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Deciding first order logic properties of matroids
[article]

2011
*
arXiv
*
pre-print

Here, we introduce an analogous notion for

arXiv:1108.5457v1
fatcat:sizqdul2nzechhgavbtpknxmby
*matroids*(locally bounded branch-width) and show the existence*of*a fixed parameter algorithm for*first**order**logic**properties*in classes*of*regular*matroids*... ACM 48 (2006), 1184-1206] introduced a notion*of*graph classes with locally bounded tree-width and established that every*first**order**logic**property*can be*decided*in almost linear time in such a graph ... Acknowledgement The authors would like to thank Ken-ichi Kawarabayashi for discussing the details*of*his fixed parameter algorithm for computing multiway cuts given in [18] during the NII workshop-Graph ...##
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Trees, grids, and MSO decidability: From graphs to matroids

2006
*
Theoretical Computer Science
*

(

doi:10.1016/j.tcs.2005.10.006
fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4
*Matroids*can be viewed as a wide generalization*of*graphs, and they seem to capture some second*order**properties*in a more suitable way than graphs themselves, cf. the recent development in*matroid*structure ... Monadic second*order*(MSO)*logic*has proved to be a useful tool in many areas*of*application, reaching from*decidability*and complexity to picture processing, correctness*of*programs and parallel processes ... Courcelle, the organizer*of*the workshop on "*Logic*and Graph Transformations" at the ICGT 2004 conference, for inviting us to present our research results there. ...##
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On Decidability of MSO Theories of Representable Matroids
[chapter]

2004
*
Lecture Notes in Computer Science
*

We show that, for every finite field , the class

doi:10.1007/978-3-540-28639-4_9
fatcat:36run6tgjrhtdpcu7ir6hdnxze
*of*allrepresentable*matroids**of*branch-width at most a constant t has a*decidable*MSO theory. ... In the other direction, we prove that every class*of*-representable*matroids*with a*decidable*MSO theory must have uniformly bounded branch-width. ... Nishimura, the organizers*of*the Dagstuhl Seminar in 2003 on "'Fixed Parameter Algorithms"' for inviting us to this interesting workshop, where the idea for this paper was born. ...##
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Monadic second-order model-checking on decomposable matroids
[article]

2011
*
arXiv
*
pre-print

We

arXiv:0908.4499v2
fatcat:zslper2a7bftbhigjrk4uv6zqy
*first*give a proof*of*the polynomial time model-checking*of*monadic second-*order*formulas on representable*matroids**of*bounded branch-width, by reduction to monadic second-*order*formulas on trees. ... We also provide a link between our*logical*approach and a grammar that allows to build*matroids**of*bounded branch-width. ... I am also very grateful to Brice Minaud and Pierre Clairambault for discussions we had respectively about*matroid*representations and categories. ...##
###
Decomposition width of matroids

2012
*
Discrete Applied Mathematics
*

B 96 (2006), 325-351] showed that every

doi:10.1016/j.dam.2011.03.016
fatcat:rl7igzidhbeino3viz3xljlc2u
*matroid**property*expressible in the monadic second-*order**logic*can be*decided*in linear time for*matroids*with bounded branch-width that are represented over finite ... Hliněný, Branch-width, parse trees, and monadic second-*order**logic*for*matroids*, J. Combin. Theory Ser. ... The comments*of*the anonymous referees on the conference version*of*the paper that helped to improve and clarify the presentation a lot and are also greatly appreciated. ...##
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Decomposition Width of Matroids
[chapter]

2010
*
Lecture Notes in Computer Science
*

B 96 (2006), 325-351] showed that every

doi:10.1007/978-3-642-14165-2_6
fatcat:5kkgy24nrnbkjchzozeiysgtmi
*matroid**property*expressible in the monadic second-*order**logic*can be*decided*in linear time for*matroids*with bounded branch-width that are represented over finite ... Hliněný, Branch-width, parse trees, and monadic second-*order**logic*for*matroids*, J. Combin. Theory Ser. ... The comments*of*the anonymous referees on the conference version*of*the paper that helped to improve and clarify the presentation a lot and are also greatly appreciated. ...##
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Amalgam width of matroids
[article]

2013
*
arXiv
*
pre-print

In particular, any

arXiv:1304.0299v2
fatcat:b7tgznfdfvbddkpboc2tadfyce
*property*expressible in the monadic second*order**logic*can be*decided*in linear time for*matroids*with bounded amalgam-width. ... We introduce a new*matroid*width parameter based on the operation*of**matroid*amalgamation, which we call amalgam-width. ... Research*of*this phenomenon culminated in proving a general theorem*of*Courcelle [2] , which asserts that any graph*property*expressible in the monadic second*order*(MSO)*logic*can be*decided*in linear ...##
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Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids
[chapter]

2003
*
Lecture Notes in Computer Science
*

We prove that if M is a family

doi:10.1007/3-540-36494-3_29
fatcat:wsejm3qjovbjbbt6nfeetf4ypm
*of**matroids*described by a sentence in the monadic second-*order**logic**of**matroids*, then there is a finite tree automaton accepting exactly those parse trees which build vector ... We introduce "*matroid*parse trees" which, using only a limited amount*of*information at each node, can build up the vector representations*of**matroids**of*bounded branch-width over a finite field. ... Applying the language*of*MSO*logic*to such*matroid*structures gives the monadic second-*order*theory M S M*of**matroids*: Definition. ...##
###
Branch-width, parse trees, and monadic second-order logic for matroids

2006
*
Journal of combinatorial theory. Series B (Print)
*

We prove that if M is a family

doi:10.1016/j.jctb.2005.08.005
fatcat:rhduqrqesjepdk7elikt2beiku
*of**matroids*described by a sentence in the monadic second-*order**logic**of**matroids*, then there is a finite tree automaton accepting exactly those parse trees which build vector ... We introduce "*matroid*parse trees" which, using only a limited amount*of*information at each node, can build up the vector representations*of**matroids**of*bounded branch-width over a finite field. ... Applying the language*of*MSO*logic*to such*matroid*structures gives the monadic second-*order*theory M S M*of**matroids*: Definition. ...##
###
Amalgam Width of Matroids
[chapter]

2013
*
Lecture Notes in Computer Science
*

In particular, any

doi:10.1007/978-3-319-03898-8_23
fatcat:yod4q3b5tfgu5mfeblrxqu7rn4
*property*expressible in the monadic second*order**logic*can be*decided*in linear time for*matroids*with bounded amalgam-width. ... We introduce a new*matroid*width parameter based on the operation*of**matroid*amalgamation, which we call amalgam-width. ... Besides MSO*properties*,*first**order**properties*for*matroids*have been studied from algorithmic point*of*view in [11] . ...##
###
Decomposition width - a new width parameter for matroids
[article]

2009
*
arXiv
*
pre-print

We introduce a new width parameter for

arXiv:0904.2785v1
fatcat:jyyg4ed2praelecuxtzim7g5gu
*matroids*called decomposition width and prove that every*matroid**property*expressible in the monadic second*order**logic*can be computed in linear time for*matroids*...*of*Hlineny [J. ... Acknowledgement The author would like to thank Bruno Courcelle for his valuable comments on the conference paper [18] and Ondřej Pangrác for his insightful comments on the content*of*this paper as well ...##
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Monadic second-order model-checking on decomposable matroids

2011
*
Discrete Applied Mathematics
*

We

doi:10.1016/j.dam.2011.02.005
fatcat:33f6zing5zg4jnuqy4arvae47a
*first*give a proof*of*the polynomial time model-checking*of*monadic second-*order*formulas on representable*matroids**of*bounded branch-width, by reduction to monadic second-*order*formulas on trees. ... We also provide a link between our*logical*approach and a grammar that allows one to build*matroids**of*bounded branch-width. ... I am also very grateful to Brice Minaud and Pierre Clairambault for early discussions on*matroids*and pushout and to my advisor Arnaud Durand for his support and help while writing this article. ...##
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Some Hard Problems on Matroid Spikes

2007
*
Theory of Computing Systems
*

xj − 1
j∈[1,n], fj ∈X
c) Otherwise, X is not a basis

doi:10.1007/s00224-007-1307-5
fatcat:m2m4ooibbne2leibtycu5xd34y
*of*S. 3 Easy and Hard MSO*Properties*Monadic second-*order*(MSO)*logic*is famous for its well-balanced ... Namely, the question whether a given spike is the free spike is co-N P -hard (though the*property*itself is definable in monadic second-*order**logic*); and the task to compute the ...##
###
Width Parameters Beyond Tree-width and their Applications

2007
*
Computer journal
*

In this contribution, we demonstrate how 'width' parameters

doi:10.1093/comjnl/bxm052
fatcat:n4nbasuvubc5lkajkskqk22fey
*of*graphs and generalized structures (such as*matroids*or hypergraphs), can be used to improve the design*of*parameterized algorithms and the ... Besides the very successful concept*of*tree-width (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs*of*bounded treewidth. ... Dermietzel for his help with generating some*of*the figures. Our last, special thanks we reserve for the editors*of*this volume for the invitation and their constant encouragement. ...##
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Vertex-minors, monadic second-order logic, and a conjecture by Seese

2007
*
Journal of combinatorial theory. Series B (Print)
*

We obtain a slight weakening

doi:10.1016/j.jctb.2006.04.003
fatcat:3dfb5tqexndablnknwykkbcdlu
*of*a conjecture by Seese stating that sets*of*graphs having a*decidable*satisfiability problem for monadic second-*order**logic*have bounded clique-width. ... We prove that one can express the vertex-minor relation on finite undirected graphs by formulas*of*monadic second-*order**logic*(with no edge set quantification) extended with a predicate expressing that ... Section 5 reviews monadic second-*order**logic*and its use for expressing*properties*and transformations*of*graphs,*matroids*, and isotropic systems. ...
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