Dot-Depth and Monadic Quantifier Alternation over Pictures — Extended Abstract

2000 ◽  
pp. 169-178
Author(s):  
Oliver Matz
Keyword(s):  
Quantifier Alternation

The tale of the quantifier alternation hierarchy of first-order logic over words

2015 ◽  
Vol 2 (3) ◽  
pp. 4-17 ◽  
Author(s):  
Thomas Place ◽  
Marc Zeitoun
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Quantifier Alternation

scholarly journals Descriptive complexity of finite structures: Saving the quantifier rank

2005 ◽  
Vol 70 (2) ◽  
pp. 419-450 ◽  
Author(s):  
Oleg Pikhurko ◽  
Oleg Verbitsky
Keyword(s):  
Automorphism Group ◽  
Descriptive Complexity ◽  
First Order ◽  
Finite Structures ◽  
Universal Quantifiers ◽  
Quantifier Alternation

AbstractWe say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M′ over the same vocabulary if Φ is true on M but false on M′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M′.We prove that every n-element structure M is identifiable by a formula with quantifier rank less than and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification.The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than quantifiers suffice to identify M and. as long as we keep the number of universal quantifiers bounded by a constant, at total quantifiers are necessary.

scholarly journals First order quantifiers in monadic second order logic

2004 ◽  
Vol 69 (1) ◽  
pp. 118-136 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wafik Boulos Lotfallah
Keyword(s):  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Reachability Problem ◽  
Open Problems ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Quantifier Alternation

AbstractThis 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].

scholarly journals Going Higher in First-Order Quantifier Alternation Hierarchies on Words

2019 ◽  
Vol 66 (2) ◽  
pp. 1-65 ◽  
Author(s):  
Thomas Place ◽  
Marc Zeitoun
Keyword(s):  
First Order ◽  
Quantifier Alternation

scholarly journals The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

2002 ◽  
Vol 179 (2) ◽  
pp. 356-383 ◽  
Author(s):  
Oliver Matz ◽  
Nicole Schweikardt ◽  
Wolfgang Thomas
Keyword(s):  
Quantifier Alternation
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