G. L. McColm, When is arithmetic possible?, Annals of Pure and Applied Logic, 50:29--51, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....VARIABLES 11 negative answer to this question is already obtained from an example appearing in [51] Any linear order is the unique, up to isomorphism, finite model of its L 3 theory. We will consider a somewhat subtler interpretation of this question in Section 5. 3. McColm s conjectures. In [48, 47], McColm initiated a study of the relationship between first order logic, LFP and L 1 on classes of finite structures. The analysis was in terms of a notion he termed proficiency . In the following definition, we write jj jj A to denote the closure ordinal of on A (see Section 1.3) ....

....ordinal of on A (see Section 1.3) Definition 16. A class C of structures is proficient , if there is some positive first order formula such that sup(fjj jj A j A 2 Cg) That is, C is proficient if there is some formula whose closure ordinal is unbounded among structures in C. McColm in [48] formulated two conjectures, which taken together state that the following three conditions are equivalent for any class of structures C. 1. C is proficient; 2. There is a formula of LFP that is not equivalent to any formula of first order logic on C. 3. There is a formula of L 1 that is ....

G. L. McColm, When is arithmetic possible?, Annals of Pure and Applied Logic, 50:29--51, 1990.

....arbitrary sub classes of the class of all finite structures (again, for some fixed signature oe) and ask what queries can be expressed over these sub classes. The question we focus on is, for which classes C are the inductive logics strictly more expressive than first order logic In this context, McColm [McColm, 1990] put forward two conjectures that would characterize such classes. Kolaitis and Vardi [Kolaitis and Vardi, 1992a] have affirmatively answered one of these conjectures. We present a perspicuous proof of this result by relating it to a natural notion of compactness derived from the idea of bounded ....

....it to the open conjecture and study this property on particular classes of structures, such as trees. In Section 4.4, we state Kolaitis and Vardi s ordered conjecture, and investigate its connections to the preservation property on the class of all finite structures. 4. 1 McColm s Conjectures In [McColm, 1990], McColm presented two conjectures which relate the behavior of inductive definitions over a class of finite structures C to the expressive power of L 1 and first order logic over C: The essence of these conjectures is that inductive logic derives its power from the possibility of forming ....

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G. L. McColm. When is arithmetic possible? Annals of Pure and Applied Logic, 50:29--51, 1990.

....on the class O of all ordered finite structures over a fixed vocabulary, as well as on the class F of all (unordered) finite structures over a fixed vocabulary. There are, however, classes of unordered finite structures on which LFP collapses to FO, and both are properly contained in PTIME. McColm [McC90] was the first 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 ....

G. L. McColm. When is arithmetic possible? Annals of Pure and Applied Logic, 50:29--51, 1990.

.... p in prenex normal form with prefix p, over a single binary relation, such that for all sentences in prenex normal form, if is equivalent to p , then p can be embedded in the prefix of . We leave its precise statement to Section 1. This also resolves a conjecture of Gradel and McColm [1], explained below. 1 Background and statement of the main theorem 1.1 Terminology and definitions We adopt the following terminology and conventions. We will consider (fragments of) first order logic (FO) and infinitary logic (L1 ) which allows infinitary conjunctions and disjunctions. ....

....prefix p, we define the prefix class FO(p) as the set f j is a FO formula in prenex normal form such that pr( 6 pg. More generally, we want to assign a quantifier structure to every (FO and) L1 formula, which will be a set of prefixes. The following definition is from Gradel and McColm [1]. Definition 1.3 The quantifier structure of a formula 2 L1 ; qs( is defined inductively as follows. i) If is a literal, then qs( ii) If = V i i or W i i , then qs( S i qs( iii) If = 9x , resp. 8x , then qs( 9 qs( qs( f9g, resp. qs( 8 qs( ....

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E. Gradel and G. McColm. Hierarchies in transitive closure logic, stratified Datalog and infinitary logic. Annals of Pure and Applied Logic, 77:166--199, 1996.

.... 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, ....

G. L. McColm. When is arithmetic possible? Annals of Pure and Applied Logic, 50:29--51, 1990.

....the logics are equivalent and both of them, indeed even L 1 , collapse to first order logic. Kolaitis and Vardi [KV92a] initiated an investigation 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 ....

....let m be the m th stage of the induction determined by : m is uniformly definable in L 2k over the class of finite structures. Hence, the least fixed point of is uniformly definable in L 2k 1 over the class of finite structures. The following definition was introduced by McColm [McC90]. Definition7. A class C of structures is proficient, if there is some positive formula such that sup(fjj jj A j A 2 Cg) McColm [McC90] formulated two conjectures, which taken together state that the following three conditions are equivalent for any class of structures C. 1. C is not ....

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G. L. McColm. When is arithmetic possible? Annals of Pure and Applied Logic, 50:29--51, 1990.

....Type k (A; s) The notion of bounded variable element types (Definition 2.3) provides a useful tool for the study of inductive logics (see [5] and [4] In particular, it can be used to identify those classes of structures where inductive definitions are no more powerful than firstorder logic. In [20], McColm conjectured that the following three conditions are equivalent for any class of finite structures C : 1. For every R positive first order formula ; sup(fjj jj A j A 2 Cg) 2. Every query on C definable in LFP is definable in FO ; 3. Every query on C definable in L 1 is ....

....are equivalent for any class of finite structures C : 1. For every R positive first order formula ; sup(fjj jj A j A 2 Cg) 2. Every query on C definable in LFP is definable 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 ....

G. L. McColm. When is arithmetic possible? Annals of Pure and Applied Logic, 50:29--51, 1990.

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