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 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].

AROUND DOT-DEPTH ONE

2012 ◽  
Vol 23 (06) ◽  
pp. 1323-1339 ◽  
Author(s):  
MANFRED KUFLEITNER ◽  
ALEXANDER LAUSER
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Membership Problem ◽  
First Order ◽  
Language Classes ◽  
One To One ◽  
Quantifier Alternation

The dot-depth hierarchy is a classification of star-free languages. It is related to the quantifier alternation hierarchy of first-order logic over finite words. We consider subclasses of languages with dot-depth 1/2 and dot-depth 1 obtained by prohibiting the specification of prefixes or suffixes. As it turns out, these language classes are in one-to-one correspondence with fragments of alternation-free first-order logic without min- or max-predicate, respectively. For all fragments, we obtain effective algebraic characterizations. Moreover, we give new proofs for the decidability of the membership problem for dot-depth 1/2 and dot-depth 1.

scholarly journals Decomposable graphs and definitions with no quantifier alternation

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Oleg Pikhurko ◽  
Joel Spencer ◽  
Oleg Verbitsky
Keyword(s):  
The Other ◽  
Connected Components ◽  
Large Graphs ◽  
First Order ◽  
Minimum Number ◽  
International Audience ◽  
Decomposable Graphs ◽  
Number Of Iterations ◽  
Quantifier Alternation ◽  
Binary Logarithm

International audience Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.

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