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

2014
*
Logical Methods in Computer Science
*

We study a natural

doi:10.2168/lmcs-10(4:3)2014
fatcat:acvyqeo7xjhqlbwoko2tyjg4sy
*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. ...##
###
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

1996
*
Annals of Pure and Applied Logic
*

For

doi:10.1016/0168-0072(95)00013-5
fatcat:b3iwj3l4sze6plwzp7jw2nmvsu
*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. ...##
###
An analysis of the W*-hierarchy

2007
*
Journal of Symbolic Logic (JSL)
*

Our second main result is a new

doi:10.2178/jsl/1185803622
fatcat:fpme3obepnenbjiwschlyrpbyi
*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] . ...##
###
A double arity hierarchy theorem for transitive closure logic

1996
*
Archive for Mathematical Logic
*

As a consequence,

doi:10.1007/bf01268616
fatcat:jzilabwfofcytdi33oab7uvm5i
*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*. ...##
###
Hierarchies in Dependence Logic

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*...

##
###
Hierarchies in Dependence Logic
[article]

2011
*
arXiv
*
pre-print

We study fragments

arXiv:1105.3324v1
fatcat:4bjkgfnwxrh6lih5cwp5tqlc4e
*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 ∀ * ∃ * . ...##
###
Hierarchies in Fragments of Monadic Strict NP
[chapter]

2007
*
Lecture Notes in Computer Science
*

We expose a

doi:10.1007/978-3-540-73001-9_56
fatcat:vswxnvm4qrdqjkav2uvxztws64
*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 ...##
###
Dependence Logic with a Majority Quantifier

2015
*
Journal of Logic, Language and Information
*

We study

doi:10.1007/s10849-015-9218-3
fatcat:a77odcs2rjhxhgx25ilpitsape
*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*...##
###
Hierarchies in inclusion logic with lax semantics
[article]

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. ...

##
###
Dependence logic with a majority quantifier
[article]

2013
*
arXiv
*
pre-print

We study

arXiv:1109.4750v6
fatcat:bxv7clcmx5h55n5foxjwpugy6e
*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*...##
###
First order quantifiers in monadic second order logic

2004
*
Journal of Symbolic Logic (JSL)
*

As a corollary, if

doi:10.2178/jsl/1080938831
fatcat:toqaorutzvgsndhvmxyucsvryu
*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*. ...##
###
Going Higher in the First-Order Quantifier Alternation Hierarchy on Words
[chapter]

2014
*
Lecture Notes in Computer Science
*

We investigate

doi:10.1007/978-3-662-43951-7_29
fatcat:euvrthf47vffhiewgk5jrapr3m
*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*. ...##
###
Expressivity and Complexity of Dependence Logic
[chapter]

2016
*
Dependence Logic
*

Section 3.3 reviews results

doi:10.1007/978-3-319-31803-5_2
fatcat:rsvfibtfwzcntandlijfcjzhxi
*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*. ...##
###
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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