Definability hierarchies of generalized quantifiers

Hella, Lauri

doi:10.1016/0168-0072(89)90070-5

Cited by 65 publications

(72 citation statements)

References 16 publications

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“…The bijective game is defined by recursion on n as follows. The importance of the bijective Ehrenfeucht-Fraïssé game stems from the following result, which is proved in [4,5].…”

Section: Optionmentioning

confidence: 99%

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A local normal form theorem for infinitary logic with unary quantifiers

Keisler¹,

Lotfallah

2005

MLQ - Math. Log. Quart.

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We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ω(Q u ) ω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ω(Q u ) ω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (∃ ≥i y) ψ(y), where ψ(y) has counting quantifiers restricted to the (2 n−1 − 1)-neighborhood of y.

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“…The bijective game is defined by recursion on n as follows. The importance of the bijective Ehrenfeucht-Fraïssé game stems from the following result, which is proved in [4,5].…”

Section: Optionmentioning

confidence: 99%

“…n is characterized by the so called bijective Ehrenfeucht-Fraïssé game, introduced in [4]. The bijective game is played on two models by two players, Spoiler and Duplicator.…”

Section: Optionmentioning

confidence: 99%

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A local normal form theorem for infinitary logic with unary quantifiers

Keisler¹,

Lotfallah

2005

MLQ - Math. Log. Quart.

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“…They believed that similar arguments can be used to show that F O(F n ) F O(F n+1 ) but they did not work this out. Meanwhile, using a general method of analyzing generalized quantifiers, Hella showed in [10] that each of the hierarchies F O(H k n ) and F O(F k n ) (fixing either n or k) does not collapse at a finite level. He also showed some undefinability results between the levels of F O(H k n ) and F O(F k n ).…”

Section: Henkin and Function Quantifiersmentioning

confidence: 99%

“…Also, if w starts with any other generalized quantifier, we just use the appropriate move sequence described in the literature of logical games (see [3,17,10,13]). Finally we have: The following theorem clarifies the role of the class games.…”

Section: Examples Identifying Equivalent Sentences First Order Logicmentioning

confidence: 99%

An Ehrenfeucht‐Fraïssé class game

Lotfallah

2004

Mathematical Logic Qtrly

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This paper introduces a new Ehrenfeucht-Fraïssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai-Fagin game to the case when there are several alternating (coloring) moves played in different models. The game allows Duplicator to delay her choices of the models till (practically) the very end of the game, making it easier for her to win. This adds on the toolkit of winning strategies for Duplicator in Ehrenfeucht-Fraïssé type games and opens up new methods for tackling some open problems in descriptive complexity theory. As an application of the class game, it is shown that, if m is a power of a prime, then first order logic augmented with function quantifiers F 1 n of arity 1 and height n < m can not express that the size of the model is divisible by m. This, together with some new expressibility results for Henkin quantifiers H 1 n gives some new separation results on the class of finite models among various F 1 n on one hand and between F 1 n and H 1 n on the other. Since function quantifiers involve a bounded type of second order existential quantifiers, the class game (in a sense) solves an open problem raised by Fagin, which asks for some inexpressibility result using a winning strategy by Duplicator in a game that involves more than one coloring round.

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