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The Power of Counting Logics on Restricted Classes of Finite Structures
[chapter]

*
Lecture Notes in Computer Science
*

that

doi:10.1007/978-3-540-74915-8_10
fatcat:vrbvnsu6gfbjjcn6abzya3ju6q
*the**logic*does capture P*on*specific*classes**of**structures*. ... Although Cai, Fürer and Immerman have shown that fixedpoint*logic*with*counting*(IFP + C) does not express all polynomialtime properties*of**finite**structures*, there have been a number*of*results demonstrating ... There is a growing body*of*work studying*the**finite*model theory*of**restricted**classes**of**structures*, where*the**restrictions*are essentially borrowed from graph*structure*theory. ...##
###
Reflections on Finite Model Theory

2007
*
22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
*

Main Finding: All major computational complexity

doi:10.1109/lics.2007.39
dblp:conf/lics/Kolaitis07
fatcat:3eulyaqcgbfdzihya6kzjxm2ru
*classes*, including P, NP, and PSPACE, can be characterized in terms*of*definability in various*logics**on**classes**of**finite**structures*. ... Reinforces*the*unity*of*computation and*logic*. Yields machine-independent characterizations*of*computational complexity*classes*. ...*Restricted**Classes**of**Finite**Structures*Progressive shift*of*emphasis from*the**class**of*all*finite**structures*to*restricted**classes**of**finite**structures*. ...##
###
On the expressive power of counting

1995
*
Theoretical Computer Science
*

It is essential in

doi:10.1016/0304-3975(95)00026-s
fatcat:vqzgtgau2ne65dtgzzv3hcbexu
*the*case*of*unordered*structures*. Our aim is to understand*the*expressive*power*gained with a limited*counting*ability. ... We investigate*the*expressive*power**of*various extensions*of*first-order, inductive, and infinitary*logic*with*counting*quantifiers. ... Acknowledgements We are greatly indebted to Guy Fayolle for*the*collaboration*on**the*proof*of*Theorem 5.3. We also thank Jianwen Su for comments*on*an earlier draft. ...##
###
On the Bisimulation Invariant Fragment of Monadic Σ1 in the Finite
[chapter]

2004
*
Lecture Notes in Computer Science
*

We investigate

doi:10.1007/978-3-540-30538-5_19
fatcat:v5ygznzd5zcprnovcpma4hm3xq
*the*expressive*power**of*existential monadic second-order*logic*(monadic Σ1)*on**finite*transition systems. ... We show that*on**finite*unary transition systems*the*bisimulation invariant fragment*of*monadic Σ1 is equivalent to bisimulation-invariant monadic second order*logic*itself or, equivalently,*the*mu-calculus ...*One*reason why*the*question*of**the*equivalence*of*these*logics*is so different in*the**finite*is that, once we*restrict*ourselves to*finite**structures*, we no longer have a tree model property. ...##
###
Page 4089 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

Results are given

*on*formulas with a bounded number*of*variables over*finite**structures*, and these results are applied to other ques- tions in*finite*model theory, such as*the**power**of**logics*with fixed ... In particular,*the**class**of*planar graphs is not in Lix,(Q,).” 98g:03083 03C13 03C75 68Q15 Otto, Martin (D-AACH-GI; Aachen)*The*expressive*power**of*fixed-point*logic*with*counting*. ...##
###
Characterising Choiceless Polynomial Time with First-Order Interpretations

2015
*
2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
*

It is a strict extension

doi:10.1109/lics.2015.68
dblp:conf/lics/GradelPSK15
fatcat:o6zudj655jdxzk54icw22ulauu
*of*fixed-point*logic*with*counting*, but to date*the*question is open whether it expresses all polynomial-time properties*of**finite**structures*. ... While this is very convenient and*powerful*for*the*design*of*abstract computations*on**structures*, it makes*the*analysis*of**the*expressive*power**of*CPT rather difficult. ... If*one*imposes no further*restriction**on*BGS-*logic*then every decidable*class**of**structures*can be defined in BGS-*logic*. ...##
###
Is Polynomial Time Choiceless?
[chapter]

2015
*
Lecture Notes in Computer Science
*

A long time ago, Yuri Gurevich made precise

doi:10.1007/978-3-319-23534-9_11
fatcat:n6rczztse5g4nezyzahgrmis3q
*the*problem*of*whether there is a*logic*capturing polynomial-time*on*arbitrary*finite**structures*, and conjectured that no such*logic*exists. ... We survey some recent results*on*this*logic*. ... It follows that*one*-dimensional PIL, when evaluated*on**the*expansions*of**finite**structures*by an ordered numerical sort, has precisely*the*expressive*power**of*FPC. ...##
###
Counting Proportions of Sets: Expressive Power with Almost Order
[chapter]

2006
*
Lecture Notes in Computer Science
*

When

doi:10.1007/11682462_14
fatcat:eyo4z4oitfg6df3xbbchve53ve
*restricted*to monadic second order variables our*logic**of*proportional quantifiers admits a semantic approximation based*on*almost linear orders, which is not as weak as other known*logics*with*counting*... When*restricted*to monadic second order variables our*logic**of*proportional quantifiers admits a semantic approximation based*on*almost linear orders, which is not as weak as other known*logics*with*counting*...*of**the*formulae*on**finite**structures*where all predicates are*restricted*to act subject to an integer modulo. ...##
###
The monadic theory of finite representations of infinite words

2007
*
Information Processing Letters
*

*classes*

*of*graphs coincide with

*classes*

*of*graphs definable by means

*of*(an extension

*of*)

*finite*state ω-word automata. ... We show that,

*on*these graphs,

*the*bisimulation invariant fragment

*of*monadic Σ1 equals

*the*bisimulation invariant fragment

*of*monadic second order

*logic*itself, and that MSO-definable bisimulation closed ... We say that a

*class*

*of*graphs is (

*counting*) bisimulation closed in

*the*

*finite*when

*the*above property holds

*restricted*to

*finite*graphs only. ...

##
###
Page 895 of Mathematical Reviews Vol. , Issue 2000b
[page]

2000
*
Mathematical Reviews
*

ISBN 3-540-62037-0
Descriptive complexity theory, a part

*of**finite*model theory, fo- cuses*on*studying*the*relationship between*the*computational complexity*of*a*class**of**finite**structures*and its definability ...*The*author diagnoses this shortfall in*the*expressive*power**of*first-order*logic*as being due to its lacking two features: recursion and*counting*. ...##
###
Generalized quantifiers and pebble games on finite structures

1995
*
Annals of Pure and Applied Logic
*

First-order

doi:10.1016/0168-0072(94)00025-x
fatcat:awup6imfonb4tfu3ovybcfyvhe
*logic*is known to have a severely limited expressive*power**on**finite**structures*. ... quantifiers*on**finite**structures*. ... We allow*the**logic*to be defined relative to a*restricted**class**of**structures*, like*the**class**of*all*finite**structures*, or*the**class**of*all ordered*finite**structures*. ...##
###
Logical Definability of Counting Functions

1996
*
Journal of computer and system sciences (Print)
*

For a

doi:10.1006/jcss.1996.0069
fatcat:fksc6h5r2jdg5nohcko3pxeawq
*logic*L, *L is*the**class**of*functions*on**finite**structures**counting**the*tuples (T , cÄ ) satisfying a given formula (T , cÄ ) in L. ...*The*most well studied*class**of**counting*functions is *P, which consists*of**the*functions*counting**the*accepting computation paths*of*a nondeterministic polynomial-time Turing machine. ...*THE**POWER**OF**LOGICAL**COUNTING**CLASSES*We now investigate*the**power**of**FO and*classes**L for*logics*L that extend FO. ...##
###
Finite Model Reasoning in Expressive Fragments of First-Order Logic

2017
*
Electronic Proceedings in Theoretical Computer Science
*

*the*image

*of*

*the*standard translation

*of*modal

*logic*to first-order

*logic*. ... This applies most notably to

*the*guarded fragment, where quantifiers are appropriately relativized by atoms, and

*the*fragment defined by

*restricting*

*the*number

*of*variables to two. ...

*Restricted*

*classes*

*of*

*structures*In modal correspondence theory various conditions

*on*

*the*accessibility relations allow

*one*to

*restrict*

*the*

*class*

*of*Kripke

*structures*considered, e.g. to transitive

*structures*...

##
###
Graded modal logic and counting bisimulation
[article]

2019
*
arXiv
*
pre-print

We focus

arXiv:1910.00039v1
fatcat:v7htrysqvbep7kx7ayxdtnytte
*on*showing expressive completeness*of*graded multi-modal*logic*for those first-order properties*of*pointed Kripke*structures*that are preserved under*counting*bisimulation equivalence among all ... or among just all*finite*pointed Kripke*structures*. ... signature*of*ϕ; (ii) over*the**class**of*all*finite*pointed Kripke*structures*ϕ is expressible in graded modal*logic*CML: ϕ ≡ fin ϕ ′ for some ϕ ∈ CML. ...##
###
Finite and Algorithmic Model Theory (Dagstuhl Seminar 17361)

2018
*
Dagstuhl Reports
*

This report documents

doi:10.4230/dagrep.7.9.1
dblp:journals/dagstuhl-reports/DawarGKS17
fatcat:teyilwvdgrd6tch2gdb4d4dd5m
*the*program and*the*outcomes*of*Dagstuhl Seminar 17361 "*Finite*and Algorithmic Model Theory". ... We formalize agent-based models as stochastic processes whose states are metafinite models, and we define a notion*of*abstraction. ... Our main results are conditions that imply an abstraction is sound, and further conditions that imply it preserves*the*Markov property. ...
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