Solvability of Matrix-Exponential Equations
We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there e...
Gespeichert in:
| Veröffentlicht in: | Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science S. 798 - 806 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Tagungsbericht |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY, USA
ACM
05.07.2016
|
| Schriftenreihe: | ACM Conferences |
| Schlagworte: | |
| ISBN: | 9781450343916, 1450343910 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, ..., tk such that
We show that this problem is undecidable in general, but decidable under the assumption that the matrices A1, ..., Ak commute. Our results have applications to reachability problems for linear hybrid automata.
Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem. |
|---|---|
| AbstractList | We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, ..., tk such that
We show that this problem is undecidable in general, but decidable under the assumption that the matrices A1, ..., Ak commute. Our results have applications to reachability problems for linear hybrid automata.
Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem. |
| Author | Sousa-Pinto, Joao Pouly, Amaury Ouaknine, Joel Worrell, James |
| Author_xml | – sequence: 1 givenname: Joel surname: Ouaknine fullname: Ouaknine, Joel email: joel@cs.ox.ac.uk organization: University of Oxford – sequence: 2 givenname: Amaury surname: Pouly fullname: Pouly, Amaury email: amaury.pouly@cs.ox.ac.uk organization: University of Oxford – sequence: 3 givenname: Joao surname: Sousa-Pinto fullname: Sousa-Pinto, Joao email: jspinto@cs.ox.ac.uk organization: University of Oxford – sequence: 4 givenname: James surname: Worrell fullname: Worrell, James email: jbw@cs.ox.ac.uk organization: University of Oxford |
| BookMark | eNqNj8FKxDAURQMqqGPXbrsUpPW9pMlrljLUURhxoa5DkjYQrY1Oq4x_b8V-gKu7uJfDPafscEhDx9g5QolYySuuhZAkyzkrKeoDlmmq5wJEJTSqY5aN4wsAcKRaA56wy8fUf1kX-zh95ynk93baxX3R7N9n8DBF2-fNx6edYhrGM3YUbD922ZIr9nzTPK1vi-3D5m59vS0s5zQV6GQLAYlICHDgvNNe1eDJe0IdWuWD9uhJSQmhIse1wrajFpEQ7Xx1xS7-uNa_GZfS62gQzK-fWfzM4jdPy39OjdvFLogftstRug |
| ContentType | Conference Proceeding |
| Copyright | 2016 Owner/Author |
| Copyright_xml | – notice: 2016 Owner/Author |
| DOI | 10.1145/2933575.2934538 |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science |
| EndPage | 806 |
| GroupedDBID | 6IE 6IF 6IG 6IL 6IN AAJGR ACM ADPZR ALMA_UNASSIGNED_HOLDINGS APO BEFXN BFFAM BGNUA BKEBE BPEOZ CBEJK GUFHI IEGSK IJVOP OCL RIB RIC RIE RIL RIO |
| ID | FETCH-LOGICAL-a227t-1b5d0f1777330b0bcb9c680c7cc719fd6cf9c1c76550f47b2961de7d11711a503 |
| ISBN | 9781450343916 1450343910 |
| ISICitedReferencesCount | 4 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000387609200081&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| IngestDate | Wed Jan 31 06:44:04 EST 2024 Wed Jan 31 06:45:21 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | false |
| IsScholarly | true |
| Keywords | matrix logarithms exponential matrices commuting matrices matrix reachability hybrid automata |
| Language | English |
| License | This work is licensed under a Creative Commons Attribution International 4.0 License. |
| LinkModel | OpenURL |
| MeetingName | LICS '16: 31st Annual ACM/IEEE Symposium on Logic in Computer Science |
| MergedId | FETCHMERGED-LOGICAL-a227t-1b5d0f1777330b0bcb9c680c7cc719fd6cf9c1c76550f47b2961de7d11711a503 |
| PageCount | 9 |
| ParticipantIDs | acm_books_10_1145_2933575_2934538_brief acm_books_10_1145_2933575_2934538 |
| PublicationCentury | 2000 |
| PublicationDate | 20160705 |
| PublicationDateYYYYMMDD | 2016-07-05 |
| PublicationDate_xml | – month: 07 year: 2016 text: 20160705 day: 05 |
| PublicationDecade | 2010 |
| PublicationPlace | New York, NY, USA |
| PublicationPlace_xml | – name: New York, NY, USA |
| PublicationSeriesTitle | ACM Conferences |
| PublicationTitle | Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science |
| PublicationYear | 2016 |
| Publisher | ACM |
| Publisher_xml | – name: ACM |
| SSID | ssj0002178901 |
| Score | 2.0122957 |
| Snippet | We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square... |
| SourceID | acm |
| SourceType | Publisher |
| StartPage | 798 |
| SubjectTerms | Computing methodologies Computing methodologies -- Symbolic and algebraic manipulation Computing methodologies -- Symbolic and algebraic manipulation -- Symbolic and algebraic algorithms Computing methodologies -- Symbolic and algebraic manipulation -- Symbolic and algebraic algorithms -- Linear algebra algorithms Mathematics of computing Mathematics of computing -- Mathematical analysis Mathematics of computing -- Mathematical analysis -- Numerical analysis Mathematics of computing -- Mathematical analysis -- Numerical analysis -- Computations on matrices |
| Title | Solvability of Matrix-Exponential Equations |
| WOSCitedRecordID | wos000387609200081&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1La9wwEBZN6KGnvlKavnCh0MPi1lpJHukYQkovGxY2hdyMJFuwtGunazts_n1Htux10kDbQy-yGYyE9I2lmdE8CPkgKKS55io2qeIxLxKIFc9FzKwBw5STOXddsQk4P5eXl2oZ3MbqrpwAlKXc7dTVf4UaaQi2D539B7jHTpGA7wg6tgg7tnck4nsPn-VIrIf7f0brZhYS6Z-cLrBrr-LNVjcb77LVbvyFga-5bPsQwL7Mw_DXT3lqVf247vN63_SuM812vYvPdldV6Z2OfIrin-3EBOiNt63-Xq7LYKkvRn-OZdX2Na5PfGKj0a6_qtpax8t12fQ3QpWuxmPD1xEJLjl6iFvxK1rUyFCLSeziLVMGTTu3VzFhvsUt7ZZykbAuMniyw0JftHo4rLt0BfecA9ynzEBRhqE4-gmfHDf2A3IAkPQxfqMdDlUyiTJRF_MXhhtTgQ3Dh5RQSPh8p0sv09jNRCK5eEKO9vON9pg_JQ-K8hl5PKAYBRSfk9kEvKhy0e_gRSN4R-Tbl7OL069xqJQR6_kcmpgakSeOAgBjiUmMNcqmMrFgLVDl8tQ6ZamFFPVRx8HMVUrzAnJKgVKN03xBDksc7SWJUip14bhjMnecaidFLiDhBarlTEstjsl7nHHmGb7O-qh2kYVVycKqHJOPf_wmM8gf7tVf9PaaPNozyxty2Gzb4i15aK-bdb1916H5Cz1mV8k |
| linkProvider | IEEE |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=proceeding&rft.title=Proceedings+of+the+31st+Annual+ACM%2FIEEE+Symposium+on+Logic+in+Computer+Science&rft.atitle=Solvability+of+Matrix-Exponential+Equations&rft.au=Ouaknine%2C+Joel&rft.au=Pouly%2C+Amaury&rft.au=Sousa-Pinto%2C+Joao&rft.au=Worrell%2C+James&rft.series=ACM+Conferences&rft.date=2016-07-05&rft.pub=ACM&rft.isbn=9781450343916&rft.spage=798&rft.epage=806&rft_id=info:doi/10.1145%2F2933575.2934538 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9781450343916/lc.gif&client=summon&freeimage=true |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9781450343916/mc.gif&client=summon&freeimage=true |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9781450343916/sc.gif&client=summon&freeimage=true |