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First order quantifiers in monadic second order logic

Published online by Cambridge University Press:  12 March 2014

H. Jerome Keisler  and
Wafik Boulos Lotfallah
H. Jerome Keisler
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison WI 53706, USA, E-mail: keisler@math.wisc.edu
Wafik Boulos Lotfallah
Affiliation:
Dept. of Eng. Math. and Phys., Cairo University, Cairo, 11451, Egypt, E-mail: lotfalla@uwalumni.com
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Abstract

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].

We introduce an operation existsn (S) on properties S that says “there are n components having S”. We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.

We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.

We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn ⊈ FO(Σn), Σn ⊈ FO(∆n). and ∆n+1 ⊈ FOB(Σn), solving some open problems raised in [Mat98].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 1 , March 2004 , pp. 118 - 136
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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