J. Barwise, On Moschovakis closureordinals, Journal of Symbolic Logic  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with at most two distinct variables [27] Many constraints considered here are not expressible in FO , including foreign keys of L, inverse constraints of L ## , and (unary) keys of all the three languages. These can be verified by using the 2 pebble Ehrenfeucht Fra ss e (EF) style game [5]. As shown by Borgida [8] FO has equivalent expressive power as the language DL##trans; compose; at least; at most#, i.e. description logic omitting the transitive closure and composition constructors as well as counting quantifiers. As an immediate result, many XML constraints considered in ....

J. Barwise. On moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

.... following fact: Theorem 3 ( 21] For any formula of IFP or PFP, there is a k such that if (A; a) b) then A j= a] if, and only if, B j= b] Moreover, the equivalence relations have an elegant characterisation in terms of a two player pebble game (essentially due to Barwise [5]) The game board consists of two structures A and B and a supply of k pairs of pebbles (p i ; q i ) 1 i k. The pebbles p 1 ; p l are initially placed on the elements of an l tuple a of elements in A, and the pebbles q 1 ; q l on a tuple b in B. There are two players, ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292-296, 1977.

.... game for P (for a definition see e.g. EF95] Variants of Ehrenfeucht games are available for proving inexpressibility results for many other logics, including second order logics [Ten75] existential second order logics [Ten75, AF90] transitive closure logics [CM91] and finite variable logics [Bar77, Imm82]. Proving the existence of a winning strategy for the duplicator is often very difficult. To simplify such proofs the following approaches have been taken. There have been developed several conditions that assure that the duplicator has a winning strategy on two given structures: the ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

....that if 8 presents 9 with a defect, and in curing it she would have to add a new node which would push the total number of nodes above n, then 8 must rst politely remove some node of the existing network to avoid this happening. Thus, we are in the familiar arena of n pebble games see, e.g. Bar77, Imm82, Poi82] for examples of back and forth n pebble games. If n = we recover the game described before. can be taken as the algebra of all sets of ultra lters of A, with algebraic operations induced from those in A. It embeds A as a subalgebra and we identify A with its image under ....

.... written with at most p variables, possibly re used, and of quanti er depth at most r. In the example above, a relevant sentence is 9x 0 ; x p i j p E(x i ; x j ) where E is the edge relation (playing the role of 6= A similar result for arbitrary L and r = can also be proved [Bar77] 5.2 The modi ed game EF r (A; B) The de nition of the Ehrenfeucht Fra ss e game EF r (A; B) above, is quite standard. We need a minor modi cation of the game to make it appropriate to our purposes. In the constructions that follow we will use a relational language L with only unary ....

J Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292-296, 1977.

....that, if A is a nite structure and A B, then k size(A) k size(B) see, for instance, 8] The equivalence relation has a natural characterisation in terms of two player pebble games in the style of Ehrenfeucht Fra ss e games. This characterisation is essentially due to Barwise [4] (see also [13, 18] The game board consists of two structures A and B and a supply of k pairs of pebbles (a i ; b i ) 1 i k. The pebbles a 1 ; a l are initially placed on the elements of an l tuple s in A, and the pebbles b 1 ; b l on a tuple t in B. There are two players, ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292{ 296, 1977.

....nitary logic 1 with nitely many variables. During the past several years, the study of 1 became a focal point of research in nite model theory. One of the reasons for the interest in L 1 is that its expressive power has an elegant characterization in terms of two player pebble games [Bar77,Imm82]. This has made it possible to apply game theoretic techniques in order to derive lower bounds for expressibility in L 1 , as well as to obtain positive structural results, such as 0 1 laws (cf. KV92c] Moreover, as demonstrated in [AV91,DLW92] there is a deeper connection between L 1 ....

.... . Moreover, it was shown in [DLW92,KV92a] and implicitly in [AV91] that for each k 1 there is a formula of xpoint logic LFP that de nes L equivalence. These results make use of the fact that L 1 equivalence has a combinatorial characterization in terms of k pebble games (cf. [Bar77,Imm82]) Abiteboul and Vianu [AV91] showed that for every k 1 there is a formula of xpoint logic that de nes a linear order k on (the representatives of) the equivalence classes of L 1 equivalence (cf. also [DLW92] This is the main technical tool used by Abiteboul and Vianu [AV91] in ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292-296, 1977.

....are among x 1 ; x 2 ; x k ; L k 1 the closure of L k under the operations of conjunction and disjunction applied to arbitrary ( nite and in nite) sets of formulae (with obvious semantics) L 1 the union S 1 k=0 L k 1 . The logic L 1 was introduced by Barwise [3] and is known as bounded variable in nitary logic. It plays an important role in nite model theory (see, for example, 9] However, we introduce L 1 here for a speci c reason. Bounded variable in nitary logic can be extended by a set of Lindstr om quanti ers (possibly in nite and not ....

J. Barwise, On Moschovakis closure ordinals, J. Symbolic Logic 42 (1997) 292-296.

....L k for the collection of all rst order formulas with at most k distinct variables. It should be pointed out that a variable may have an in nite number of occurrences in a formula of L 1 . The family L 1 of the in nitary languages L k 1 , k 1, was introduced rst by Barwise [Bar77], as a tool for studying xpoint logic on in nite structures. It turned out, however, that these languages have in addition interesting uses in theoretical computer science. Indeed, they underlie much of the work in [Imm82, dR87, LM89] and they have been also studied in their own right in [Kol85, ....

....the expressibility of a query in L and its computational complexity. The preceding Theorem 3.6 constitutes a re nement of an earlier result to the e ect that on every xed structure the in nitary logic L 1 can express every xpoint query. That result appeared rst in print in Barwise [Bar77] and Immerman [Imm82] but is actually anticipated in the unpublished Ph.D. thesis of A. Rubin [Rub75] 11 4 Expressiveness and Pebble Games The results of the previous section imply that, in order to show that a particular query Q is not expressible in Datalog(6= it is enough to establish ....

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J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292-296, 1977.

....is P [ fabg. The duplicator wins the game if each position P that occurs is a partial isomorphism between A and B (that is, an isomorphism whose domain is the substructure of A with universe fa 2 A j 9b 2 B : ab 2 Pg) If A = B, we refer to the game on AB as the game on A. Theorem 7 (Barwise [2], Immerman [11] Poizat [17] Let k 1 and a vocabulary. Furthermore, let A; B be structures and a = a 1 : a l 2 A l , b = b 1 : b l 2 B l l tuples, for an l 2 [1; k] Then we have: i) A L k B if, and only if, the duplicator has a winning strategy for the k pebble game on AB ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

....MODEL CHECKING FOR IFP is EXPTIME complete. 3.2. Infinitary finite variable logics. There is a very fruitful connection between fixedpoint logics and finite variable logics. It is best to make this connection through infinitary finite variable logics, 3 which have been introduced by Barwise [6]. Kolaitis and Vardi [51] were the first to realize their importance for finite model theory. As usual, L1 denotes the infinitary first order logic allowing conjunctions and disjunctions over arbitrary sets. L k 1 is the fragment of L1 consisting of all formulas with at most k variables. ....

....is P [ fabg. The duplicator wins the game if each position P that occurs is a partial isomorphism between A and B (that is, an isomorphism whose domain is the substructure of A with universe fa 2 A j 9b 2 B : ab 2 Pg) If A = B, we refer to the game on AB as the game on A. Theorem 3. 7 (Barwise [6], Immerman [45] Let k 1 and a vocabulary. Furthermore, let A; B be structures and a = a 1 : a l 2 A l , b = b 1 : b l 2 B l l tuples, for an l 2 [1; k] Then we have: i) A L k 1 B if, and only if, the duplicator has a winning strategy for the k pebble game on AB with ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

....P [ fabg. The duplicator wins the game if each position P that occurs is a partial isomorphism between A and B (that is, an isomorphism whose domain is the substructure of A with universe fa 2 A j 9b 2 B : ab 2 Pg) If A = B, we refer to the game on (A; B) as the game on A. Theorem 4 (Barwise [5], Immerman [17] Let k 1 and a vocabulary. Furthermore, let A; B be structures and a = a 1 : a l 2 A l , b = b 1 : b l 2 B l l tuples, for an l k. Then we have: i) A and B are L k equivalent if, and only if, the duplicator has a winning strategy for the k pebble game on ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

....increasing, and therefore the closure ordinal is bounded by l , where = card(A) Of course, if A is infinite, the exponent l is superfluous. On the other hand, if A is finite, the closure ordinal must be finite, and the third clause in the definition above is unnecessary. Rubin [54] see also [10]) observed that for every formula (of first order logic) there is a k such that for all ordinals ff, there is a formula of L k 1 that defines the relation ff on all structures. Without fear of confusion, we call this formula ff as well. One can further note that when ff is finite, then ....

....that as with LFP, on the class of finite structures, every formula of PFP is equivalent to a formula of L 1 . 2. Pebble games and their uses. The equivalence relation j k has an elegant characterisation in terms of Ehrenfeucht Fraiss e style two player games, essentially given by Barwise [10]. The game board consists of two structures A and B and a supply of k pairs of pebbles (a i ; b i ) 1 i k. The pebbles a 1 ; a l are initially placed on the elements of an l tuple s of elements in A, and the pebbles b 1 ; b l on a tuple t in B. There are two players, Spoiler and ....

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J. Barwise, On Moschovakis closure ordinals, Journal of Symbolic Logic, 42:292--296, 1977.

....answers by selecting an a 2 A, and again the new position is P [ fabg. The duplicator wins the game if each position P that occurs is a partial isomorphism between A and B (that is, an isomorphism whose domain is the substructure of A with universe fa 2 A j 9b 2 B : ab 2 Pg) Theorem 6 (Barwise [2], Immerman [11] Poizat [18] For all structures A; B we have (i) A and B are L k equivalent if, and only if, the duplicator has a winning strategy for the k pebble game on (A; B) with initial position ; Furthermore, for l k and l tuples a = a 1 : a l 2 A, b = b 1 : b l 2 B, ....

J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

....any move preserving the isomorphism. We say that Duplicator has a winning strategy if he can play forever despite of the moves of Spoiler, preventing him from winning. This game characterizes the expressive power of the logics we have introduced in the following sense: 8 Theorem 4. 4 (Barwise [6], Immerman [12] Poizat [20] Let A; B be any two structures and let a 2 A k ; b 2 B k : 1. The Duplicator has a winning strategy in the second phase of the game when after the first one the pebbles are placed on a and b in A and B; respectively, if and only if hA; ai j k hB; bi: 2. ....

J. Barwise, `On Moschovakis closure ordinals', Journal of Symbolic Logic 42(1977), pp. 292-- 296. 35

....in L 1 (p) The embedding of logics and database query languages into infinitary logic is not new. Inexpressibility results for fixpoint logic and for Datalog using game theoretic arguments that actually apply to infinitary logic have been used by several authors since the original work in [3] and [19] Kolaitis and Vardi showed that certain queries cannot be defined in the existential negation free 2 fragment of L 1 and are therefore not computable by Datalog programs [27] Our more general treatment of quantifier classes in L 1 yields structural results on certain ....

....k 1 (P ) Ehrenfeucht Fraiss e games [13, 11] provide a powerful tool for proving inexpressibility results for various logics. In their classical form they give a criterion for the indistinguishability of two (classes of) structures by means of first order formulae. They were modified by Barwise [3] and Immerman [19] to deal with other logics, especially infinitary logic and fixpoint logic. Using these games, Kolaitis and Vardi [27] have established, e.g. the 0 1 law for infinitary logic. Also, variants of Ehrenfeucht Fraiss e games have been designed and used for monadic second order logic ....

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J. Barwise, On Moschovakis closure ordinals, J. Symbolic Logic 42 (1977), 292--296.

....in descriptive complexity, such as the various forms of fixed point logics. On the other hand the Ehrenfeucht Fraiss e method that characterizing elementary equivalence generalizes to a very elegant game theoretic characterisation of the distinctive power of L 1 in terms of pebble 6 games [3, 20, 31]. Most of the inexpressibility results for fixed point logics are proved using these games, so they establish in fact an inexpressibility result for L 1 . Atomic types and equality types. Definition 2.5 Let be a relational vocabulary and x 1 ; x k be distinct variables. A maximally ....

J. Barwise, On Moschovakis closure ordinals, Journal of Symbolic Logic, 42 (1977), pp. 292-- 296.

....relations, i.e. the relational store has a variable, unbounded number of relations of bounded arity. Neither extension affects expressive power. Finally, there is a natural correspondence between RM and infinitary logic. Infinitary logic with finitely many variables (denoted L 1 ) see [Ba77]) is first order logic (FO) extended by allowing disjunctions and conjunctions of infinite sets of formulas. Although it generally has non effective syntax, and can define non computable queries, L 1 subsumes most query languages and thus provides an elegant unifying formalism for a wide ....

Barwise, J., On Moschovakis closure ordinals, J. Symbolic Logic, 42 (1977), pp. 292--296.

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J. Barwise, On Moschovakis closureordinals, Journal of Symbolic Logic

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J. Barwise, On Moschovakis closure ordinals, J. Symbolic Logic 42, 1977, pp. 292-296.

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J. Barwise, "On Moschovakis Closure Ordinals, " J. Symb. Logic 42 (1977), 292-296.

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J. Barwise, On Moschovakis closure ordinals, Journal of Symbolic Logic 42 (1977), 292--296.

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J. Barwise, On Moschovakis closure ordinals, J. Symbolic Logic 42 (1977), 292--296.

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J. Barwise, On Moschovakis Closure Ordinals, Journal of Symbolic Logic 42 (1977), 292-296.

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J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292--296, 1977.

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J. Barwise. On Moschovakis closure ordinals. Journal of Symbolic Logic, 42:292{ 296, 1977.

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