R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41:50--58, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The Ehrenfeucht game was first given in [5] It was essentially found in earlier work by Fraisse and is sometimes referred to as the Ehrenfeucht Fraisse game. The classic Zero One law for random graphs with p = often called the uniform distribution) are due to Glebskii et al. 8] and Fagin [7]. The classic paper that began the theory of random graphs is by Paul Erdos and Alfred Renyi [6] The basic text on random graphs is Bollobas [2] The Zero One Law for p = n # appeared first in Shelah, Spencer [17] An approach using the Ehrenfeucht game is given in Spencer [21] A syntactic 27 ....

R. Fagin, Probabilities in Finite Models, J. Symbolic Logic 41 (1976), 50-58

....Monadic Second Order (with built in linear ordering) queries over random free trees, proving that for any MSO query , the probability that holds on a random (labelled) free tree of n vertices is an eventually periodic function of n. 1 Introduction Since Glebski et al. [14] and Fagin [13] showed that zero one laws held for all First Order (FO) queries over random graphs, the study of asymptotic This research was partially supported by NSF grant CCR 940 3463. probabilities of logical queries has become quite active. There have been many papers saying that asymptotic ....

R. Fagin, Probabilities on Finite Models, J. Symbolic Logic 41:1 (1976), 50--58.

....3CNF formula, the bipartite graph that relates clauses with the variables that occur in them, satisfies certain extension axioms that we introduce based on the expander properties of the graph. The use of extension axioms is a recurrent theme in the analysis of pebble games on random structures [Fag76, Lyn80, SS88, KV90]. On the other hand, the expander properties of these graphs have also been used in the context of propositional proof complexity to prove lower bounds in the size of Resolution refutations [BSW01, BSG01] This establishes an interesting link between the fields of Propositional Proof Complexity ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41:50--58, 1976.

....the full infinite clique is highly symmetric. Here is an example of another highly symmetric graph: A particularly interesting example of highly symmetric data bases are the (not necessarily recursive) countable random structures. These constitute a natural generalization of the Rado graph (cf. [ Ra, Fa] ) They are characterized by an infinite set of extension axioms, which say that for each finite set X of points in the domain, and for each possible way that a point not in X can be related to X in terms of atomic In the figure, a line between i andj represents the two edges (i,j) and (j, i) ....

R. Fagin, Probabilities on finite models, J. Symbolic Logic 41 (1976), 558.

....and a SRG(q, q 1) 2, q 5) 4, q 1) 4) In [1] and [4] it was shown that for a fixed n, su#ciently large Paley graphs are n e.c. Few examples of strongly regular non Paley n e.c. graphs are known, despite the fact that for a fixed n almost all graphs are n e.c. see [3] and [9]) The exception is when n = 1 or 2; see [5] and [6] Even for n = 3 it has proved di#cult to find strongly regular n e.c. graphs that are not Paley graphs. In [1] it was shown that P 29 is the minimal order 3 e.c. Paley graph. As reported in [5] a 3 e.c. graph has order at least 20, and a ....

R. Fagin, Probabilities on finite models, J. Symbolic Logic 41 (1976), 50--58.

....a SRG(q; q Gamma 1) 2; q Gamma 5) 4; q Gamma 1) 4) In [1] and [4] it was shown that for a fixed n, sufficiently large Paley graphs are n e.c. Few examples of strongly regular non Paley n e.c. graphs are known, despite the fact that for a fixed n almost all graphs are n e.c. see [3] and [9]) The exception is when n = 1 or 2; see [5] and [6] Even for n = 3 it has proved difficult to find strongly regular n e.c. graphs that are not Paley graphs. In [1] it was shown that P 29 is the minimal order 3 e.c. Paley graph. As reported in [5] a 3 e.c. graph has order at least 20, and a ....

R. Fagin, Probabilities on finite models, J. Symbolic Logic 41 (1976), 50--58.

....: xm ) and 0 (x 1 ; xm 1 ) be two atomic types such that 0 is consistent with . The ; 0 extension axiom, j ; 0 is the sentence: 8x 1 : 8xm9xm 1 ( 0 ) That is, j ; 0 asserts that every tuple of type can be extended to a tuple of type 0 . 8 A. DAWAR Fagin [25] proved that every extension axiom has asymptotic probability 1 on the class of finite structures. The asymptotic probability of a sentence is defined as the limit lim n 1 jMod( Sn j jSn j , where Sn is the collection of all oe structures with universe f0; n Gamma 1g. Fagin used ....

....all models of k are models of , or none are. In the former case, has asymptotic probability 1, and in the latter case it has asymptotic probability 0. We thus obtain: Theorem 7 (Kolaitis Vardi [41] L 1 has a 0 1 law. This subsumes earlier results on 0 1 laws for first order logic [27, 25], LFP [11] and PFP [39] Theorem 7 is a 0 1 law under what is called the uniform probability measure. That is, each structure of size n is assigned equal probability. A variety of other ways of assigning probabilities to structures are obtained, particularly in the case of graphs, by varying ....

R. Fagin, Probabilities on finite models, Journal of Symbolic Logic, 41(1):50--58, March 1976.

....otherwise (in this case we write without equality ) the equality symbol may occur. A 0 1 law holds for a logic if the asymptotic probability of any property which is expressible in this logic, is either 0 or 1. In 1967 Gleskii, Kogan, Liogonkii et Talanov [GKLT69] and independently Fagin in 1976 [Fag76] have shown that the 0 1 law holds for FO, the first order logic. We denote by Sigma 1 1 the existential second order logic. A Sigma 1 1 sentence over a vocabulary R is an expression of the form 9S (R; S) where S is a set of relation variables and (R; S) is a first order sentence on ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41:50--58, 1976.

....called an extension axiom: 8x 0 ; x k Gamma1 Gamma [fx 0 ; x k Gamma1 g G] Gamma 9x k ; x Gamma1 [fx 0 ; x Gamma1 g H] Delta Delta ; i.e. the one expressing every copy of G extends to a copy of H . It is well known since the paper of Fagin [4] that for arbitrary two graphs G H the equality (Ext(G; H) 1 holds. In particular (Ext( G) 1 and (Ext(G; H) 1: Then, assuming additionally that jHj is of much greater cardinality than jGj; the graph of the function n 7 n (Ext(G; H) looks like the one on figure 1, at the end of the ....

Fagin, R., Probabilities on finite models, J. Symbolic Logic 41(1976), pp. 50--58.

....oe (n) is the number of atoms to be decided to be false or true to get a complete n element structure of signature oe: It gives the theoretical upper bound on the Kolmogorov complexity of n element ordered structure over oe; as well. The first result we will need is known from the work of Fagin [11] (in the case of graphs it has been shown already by P olya [21] All the items are tightly related to each other by Burnside s Lemma. Let us recall the standard definition, that a structure is rigid if it has no nontrivial automorphism. Theorem 6.5. 1. The set R Sigma 0 of isomorphism ....

....let the following sentence be denoted Ext k;U (t) 8x 1 : x k Gamma1 V i6=j x i 6= x k V i x i = 2 U 9x k 0 B B x k 6= x 1 ; x k Gamma1 x k = 2 U ( V t) W At k;U n t) 1 C C A : All such sentences are called k; U extension axioms. Lemma 6. 8 (Fagin [11]) Let i(n) be the constant function 0. Then for every k; extension type t Sigma n (fA 2 Sigma : A j= Ext k; t)g) 1 Gamma c n for some 0 c 1: We will need an extension of this lemma for other not too fast growing functions i: Lemma 6.9. 1. Let lim sup n 1 i(n) log n 1: Then ....

R. Fagin, `Probabilities on finite models', Journal of Symbolic Logic 41(1976), pp. 50--58.

....formalism, in contrary to the classical complexity theory question, how difficult it is to decide this problem in a given computational model. Although these approaches seem quite different, they are sometimes even equivalent: the famous and seminal for descriptive complexity result of Fagin [4] states, that a problem, i.e. a class of finite structures, is computable in NP iff it can be defined by a sentence of existential second order logic Sigma 1 1 : In the descriptive complexity the resources used to define problems, and thus to measure their difficulty, are e.g. kind of ....

....of formulae. What counts, however, are all the parentheses, variables, formulas, etc. which appear right after Q; as well as the other new symbol we use: We need an information about the cardinalities of sets Fin n (oe) fA 2 Fin(oe) jAj = ng: The following well known theorem of Fagin [4] gives an asymptotic estimate of this value: Theorem 5 For any sequence foe n g n2N of non unary signatures with fixed maximal arity holds jFin n (oe)j = 1 o(1) cp oe n (n) n : The value jFin n (oe)j we denote by ns oe (n) Lemma 6 For every nonunary type oe there is a Lindstrom quantifier Q ....

R. Fagin, Probabilities on finite models, Journal of Symbolic Logic 41(1976), pp. 50--58.

....n) An interesting question is, whether the limit ( lim n 1 n ( exists and to determine the possible values of ( as ranges over the sentences in L. The first result of this kind is the 0 1 law for first order logic which was proved independently by Glebskii et al. 15] and by Fagin [13]. 2 Not all fEg structures are graphs. However, the results on asymptotic probabilities go through for the class of all graphs since it is a parametric class, as are the classes of digraphs or tournaments) cf. 10] 7 Theorem 3.1 For every first order sentence over a relational vocabulary, ....

....types s(x 1 ; x k ) and t(x 1 ; x k ; x k 1 ) such that s ae t. Then we can formulate the extension axiom oe s;t = 8x 1 Delta Delta Delta 8x k (s(x) 9x k 1 t(x; x k 1 ) It states that every realization of s can be extended to a realization of t. It was observed by Fagin [13] that every extension axiom is almost surely true on random finite structures, and that the rate of convergence is exponential. Lemma 3.2 For every extension axiom oe s;t , and all sufficiently large n n (oe s;t ) 1 Gamma c Gamman for some constant c 1. Using a common model theoretic ....

R. Fagin, Probabilities on finite models, Journal of Symbolic Logic, 41 (1976), pp. 50--58.

....Very recently, generalized quantifiers have been studied in the realm of finite structures [KV92a, Hel92] The restriction to finite structures also enabled the design and development of specific methods, among which 0 1 laws appear as central. This line of research was initiated by Fagin [Fag76] and Glebski et al..ii [GKLT69] who independently proved the following startling result: given any FO sentence , if all structures of size n are considered equiprobable, then the limit, as n 1, of the probability that is satisfied by a random structure of size n, always exists and is equal to ....

....(a.s. true (resp. almost surely false) if its asymptotic probability is 1 (resp. 0) If the asymptotic probability is defined for every sentence of L[ L[ is said to have a limit law. If, in addition, the asymptotic probability is either 0 or 1, L[ is said to have a (labelled) 0 1 law. In [Fag76], Fagin proved that there is a structure over the set of all integers, called the random countable structure, which plays a key role in the study of the asymptotic probabilities of FO[ sentences. In the sequel, Omega Gamma ] will denote the random countable structure. The following ....

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R. Fagin. Probabilities on finite models. Journal of Symbolic Logic 41(1) (1976) 50-- 58.

....for finitely representable structures. The failure of most of the known theorems of classical model theory, shows the need to develop new tools meaningful for finitely representable structures. Powerful tools have been developed, such as the locality by Gaifman [Gai81] or the 0 1 laws by Fagin [Fag76] in the case of finite model theory. The model theory of finitely representable structures differs also from finite model theory. None of these tools is known to work in the presence of some regular mathematical structure such as an order relation for instance. The problem of the definability of ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50--58, 1976.

....though his notion is more general. The definition used here comes from logic programming theory. Theorem 4.17 goes back to Los (1955) Two sources for Theorem 4.28 are Ryll Nardzewski 1959 and Svenonius 1959. A elegant introduction to 0 1 laws is Gurevich 1992. An original source is Fagin 1976. For the 0 1 law for fixed point logic, see, for instance, Blass et al. 1985. On the class of finite models with ordering, the subclasses that have a polynomial time decision problem coincide with those definable by a fixed point formula. Fagin and Immerman, independently. For the curious ....

Fagin, R. 1976. Probabilities on finite models. Journal of Symbolic Logic 41:50-- 58.

....Laws for FO We present here a powerful tool that provides a uniform approach to resolving in the negative a large spectrum of expressibility problems. It is based on the probability that a property is true in structures of a given size. This study, initiated by Fagin [Fagin76] and Glebski i [Glebskii 69] shows a very surprising fact: all properties of finite structures definable in FO are almost surely true, or almost surely false. Let oe be a sentence over some relational vocabulary R. For each n, let n (oe) denote the fraction of finite structures over R with ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1), 50--58, 1976.

....0 or 1. In finite model theory one studies 0 1 laws for asymptotic probabilities. Fix a finite relational vocabulary and let n ( denote the fraction of the structures with universe f0; n Gamma 1g that satisfy . The 0 1 law for first order logic, proved independently by Fagin [2] and Glebskii et al. 4] states that for first order sentences the limit ( lim n 1 n ( always exists and is either 0 or 1. Gaifman [3] considered countable structures with universe and finite relational vocabulary. He proved that with respect to the Lebesgue measure every property of ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41:50--58, 1976.

....structures. If C is a class of finite structures over some vocabulary oe and if P is a property of some structures in C, then the asymptotic probability (P ) on C is the limit as n 1 of the fraction of the structures in C with n elements that satisfy P , provided that the limit exists. Fagin [F2] and Glebskii et al. GKLT] were the first to discover the connection between logical definability and asymptotic 6 In the direction going from expressibility in E Sigma 1 k to computability in Sigma 1 k , the secondorder quantifiers are used to define a total order and predicates and , ....

....in a logic and asymptotic probabilities. Several additional logics, such as fixpoint logic, iterative logic and strict E Sigma 1 1 , have been shown by various authors to satisfy the 0 1 law too. A standard method for establishing 0 1 laws on finite structures, originating in Fagin [F2], is to prove that the following transfer theorem holds: there is an infinite structure A over oe such that for any property P expressible in L: A j= P iff (P ) 1 on C: It turns out that there is a single countable structure A that satisfies this equivalence for all the logics mentioned above. ....

R. Fagin, "Probabilities on Finite Models", J. of Symbolic Logic, 41, (1976), 50 -- 58.

....structures of a given signature, and M accepts or rejects depending on whether or not the given structure satisfies a fixed first order formula. If A n is the set of n element structures accepted, and a n = jA n j=jI n j, then lim n 1 a n 2 f0; 1g. This was proven independently in [GlKLT69] and [Fa76]; see [EbF95] I99] There are at least two applications of these kinds of results: ffl To prove that certain properties of inputs cannot be expressed as accept reject problems of a certain kind of machine. For example, I = f0 n : n is primeg is not accepted by a DFA because there are ....

R. Fagin, Probabilities on Finite Models, J. Symbolic Logic 41:1 (1976), 50--58.

....How does one analyze this very natural extension of the standard relational calculus We know many results about expressivity of FO without the interpreted structure M; in fact, much of the development of finite model theory was motivated by database problems. Standard techniques such as 0 1 laws [15] and locality [16] tell us that queries such as parity of cardinality or the transitive closure of a graph are inexpressible in FO. However, these techniques become inapplicable in the embedded setting. Other techniques, such as Ehrenfeucht Fraiss e games, become extremely awkward to apply with ....

....on a countably infinite set U : that is, any model that satisfies every sentence that is true in almost all finite graphs. Here almost all is with respect to the uniform probability distribution: E(a; b) holds of nodes a; b with probability one half, independently for each pair a; b. It is known [15] that the set of all such sentences forms a complete theory with infinite models, and that this theory is decidable. Proposition 1 (see [27] The natural active collapse fails over the random graph. 2 The idea of the proof is to use the extension axioms to simulate monadic second order logic over ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic 41 (1976), 50--58.

....they involve two basic primitives, recursion and counting, and because it is known that they cannot be expressed by the relational calculus. It was noted in [16] that useful properties for proving the inexpressibility of these queries in the relational calculus, such as locality [14] and 0 1 law [13], do not carry over to constraint query languages. Nevertheless, a number of inexpressibility results were established recently. In [18] it is shown, via an AC 0 data complexity result, that the parity query cannot be expressed if only linear constraints are added to the relational calculus. In ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50-58, 1976.

....set U : that is, any model that satisfies every sentence that is true in almost all finite 3 hypergraphs. Here almost all is with respect to the uniform probability distribution: R(a; b; c) holds of nodes a; b; c with probability one half, independently for each triple a; b; c. It is known [24] that the set of all such sentences forms a complete theory with infinite models, and that this theory is decidable. Proposition 7 There is a first order natural semantics sentence that defines parity over RT . Proof: A model of the random ternary relation has the property [24] that for every ....

....b; c. It is known [24] that the set of all such sentences forms a complete theory with infinite models, and that this theory is decidable. Proposition 7 There is a first order natural semantics sentence that defines parity over RT . Proof: A model of the random ternary relation has the property [24] that for every quantifier free formula ( x; y) that is consistent in predicate logic, we have RT j= 8 x 9 y ( x; y) We next consider, for every 1 k n , a formula n;k (x 1 ; x n ; y 1 ; y n ; x; y; w) defined as [ x; y) x 1 ; y 1 ) x; y) x n ; y n ) ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic 41 (1976), 50--58.

....0 or 1 [28] In the study of random structures there are analogous 0 1 laws which say that in certain classes of structures, all properties expressible in some logic have an asymptotic probability of 0 or 1. The first significant result of this kind was due to Glebskii et al. 23] and Fagin [18]. They showed that the 1 Center for Communications Research, 4320 Westerra Drive, San Diego, CA 92121; email: ed ccrwest.org 2 EECS Department, University of Michigan, Ann Arbor, MI 48109 2122; email: kjc umich.edu 3 Department of Combinatorics and Optimization, University of Waterloo, ....

Ronald Fagin. Probabilities on finite models. J. Symbolic Logic, 41:50--58, 1976.

....Monadic Second Order (with built in linear ordering) queries over random free trees, proving that for any MSO query , the probability that holds on a random (labelled) free tree of n vertices is an eventually periodic function of n. 1 Introduction Since Glebski et al. [14] and Fagin [13] showed that zero one laws held for all First Order (FO) queries over random graphs, the study of asymptotic This research was partially supported by NSF grant CCR 940 3463. 1 probabilities of logical queries has become quite active. There have been many papers saying that asymptotic ....

R. Fagin, Probabilities on Finite Models, J. Symbolic Logic 41:1 (1976), 50--58.

.... involve two basic primitives, recursion and counting, and because it was known that they cannot be expressed by the relational calculus [4] In [10] it was shown that useful properties for proving the inexpressibility of these queries in the relational calculus, such as locality [8] and 0 1 law [7], do not carry over to the constraint query languages. Nevertheless, with a considerable amount of effort, a number of inexpressibility results were established recently. In [12] it is shown, via an AC 0 data complexity result, that the parity query cannot be expressed if only linear constraints ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50-58, 1976.

....of interesting properties on the class G of all finite graphs. It is, for example, well known and easy to prove that (connectivity) 1, while (k colorabilty) 0, for every k 0 [Bol85] A theorem of Posa [Pos76] implies that (Hamiltonicity) 1. Glebskii et al. GKLT69] and independently Fagin [Fag76] were the first to establish a fascinating connection between logical definability and asymptotic probabilities. More specifically, they showed that if C is the class of all finite structures over some Work partially supported by NSF grants CCR 9610257 and CCR 9732041. Work partially ....

....P expressible in the logic L. We write (L) for the collection of all sentences P in L with (P ) 1. Notice that if L is first order logic, then the existence of the 0 1 law is equivalent to stating that (L) is a complete theory. A standard method for establishing 0 1 laws, originating in Fagin [Fag76], is to prove that the following transfer theorem holds: there is an infinite structure A over the vocabulary R such that for every property P expressible in L we have: A j= P ( P ) 1: It turns out that there is a unique (up to isomorphism) countable structure A that satisfies the above ....

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Fagin, R.: Probabilities on finite models. J. Symbolic Logic 41(1976), pp. 50--58.

....size of the structures tends to infinity. The relational signature of a class of structures over Sigma is the common set of relation symbols in the vocabulary Sigma, together with their arities. A famous theorem by Glebskii, Kogan, Liogon kii and Talanov [9] and proved independently by Fagin [7], states that if C is the class of all structures for a given relational signature, then C has a first order 0 1 law. However, deciding the limiting probability of a given property is a difficult problem, as formalized by a theorem by Grandjean: when a class C has a 0 1 law, the set of ....

Fagin (Ronald). -- Probabilities on finite models. Journal of Symbolic Logic, vol. 41, 1976, pp. 50--58.

....drastically from the initial one, since most of the powerful tools of classical logic fail when the semantics is restricted to classes of models, such as finite models for instance. New techniques have been developed such as the study of the asymptotic probabilities of the truth of sentences [Fag76]. Finite model theory is strongly motivated by database theory. A relational database is just a finite structure. The new applications of database systems (e.g. geographical information) have lead to new data models, appropriate for the manipulation of geometrical, topological, and arithmetical ....

R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50--58, 1976.

....of the work on extended logics over finite structures was related to important problems of descriptive complexity. The restriction to finite structures also enabled the design and development of specific methods, among which 0 1 laws appear as central. This line of research was initiated by Fagin [Fag76] and Glebski et al. GKLT69] who independently proved the following startling result: given any FO sentence , if all structures of size n are considered equiprobable, then the limit, as n 1, of the probability that is satisfied by a random structure of size n, always exists and is equal to ....

....is defined for every sentence of L[ L[ is said to have a limit law. If, in addition, the asymptotic probability is either 0 or 1, L[ is said to have a (labeled) 0 1 law. First order logic without constant or function symbols (FO) was the first logic to be proved to enjoy a 0 1 law [GKLT69, Fag76]. Moreover, Fagin considered the (infinite) set of all extension axioms, which constitute an categorical and complete theory Theta, whose unique countable model, the random countable structure, denoted by Omega Gamma satisfies: a sentence 2 FO has asymptotic probability 1 iff it is true ....

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R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50--58, 1976.

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R. Fagin. Probabilities on finite models. Journal of Symbolic Logic 41(1), pages 50--58, 1976.

....example (discussed in Section 5) is the relationship to complexity classes. Another example (discussed in Section 6) is the idea of considering the asymptotic probabilities of sentences as the cardinality of the universe grows. My 1973 Ph.D. thesis [Fag73] which was later published as the papers [Fag74, Fag75a, Fag75b, Fag75c, Fag76]) dealt exclusively with finite model theory. I was disappointed that the field languished for years afterwards: very few papers were published in the area in the next ten years or so. However, in the mid 1970 s, H ajek [H aj75, H aj77] discussed the importance of finite model theory. In fact, in ....

....appeared in 1972. Their proof uses an elimination of quantifiers argument. Without knowing about their results, I proved the 0 1 law in 1971, and wrote a summary for the American Mathematical Society Notices in 1972 [Fag72] Although it appeared in my thesis in 1973, the journal version [Fag76] did not appear until 1976, due to refereeing delays. I now describe my proof, which is quite different from that of Glebskii et al. For ease in description, let us assume for now that the language consists of a single binary relation symbol P . Thus, we can view the language as talking about ....

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R. Fagin. Probabilities on finite models. Journal of Symbolic Logic, 41(1):50-- 58, 1976.

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Fagin, R. (1976). Probabilities on finite models. Journal of Symbolic Logic, 41(1):50--58.

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