A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In H. K. Buning, editor, Computer Science Logic '95, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....R S(n) admits quantifier elimination for L 3 . Remark 11 The property of an L k theory T that there is a finite collection of definable relations such that expanding the models of T by these relations yields a theory admitting quantifier elimination has been called k compactness in [5]. Thus our lemma implies that the L 3 theory T s is 3 compact. However, this alone does not suffice for our applications of the lemma. We are going to use the additional information that R S(n) F S(n) 2 and that R S(n) is anti reflexive. Addition scales We can consider the theory ....

A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In H. Klein-B"uning, editor, Computer Science Logic, CSL `95, Selected Papers, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.

....any way of solving it will have Supported by the CUR, Generalitat de Catalunya, through grant 1999FI 00532, and partially supported by ALCOM FT, IST 99 14186. 2 important consequences in Complexity Theory. A refutation would imply that P 6= PSPACE [12] and a proof would imply that LINH 6= E [13]. Here, LINH is the linear time hierarchy of Wrathall [35] and E is the usual complexity class that consists of all languages that are accepted by deterministic Turing machines in time 2 O(n) There is a special case of the conjecture, singled out by Gurevich, Immerman, and Shelah [15] that ....

A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In Computer Science Logic '95, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.

....concerning necessary and sufficient conditions for the collapse of LFP to FO on an arbitrary class of finite structures. Although in its full generality McColm s conjecture was refuted by Gurevich, Immerman and Shelah [GIS94] it sparked a sequence of related investigations in finite model theory [KV92, DH95, KV96, DLW96]. Moreover, the following special case of McColm s conjecture still remains open: Conjecture 1 If C is an infinite class of ordered finite structures, then first order logic FO is properly contained in least fixed point logic LFP on C. This conjecture, which is often called the Ordered ....

....same time, researchers have established that either way of settling the ordered conjecture will have sig nificant complexity theoretic consequences. Specifically, Dawar and Hella [DH95] showed that if the ordered conjecture is false, then PTIME 6= PSPACE. Furthermore, Dawar, Lindell and Weinstein [DLW96] pointed out that if the ordered conjecture holds, then LINH 6= ETIME, where LINH (the Linear Time Hierarchy) is the class of languages computable by alternating Turing machines in linear time using a constant number of alternations, and ETIME is the class of languages computable by deterministic ....

A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In Computer Science Logic '95, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.

....logic, but is not definable in first order logic. It has turned out that any way of resolving the question would have important consequences in complexity theory. Dawar and Hella [DH95] have shown that if the conjecture fails, then P 6= PSPACE. In addition, Dawar, Lindell and Weinstein [DLW96] pointed out that if the conjecture holds, then LINH 6= E. Here, LINH is the Linear time Hierarchy introduced by Wrathall [Wra78] and E is the class of languages accepted by deterministic Turing machines running in linear exponential time, that is, time bounded by 2 cn where c is a constant, ....

....orders, 9 fixed point logic is more powerful than first order logic. We give a complete proof of this result, and its easy extension to the class of all ordered structures for any particular vocabulary. Then we focus on a different restriction: unary vocabularies. Dawar, Lindell, and Weinstein [DLW96] showed that the Ordered Conjecture holds when restricted to unary vocabularies, that is, vocabularies that only contain unary relation symbols. More precisely, for every infinite class of finite ordered structures for a unary vocabulary, least fixed point logic is more expressive than ....

[Article contains additional citation context not shown here]

A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In Computer Science Logic '95, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.

....S(n) Delta admits quantifier elimination for L 3 . Remark 11. The property of an L k theory T that there is a finite collection of definable relations such that expanding the models of T by these relations yields a theory admitting quantifier elimination has been called k compactness in [5]. Thus our lemma implies that the L 3 theory T s is 3 compact. However, this alone does not suffice for our applications of the lemma. We are going to use the additional information that R S(n) F S(n) 2 and that R S(n) is anti reflexive. Addition scales We can consider the theory ....

A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In H. Klein-B"uning, editor, Computer Science Logic, CSL `95, Selected Papers, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.

....(2) above was refuted by Gurevich et al. 32] Some questions arising from it are discussed in Section 3.2. 3.1. Compactness. The equivalence of (1) and (3) is perhaps best understood in terms of the number of L k types that are realised among structures in C. The following definition is from [14, 20]. Definition 17. We say that a class of structures C is k compact if fType k (A; s) j A 2 C; s 2 A k g is a finite set. It turns out that a class C is k compact (for all k) if, and only if, it is not proficient. We will see why this is the case through a series of facts which in themselves ....

....to a first order formula, but one with possibly many more variables. Another question related to McColm s first conjecture that remains open is whether it holds of classes of structures C where structures are linearly ordered. This form of the question was posed in [40] It was argued in [20] that a resolution of the question either way would solve outstanding complexity theoretic questions. In particular, if it can be shown that there is an infinite class of ordered structures on which every formula of LFP is equivalent to a first order formula, then P6=PSPACE (this was proved in ....

[Article contains additional citation context not shown here]

A. Dawar, S. Lindell, and S. Weinstein, First order logic, fixed point logic and linear order, in: H. Kleine-Buning, editor, Computer Science Logic '95, volume 1092 of LNCS, pages 161--177. Springer, 1996.

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A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In H. K. Buning, editor, Computer Science Logic '95, volume 1092 of Lecture Notes in Computer Science, pages 161--177. Springer-Verlag, 1996.

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