Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification

Autores
Allamigeon, Xavier; Fahrenberg, Uli; Gaubert, Stéphane; Katz, Ricardo David; Legay, Axel
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata). 
Fil: Allamigeon, Xavier. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; Francia. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia
Fil: Fahrenberg, Uli. Institut National de Recherche en Informatique et en Automatique; Francia
Fil: Gaubert, Stéphane. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; Francia
Fil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y Sistemas; Argentina
Fil: Legay, Axel. Institut National de Recherche en Informatique et en Automatique; Francia
Materia
Tropical Polyhedra
Strict Inequalities
Fourier-Motzkin Elimination
Mean Payoff Games
Real-Time Verification
Timed Automata
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/4812
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/4812
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Tropical Fourier-Motzkin Elimination, with an Application to Real-Time VerificationAllamigeon, XavierFahrenberg, UliGaubert, StéphaneKatz, Ricardo DavidLegay, AxelTropical PolyhedraStrict InequalitiesFourier-Motzkin EliminationMean Payoff GamesReal-Time VerificationTimed Automatahttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata). Fil: Allamigeon, Xavier. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; Francia. Ecole Polytechnique. Centre de Mathematiques Appliquees; FranciaFil: Fahrenberg, Uli. Institut National de Recherche en Informatique et en Automatique; FranciaFil: Gaubert, Stéphane. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; FranciaFil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y Sistemas; ArgentinaFil: Legay, Axel. Institut National de Recherche en Informatique et en Automatique; FranciaWorld Scientific2014-08info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/4812Allamigeon, Xavier; Fahrenberg, Uli; Gaubert, Stéphane; Katz, Ricardo David; Legay, Axel; Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification; World Scientific; International Journal of Algebra and Computation; 24; 5; 8-2014; 569-6070218-1967enginfo:eu-repo/semantics/altIdentifier/url/http://www.worldscientific.com/doi/abs/10.1142/S0218196714500258info:eu-repo/semantics/altIdentifier/url/http://arxiv.org/abs/1308.2122info:eu-repo/semantics/altIdentifier/url/https://hal.inria.fr/hal-01087367info:eu-repo/semantics/altIdentifier/doi/info:eu-repo/semantics/altIdentifier/doi/10.1142/S0218196714500258info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:52:04Zoai:ri.conicet.gov.ar:11336/4812instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:52:04.74CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
title Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
spellingShingle Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
Allamigeon, Xavier
Tropical Polyhedra
Strict Inequalities
Fourier-Motzkin Elimination
Mean Payoff Games
Real-Time Verification
Timed Automata
title_short Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
title_full Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
title_fullStr Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
title_full_unstemmed Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
title_sort Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification
dc.creator.none.fl_str_mv Allamigeon, Xavier
Fahrenberg, Uli
Gaubert, Stéphane
Katz, Ricardo David
Legay, Axel
author Allamigeon, Xavier
author_facet Allamigeon, Xavier
Fahrenberg, Uli
Gaubert, Stéphane
Katz, Ricardo David
Legay, Axel
author_role author
author2 Fahrenberg, Uli
Gaubert, Stéphane
Katz, Ricardo David
Legay, Axel
author2_role author
author
author
author
dc.subject.none.fl_str_mv Tropical Polyhedra
Strict Inequalities
Fourier-Motzkin Elimination
Mean Payoff Games
Real-Time Verification
Timed Automata
topic Tropical Polyhedra
Strict Inequalities
Fourier-Motzkin Elimination
Mean Payoff Games
Real-Time Verification
Timed Automata
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata). 
Fil: Allamigeon, Xavier. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; Francia. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia
Fil: Fahrenberg, Uli. Institut National de Recherche en Informatique et en Automatique; Francia
Fil: Gaubert, Stéphane. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; Francia
Fil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y Sistemas; Argentina
Fil: Legay, Axel. Institut National de Recherche en Informatique et en Automatique; Francia
description We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata). 
publishDate 2014
dc.date.none.fl_str_mv 2014-08
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/4812
Allamigeon, Xavier; Fahrenberg, Uli; Gaubert, Stéphane; Katz, Ricardo David; Legay, Axel; Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification; World Scientific; International Journal of Algebra and Computation; 24; 5; 8-2014; 569-607
0218-1967
url http://hdl.handle.net/11336/4812
identifier_str_mv Allamigeon, Xavier; Fahrenberg, Uli; Gaubert, Stéphane; Katz, Ricardo David; Legay, Axel; Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification; World Scientific; International Journal of Algebra and Computation; 24; 5; 8-2014; 569-607
0218-1967
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.worldscientific.com/doi/abs/10.1142/S0218196714500258
info:eu-repo/semantics/altIdentifier/url/http://arxiv.org/abs/1308.2122
info:eu-repo/semantics/altIdentifier/url/https://hal.inria.fr/hal-01087367
info:eu-repo/semantics/altIdentifier/doi/
info:eu-repo/semantics/altIdentifier/doi/10.1142/S0218196714500258
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv World Scientific
publisher.none.fl_str_mv World Scientific
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1844613598439014400
score 13.070432

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