounded degree and every isomorphism type appears either in both structures equally often or in both structures very often then Duplicator has a winning strategy. Let A be a finite structure over some signature S containing only ....

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.

.... compensate for the lack of expressive power In our most recent work [Sch97] we have answered this question negatively and thereby sharpened results from [BT87] and [TWW82] The idea was to make use of the insight that with first order logic formulas only local properties can be expressed [Han65, Gai82] a phenomenon normally exploited to study the descriptive complexity in finitemodel theory; see e.g. FSV95] A remaining question is, whether first order specifications are properly more expressive than conditional equational specifications (both without hiding facilities) 12 This ....

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

....Wong [LW97] Despite its importance as an ingredient for more expressive logics, it is wellknown 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 substructure of A which is induced by all elements of A ....

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.

....calculus and algebra cannot express the transitive closure of a graph or the parity test. A large number of tools have been developed for first order logic (or equivalently, the relational calculus) these include Ehrenfeucht Fraiss e games [13, 17] locality [18] 0 1 laws [15] Hanf s technique [16, 24], the bounded degree property [36] etc. We are especially interested in local properties of queries, first introduced by Gaifman [18] These state that the result of a query can be determined by looking at small neighborhoods of its arguments. Expressiveness of database query languages ....

....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 one to prove locality without a recourse to Hanf s theorem. The same paper showed a version 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.

....least m 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, FV k (Q 1 ) has the same expressive power as the logic FV k (C) see also [KV95] 2. 2 Hanf s technique and unary quantifiers Hanf [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] ....

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.

....relation, on the other hand, if distributed carefully over the whole graph, cannot help in testing whether a clique has even size or not. We note that Theorem 4. 1 could also be used in several other proofs, like [5] and [11] But it seems to be, in general, incompatible with the technique of Hanf [12], which is described in [11] and the related technique of [2] We hope that our technique will be useful to obtain other inexpressibility results. Maybe with its help or with the help of some other techniques developed recently (cf. 11, 2] it will be possible to attack the case of existential ....

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.

....) N t (c B ) Although these models have isomorphic t neighborhoods of c, we still know nothing about the other part of each model, which might make A and B look very different in FO. The final step of the proof takes care of this by using a version of Hanf s lemma. Proposition 5 (Hanf [4]) For each signature oe, there is a function f(x) with the following property. For all n, A and B, if there is a bijection h : A B such that for all a 2 A, N f(n) a) N f(n) h(a) with a and h(a) distinguished) then Aj n B. Lemma 4 Let A and B be (2f(n) pseudotrees with N 2f(n) ....

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.

....) N t (c B ) Although these models have isomorphic t neighborhoods of c, we still know nothing about the other part of each model, which might make A and B look very different in FO. The final step of the proof takes care of this by using a version of Hanf s lemma. Proposition 6 (Hanf [4]) For each signature oe, there is a function f(x) with the following property. For all n, A and B, if there is a bijection h : A 7 B such that for all a 2 A, N f(n) a) N f(n) h(a) with a and h(a) distinguished) then Aj n B. Lemma 4 Let A and B be (3f(n) pseudotrees with N 3f(n) c ....

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 directed graphs, the problem is called the directed reachability problem. 3 First order games In this section, we focus on first order Ehrenfeucht Fraiss e games, and give three sufficient conditions for one player (the duplicator) to win. These conditions are based on techniques of Hanf [Han65] (and given a new interpretation by Fagin, Stockmeyer and Vardi [FSV95] Arora and Fagin [AF94] and Schwentick [Sch96b] As we shall discuss, such techniques and conditions are valuable tools for obtaining inexpressibility results. We begin with an informal definition of an r round first order ....

....r game, that is, for showing that G 0 r G 1 for two structures G 0 ; G 1 . 3.1 Hanf s condition Fagin, Stockmeyer and Vardi [FSV95] provide a simple but very useful sufficient condition for guaranteeing that G 0 r G 1 for two structures G 0 ; G 1 . The proof is based on a technique of Hanf [Han65]. They used this condition as a part of a simple proof that connectivity is not in monadic NP (much simpler than the author s original proof [Fag75] Let G be an L structure, where L consists of the relation symbols P 1 ; P p , possibly along with some constant symbols c 1 ; c z , ....

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.

....logic and monadic second order logic. Some easy propositions are listed which illustrate the expressive power of these logics. In a section on first order logic we present the key theorem which supplies a bridge to automata theory. It is a classical result of first order model theory, due to Hanf [Hnf65], but not well known in the community of theoretical computer science. Automata over acyclic graphs are introduced in Section 5. Some special forms are presented, and classes of partial orders are singled out over which these special forms are no restriction (i.e. normal forms of automata) In ....

....The last two equalities are clear from Proposition 3.2. In Section 6 we shall see that over Grids, FO[ logic and EMSO logic (or (mon Sigma 1 1 ) logic) are incompatible in expressive power, and that the last two equalities of Proposition 3.4 turn into strict inclusions. 4. Hanf s Theorem In [Hnf65], Hanf showed that in the first order language of graphs only local properties can be specified. A property is local if it depends only on the occurrence (or non occurrence) of certain local neighbourhoods around vertices. More precisely, call (for r 0) r sphere around vertex v in the graph G ....

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W. Hanf, Model-theoretic methods in the study of elementary logic, in: The Theory of Models (J. Addison, L. Henkin, P. Suppes, Eds.), North-Holland, Amsterdam 1965, pp. 132-145.

....) N t (c B ) Although these models have isomorphic t neighborhoods of c, we still know nothing about the other part of each model, which might make A and B look very different in FO. The final step of the proof takes care of this by using a version of Hanf s lemma. Proposition 35 (Hanf [16]) For each signature oe, there is a function f(x) with the following property. For all n, A and B, if there is a bijection h : A 7 B such that for all a 2 A, N f(n) a) N f(n) h(a) with a and h(a) distinguished) then Aj n B. Lemma 10 Let A and B be (3f(n) pseudotrees with N 3f(n) c ....

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.

....that r;q equivalence (for suitable r; q) is fine enough to capture m equivalence (i.e. indistinguishability by formulas of quantifier depth m) More general formulations are possible, but for simplicity we stay with the case of graphs of degree bounded by d. Theorem 4. 7 ( Sphere Theorem , [Hnf65]) For any m 0 there are r; q 0 such that for any two graphs S; T (finite or infinite, but of degree d) we have: If S r;q T then S jm T . Proof. By Theorem 4.4, it suffices to ensure S =m T from S r;q T for suitably chosen r; q. Set r = 3 m and q = m Delta c where c is the maximal ....

W. Hanf, Model-theoretic methods in the study of elementary logic, in: The Theory of Models (J. Addison, L. Henkin, P. Suppes, Eds.), NorthHolland, Amsterdam 1965, pp. 132-145.

....aggregation. As the numerical domain, we choose the set of rational numbers Q, although any other domain 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 [9, 13] and Gaifman s [11] 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 ....

....(Aggr 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 [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 ....

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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.

....more expressive logics, it is well known that the expressive power of first order logic is rather limited. One of its limitations is that, intuitively, 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 substructure of A which is induced by all elements of A that ....

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 ....

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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.

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. 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.

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.

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.

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.

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.

No context found.

W. Hanf, "Model-Theoretic Methods in the Study of Elementary Logic", in the Theory of Models (J. Addison, L. Henkin and A. Tarski, eds.), North-Holland, Amsterdam, 1965, pp. 132--145.

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