Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In 9th IEEE Symposium on Logic in Computer Science, pages 10--19, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and Vardi [40] showed that (3) implies (1) establishing the equivalence of (1) and (3) which was the second of McColm s two conjectures. A proof of this is outlined in Section 3.1 below. The first of McColm s conjectures, the equivalence of (1) and (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 ....

....to a finite disjunction of formulas of quantifier rank m. 3.2. LFP on proficient classes. As mentioned above, McColm s first conjecture, the equivalence of proficiency with the existence of formulas of LFP that are not equivalent to any first order formula was refuted by Gurevich et al. [32]. They construct two distinct classes of structures on which every LFP formula is equivalent to a first order formula but which are nonetheless proficient. The resolution of McColm s conjectures, one positively and the other negatively still leaves a number of questions open. In particular, ....

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Y. Gurevich, N. Immerman, and S. Shelah, McColm's conjecture, in: Proc. 9th IEEE Symp. on Logic in Computer Science, pages 10--19, 1994.

.... 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 is of particular interest. Namely, it is unknown whether LFP collapses to FO on the class of all finite structures of the form (f0; n Gamma 1g; BIT) where is the usual linear ordering, and BIT is the binary relation that consists of all pairs (p; q) of natural numbers such ....

....to FO on the class of all finite structures of the form (f0; n Gamma 1g; BIT) where is the usual linear ordering, and BIT is the binary relation that consists of all pairs (p; q) of natural numbers such that the p th bit in the binary expansion of q is one. As pointed out in [15], the collapse happens if and only if DLOGTIME uniform AC 0 = P uniform AC 0 (see Section 2 for definitions) or equivalently, if and only if LINH = E. Motivated by this interesting connection, Atserias and Kolaitis [3] investigated the difficulty of settling this special case of the Ordered ....

Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In 9th IEEE Symposium on Logic in Computer Science, pages 10--19, 1994.

....to focus attention on this phenomenon and to formulate a certain conjecture 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 ....

....be the class of all ordered finite BIT structures Bn = f0; 1; n Gamma 1g; BITn ) where is the standard linear order and BITn = BIT f0; 1; n Gamma 1g 2 . Question 1 Is FO 6= LFP on B In other words, does the ordered conjecture hold on B Gurevich, Immerman and Shelah [GIS94] raised this question and stated that it is a fascinating question in complexity theory and logic related to uniformity of circuits and logical descriptions. Indeed, it can be shown that FO 6= LFP on B if and only if log time uniform AC 0 is different than polynomial time uniform AC 0 (see ....

Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In Proc. 7th IEEE Symp. on Logic in Comp. Sci., pages 10--19, 1994.

....vocabulary, then LINH 6= E. This was already observed by Dawar, Lindell, and Weinstein [DLW96] In fact, we show that the two questions are literally equivalent, and in turn, they are equivalent to an important question on circuit uniformity that we discuss next. Gurevich, Immerman, and Shelah [GIS94] pointed out without proof that LFP is more expressive than FO on the class of finite ordered structures with only the BIT predicate if and only if DLOGTIME uniform AC 0 = P uniform AC 0 . Here C uniform AC 0 is the class of languages that are accepted by constant depth, unbounded fan in, ....

.... For example, it is an open question whether integer division can be done by a DLOGTIME uniform family of NC 1 circuits, while it is known 67 that a P uniform such family exists (see [BCH86] This connection with uniform circuits was pointed out without proof by Gurevich, Immerman, and Shelah [GIS94] They isolated the class of finite structures of the form B n = f0; n Gamma 1g; Bn ; BIT Bn g) where Bn is the standard linear order, and BIT Bn is the binary relation that contains all pairs (a; b) 2 f0; n Gamma 1g 2 such that the a th bit of the binary ....

Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In 9th IEEE Symposium on Logic in Computer Science, pages 10--19, 1994. 108

.... from questions and results about the class of all finite structures or about arbitrary classes of ordered structures to results about arbitrary classes of unordered finite structures or about specific restricted classes of finite structures that arise in combinatorics and database theory (see [McC90, KV92a, GIS94]) A promising approach to understanding when least fixpoint logic can capture a complexity class is to analyze how difficult it is to define a linear order on each member of a given class of finite unordered structures. In turn, this moves the focus on classes of finite rigid structures, that is, ....

Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In Proc. 7th IEEE Symp. on Logic in Comp. Sci., pages 10--19, 1994.

....of which classes of structures C have the property that LFP and L 1 collapse to first order logic on C. They proved a conjecture of McColm [McC90] showing that L 1 collapses to FO if, and only if, every positive, first order induction is bounded. Gurevich, Immerman and Shelah [GIS94] refuted another conjecture due to McColm by constructing a class of structures on which LFP collapses to FO, but L 1 does not. Kolaitis and Vardi [KV92b] conjectured the following weaker version of McColm s conjecture, which remains open: Conjecture 1 (Kolaitis Vardi) On every infinite ....

....that m is uniformly defined by a first order formula. McColm [McC90] also showed that condition (3) implies (1) Kolaitis and Vardi [KV92a] showed that (1) implies (3) thereby establishing the equivalence of (1) and (3) and resolving the second of McColm s two conjectures. Gurevich et al. [GIS94] construct an example of a class of structures where (2) holds but (1) fails, refuting the first of the two conjectures. While McColm s first conjecture has been refuted in the general case, it remains open whether it nonetheless holds on classes of ordered structures, i.e. for any class C that is ....

Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In Proc. 9th IEEE Symp. on Logic in Computer Science, 1994.

....in FO ; 3. Every query on C definable in L 1 is definable in FO . It is easily seen that (1) implies (2) McColm [20] showed that (3) implies (1) Kolaitis and Vardi in [16] showed that (1) implies (3) hence establishing that (1) and (3) are, indeed, equivalent. Gurevich, Immerman and Shelah [7] construct counterexamples showing that (2) does not imply (1) A discussion of McColm s conjecture based on the notions of element types was presented in [4] where it was shown that conditions (1) and (3) are equivalent to the following compactness condition: for all k , the set fType k (A; s) j ....

Y. Gurevich, N. Immerman, and S. Shelah. McColm 's conjecture. In Proc. 9th IEEE Symp. on Logic in Computer Science, 1994.

....L 1 has the same expressive power as FO on C if and only if LFP is bounded on C. Note that the two conjectures combined together assert that LFP collapses to FO on a class C of finite structures if and only if L 1 collapses to FO on C. McColm s conjectures were subsequently studied in [KV92a, GIS94], where the second conjecture was confirmed and the first conjecture was refuted. First, in [KV92a] it was shown that if LFP is bounded on C, then L 1 collapses to FO on C. Then, in [GIS94] it was shown that there is a class C on which LFP does collapse to FO though LFP is not bounded on ....

....and only if L 1 collapses to FO on C. McColm s conjectures were subsequently studied in [KV92a, GIS94] where the second conjecture was confirmed and the first conjecture was refuted. First, in [KV92a] it was shown that if LFP is bounded on C, then L 1 collapses to FO on C. Then, in [GIS94] it was shown that there is a class C on which LFP does collapse to FO though LFP is not bounded on C. The results in both [KV92a] and [GIS94] take advantage of the fact that we are considering logics whose formulas have a finite but arbitrarily large number of variables. Indeed, boundedness of ....

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Y. Gurevich, N. Immerman, and S. Shelah. Mccolm's conjecture. In Proc. 7th IEEE Symp. on Logic in Comp. Sci., pages 10-- 19, 1994.

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Y. Gurevich, N. Immerman, and S. Shelah. McColm's conjecture. In 9th IEEE Symposium on Logic in Computer Science, pages 10--19, 1994.

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Gurevich, Y., Immerman, N., and Shelah, S. (1994). McColm's conjecture. In Proceedings of the 9th IEEE Symposium on Logic in Computer Science, pages 10--19.

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