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On the strictness of the quantifier structure hierarchy in first-order logic

Yuguo He, Martín Escardó
2014 Logical Methods in Computer Science  
We study a natural hierarchy in first-order logic, namely the quantifier structure hierarchy, which gives a systematic classification of first-order formulas based on structural quantifier resource.  ...  Moreover, we prove that this hierarchy is strict even over ordered finite structures, which is interesting in the context of descriptive complexity.  ...  Acknowledgement I am grateful to Anuj Dawar for giving me a lot of suggestions and ideas which improve the paper significantly and to Bjarki Holm for useful discussions and proofreading.  ... 
doi:10.2168/lmcs-10(4:3)2014 fatcat:acvyqeo7xjhqlbwoko2tyjg4sy

Page 8497 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
Queries might be expressed in one of the many logics that have been considered in database theory: first-order logic, fixpoint logic, second-order logic, etc.  ...  In the present paper, the authors extend this to logics beyond first-order, such as fixpoint logic, first-order logic with counting, and second-order logic.  ... 

Arity and alternation in second-order logic

J.A. Makowsky, Y.B. Pnueli
1996 Annals of Pure and Applied Logic  
For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT(FOL) is PSpacrcomplete.  ...  We show that this imposes a proper hierarchy on second-order logic, i.e. for every k, n there are problems not definable in AA(k, n) but definable in AA(k + cl, n + d,) for some cl, d,.  ...  Fagin, who provided us with Proposition 2, and to the participants of the 1994 Oberwolfach Meeting on Finite Model Theory for various useful comments. We are also indebted to B.  ... 
doi:10.1016/0168-0072(95)00013-5 fatcat:b3iwj3l4sze6plwzp7jw2nmvsu

An analysis of the W*-hierarchy

Yijia Chen, Jörg Flum, Martin Grohe
2007 Journal of Symbolic Logic (JSL)  
Our second main result is a new logical characterization of the W*-hierarchy in terms of "Fagin-definable problems."  ...  As a by-product, we also obtain an improvement of our earlier characterization of the hierarchy in terms of model-checking problems.  ...  Characterizations of the W-hierarchy in terms of first-order logic were first obtained by Downey, Fellows, and Regan [6] and later improved by Flum and Grohe [9, 10] .  ... 
doi:10.2178/jsl/1185803622 fatcat:fpme3obepnenbjiwschlyrpbyi

A double arity hierarchy theorem for transitive closure logic

Martin Grohe, Lauri Hella
1996 Archive for Mathematical Logic  
As a consequence, the arity hierarchies of all the familiar forms of fixed point logic are strict simultaneously with respect to the arity of the induction predicates and the arity of generalized quantifiers  ...  In this paper we prove that the k-ary fragment of transitive closure logic is not contained in the extension of the (k − 1)-ary fragment of partial fixed point logic by all (2k − 1)-ary generalized quantifiers  ...  The arity hierarchies of the corresponding quantifiers are strict by our theorem. • The arity hierarchy of the Hamiltonian path quantifier Q HP is strict.  ... 
doi:10.1007/bf01268616 fatcat:jzilabwfofcytdi33oab7uvm5i

Hierarchies in Dependence Logic

Arnaud Durand, Juha Kontinen
2012 ACM Transactions on Computational Logic  
The idea of generalizing first-order logic in this respect goes back to Henkin who introduced the so-called partially ordered quantifiers (Henkin quantifiers) in [Henkin 1961] .  ...  While in first-order logic the order of quantifiers solely determines the dependence relations between variables, in dependence logic more general dependencies between variables can be expressed.  ...  Hierarchies in Dependence Logic  ... 
doi:10.1145/2362355.2362359 fatcat:heolaozcrbg6lnxyhwem6l4j2y

Hierarchies in Dependence Logic [article]

Arnaud Durand, Juha Kontinen
2011 arXiv   pre-print
We study fragments of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas.  ...  We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic.  ...  In Theorem 2.17, the first-order part of φ has the quantifier prefix ∀ * ∃ * .  ... 
arXiv:1105.3324v1 fatcat:4bjkgfnwxrh6lih5cwp5tqlc4e

Hierarchies in Fragments of Monadic Strict NP [chapter]

Barnaby Martin, Florent Madelaine
2007 Lecture Notes in Computer Science  
We expose a strict hierarchy within monotone monadic strict NP without inequalities (MMSNP), based on the number of second-order monadic quantifiers.  ...  We do this by studying a finer strict hierarchy within a class of forbidden patterns problems (FPP), based on the number of permitted colours.  ...  In fact, it is the case that MNP with a single second-order quantifier is as powerful as the whole of monadic second-order logic -not just its existential fragment -on word structures, capturing exactly  ... 
doi:10.1007/978-3-540-73001-9_56 fatcat:vswxnvm4qrdqjkav2uvxztws64

Dependence Logic with a Majority Quantifier

Arnaud Durand, Johannes Ebbing, Juha Kontinen, Heribert Vollmer
2015 Journal of Logic, Language and Information  
We study the extension of dependence logic D by a majority quantifier M over finite structures.  ...  We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities.  ...  In order to achieve this, we might have to introduce new existentially quantified functions and also universal first-order quantifiers (see Theorem 6.15 in [19] ), but the quantifier structure of the  ... 
doi:10.1007/s10849-015-9218-3 fatcat:a77odcs2rjhxhgx25ilpitsape

Hierarchies in inclusion logic with lax semantics [article]

Miika Hannula
2014 arXiv   pre-print
In case of universal quantifiers, the corresponding hierarchy collapses at the first level.  ...  Arity hierarchy is shown to be strict by relating the question to the study of arity hierarchies in fixed point logics.  ...  Inclusion logic, instead, extends first-order logic with inclusion atoms x ⊆ y which express that the set of values of x is included in the set of the values of y.  ... 
arXiv:1401.3235v1 fatcat:tkxqiebtnbfnbhzd7ur25inw2e

Dependence logic with a majority quantifier [article]

Arnaud Durand, Johannes Ebbing, Juha Kontinen, Heribert Vollmer
2013 arXiv   pre-print
We study the extension of dependence logic D by a majority quantifier M over finite structures.  ...  We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities.  ...  In order to achieve this, we might have to introduce new existentially quantified functions and also universal first-order quantifiers (see Theorem 6.15 in [19] ), but the quantifier structure of the  ... 
arXiv:1109.4750v6 fatcat:bxv7clcmx5h55n5foxjwpugy6e

First order quantifiers in monadic second order logic

H. Jerome Keisler, Wafik Boulos Lotfallah
2004 Journal of Symbolic Logic (JSL)  
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  ...  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].  ...  This implies the strictness of all the natural quantifier hierarchies in the positive first order closure of a fragment of monadic second order logic.  ... 
doi:10.2178/jsl/1080938831 fatcat:toqaorutzvgsndhvmxyucsvryu

Going Higher in the First-Order Quantifier Alternation Hierarchy on Words [chapter]

Thomas Place, Marc Zeitoun
2014 Lecture Notes in Computer Science  
We investigate the quantifier alternation hierarchy in firstorder logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas.  ...  We prove that one can decide membership of a regular language to the levels BΣ2 (boolean combination of formulas having only 1 alternation) and Σ3 (formulas having only 2 alternations beginning with an  ...  The most prominent fragment of MSO is first-order logic (FO) equipped with a predicate "<" for the linear-order.  ... 
doi:10.1007/978-3-662-43951-7_29 fatcat:euvrthf47vffhiewgk5jrapr3m

Expressivity and Complexity of Dependence Logic [chapter]

Arnaud Durand, Juha Kontinen, Heribert Vollmer
2016 Dependence Logic  
Section 3.3 reviews results on the complexity of satisfiability and model checking in the (first-order) dependence logic context.  ...  We will first define the lax version of the team semantics for first-order formulas in negation normal form. Below M |= s α refers to satisfaction in first-order logic.  ... 
doi:10.1007/978-3-319-31803-5_2 fatcat:rsvfibtfwzcntandlijfcjzhxi

Page 8387 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
alternation-depth hierarchy is strict on binary trees.  ...  Summary: “In this paper we give a simple proof that the alternation-depth hierarchy of the uw-calculus for binary trees is strict.  ... 
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