A type-based analysis of causality loops in hybrid systems modelers
Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between them. An important step in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to dead...
Uloženo v:
| Vydáno v: | Nonlinear analysis. Hybrid systems Ročník 26; s. 168 - 189 |
|---|---|
| Hlavní autoři: | , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Ltd
01.11.2017
Elsevier |
| Témata: | |
| ISSN: | 1751-570X |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between them. An important step in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to deadlock and prevent the generation of statically scheduled code.
This paper addresses this issue for a hybrid modeling language that combines synchronous data-flow equations with Ordinary Differential Equations (ODEs). We introduce the operator lastx for the left-limit of a signal x. The lastx operator is used to break causality loops and permits a uniform treatment of discrete and continuous state variables. The semantics of the language relies on non-standard analysis, defining an execution as a sequence of infinitesimally small steps. A signal is deemed causally correct when it can be computed sequentially and only changes infinitesimally outside of announced discrete events like zero-crossings. The causality analysis takes the form of a type system that expresses dependencies between signals. In well-typed programs, (i) signals are provably continuous during integration provided that imported external functions are also continuous, and (ii) sequential code can be generated.
The effectiveness of the system is illustrated with several examples written in Zélus, a Lustre-like synchronous language extended with ODEs. |
|---|---|
| AbstractList | Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between them. An important step in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to deadlock and prevent the generation of statically scheduled code.
This paper addresses this issue for a hybrid modeling language that combines synchronous data-flow equations with Ordinary Differential Equations (ODEs). We introduce the operator lastx for the left-limit of a signal x. The lastx operator is used to break causality loops and permits a uniform treatment of discrete and continuous state variables. The semantics of the language relies on non-standard analysis, defining an execution as a sequence of infinitesimally small steps. A signal is deemed causally correct when it can be computed sequentially and only changes infinitesimally outside of announced discrete events like zero-crossings. The causality analysis takes the form of a type system that expresses dependencies between signals. In well-typed programs, (i) signals are provably continuous during integration provided that imported external functions are also continuous, and (ii) sequential code can be generated.
The effectiveness of the system is illustrated with several examples written in Zélus, a Lustre-like synchronous language extended with ODEs. Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete-and continuous-time behaviors with complex interactions between them. An important step in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to deadlock and prevent the generation of statically scheduled code. This paper addresses this issue for a hybrid modeling language that combines synchronous data-flow equations with Ordinary Differential Equations (ODEs). We introduce the operator last x for the left-limit of a signal x. The last x operator is used to break causality loops and permits a uniform treatment of discrete and continuous state variables. The semantics of the language relies on non-standard analysis, defining an execution as a sequence of infinitesimally small steps. A signal is deemed causally correct when it can be computed sequentially and only changes infinitesimally outside of announced discrete events like zero-crossings. The causality analysis takes the form of a type system that expresses dependencies between signals. In well-typed programs, signals are provably continuous during integration provided that imported external functions are also continuous, and sequential code can be generated. The effectiveness of the system is illustrated with several examples written in Zélus, a Lustre-like synchronous language extended with ODEs. |
| Author | Bourke, Timothy Caillaud, Benoit Pouzet, Marc Pagano, Bruno Benveniste, Albert |
| Author_xml | – sequence: 1 givenname: Albert surname: Benveniste fullname: Benveniste, Albert email: Albert.Benveniste@inria.fr organization: Inria Rennes-Bretagne Atlantique, France – sequence: 2 givenname: Timothy orcidid: 0000-0003-0601-4182 surname: Bourke fullname: Bourke, Timothy email: Timothy.Bourke@inria.fr organization: Inria Paris, France – sequence: 3 givenname: Benoit orcidid: 0000-0002-3234-5033 surname: Caillaud fullname: Caillaud, Benoit email: Benoit.Caillaud@inria.fr organization: Inria Rennes-Bretagne Atlantique, France – sequence: 4 givenname: Bruno surname: Pagano fullname: Pagano, Bruno email: Bruno.Pagano@ansys.com organization: ANSYS/Esterel Technologies, France – sequence: 5 givenname: Marc surname: Pouzet fullname: Pouzet, Marc email: Marc.Pouzet@ens.fr organization: Inria Paris, France |
| BackLink | https://inria.hal.science/hal-01549183$$DView record in HAL |
| BookMark | eNp9kDtrwzAUhTWk0CTtH-iktYNdyfJL0CWEviDQpYVuQo9rouBYQVcN-N83JqVDh0z3cDnfGb4FmQ1hAELuOMs54_XDLh_0FvOC8SZnZc5YOSNz3lQ8qxr2dU0WiDvGKlm05ZysVzSNB8iMRnBUD7of0SMNHbX6G3Xv00j7EA5I_UC3o4neURwxwR7pPjjoIeINuep0j3D7e5fk8_npY_2abd5f3tarTWZF06ZMcyNrKIR10nBgotOVkaITTe0aMI0FUwKvqo7XU2RtbS2Xsma8cE4YWYgluT_vbnWvDtHvdRxV0F69rjZq-jFelZK34jh1i3PXxoAYofsDOFOTJrVTkyY1aVKsVCdNJ6j9B1mfdPJhSFH7_jL6eEbhJODoISq0HgYLzkewSbngL-E_RYKH1A |
| CitedBy_id | crossref_primary_10_1145_3342997 crossref_primary_10_1109_JPROC_2018_2858016 |
| Cites_doi | 10.1006/inco.2001.2963 10.1016/S0019-3577(00)88579-1 10.1007/978-3-540-31954-2_2 10.1145/1967677.1967687 10.1007/978-3-662-46663-6_4 10.1109/JPROC.2002.805829 10.1109/5.97300 10.1016/j.jcss.2011.08.009 |
| ContentType | Journal Article |
| Copyright | 2017 Distributed under a Creative Commons Attribution 4.0 International License |
| Copyright_xml | – notice: 2017 – notice: Distributed under a Creative Commons Attribution 4.0 International License |
| DBID | AAYXX CITATION 1XC VOOES |
| DOI | 10.1016/j.nahs.2017.04.004 |
| DatabaseName | CrossRef Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access) |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| EndPage | 189 |
| ExternalDocumentID | oai:HAL:hal-01549183v2 10_1016_j_nahs_2017_04_004 S1751570X17300286 |
| GroupedDBID | --K --M .~1 0R~ 123 1B1 1~. 1~5 4.4 457 4G. 5VS 7-5 71M 8P~ AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAXUO ABAOU ABMAC ABXDB ABYKQ ACAZW ACDAQ ACGFS ACNNM ACRLP ADBBV ADEZE ADMUD ADTZH AEBSH AECPX AEKER AFKWA AFTJW AGHFR AGUBO AGYEJ AHJVU AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR AXJTR BJAXD BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 FDB FIRID FNPLU FYGXN GBLVA HZ~ IHE J1W J9A JJJVA KOM M41 MHUIS MO0 N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. Q38 RIG ROL RPZ SDF SDG SES SPC SPCBC SPD SST SSW SSZ T5K ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO AEIPS AEUPX AFJKZ AFPUW AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD 1XC VOOES |
| ID | FETCH-LOGICAL-c378t-a1b96e23cd9b1e03fa5b93f376d7eb7ceb4e155f16ceb4086cc1996012dd3b923 |
| ISICitedReferencesCount | 3 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000410903700011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 1751-570X |
| IngestDate | Sat Oct 25 11:29:02 EDT 2025 Sat Nov 29 07:01:51 EST 2025 Tue Nov 18 21:58:30 EST 2025 Fri Feb 23 02:29:10 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Type systems Hybrid systems Synchronous programming languages Numerical solver HybridLang Mathematical modelling language |
| Language | English |
| License | Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c378t-a1b96e23cd9b1e03fa5b93f376d7eb7ceb4e155f16ceb4086cc1996012dd3b923 |
| ORCID | 0000-0003-0601-4182 0000-0002-3234-5033 |
| OpenAccessLink | https://inria.hal.science/hal-01549183 |
| PageCount | 22 |
| ParticipantIDs | hal_primary_oai_HAL_hal_01549183v2 crossref_primary_10_1016_j_nahs_2017_04_004 crossref_citationtrail_10_1016_j_nahs_2017_04_004 elsevier_sciencedirect_doi_10_1016_j_nahs_2017_04_004 |
| PublicationCentury | 2000 |
| PublicationDate | November 2017 2017-11-00 2017-11-01 |
| PublicationDateYYYYMMDD | 2017-11-01 |
| PublicationDate_xml | – month: 11 year: 2017 text: November 2017 |
| PublicationDecade | 2010 |
| PublicationTitle | Nonlinear analysis. Hybrid systems |
| PublicationYear | 2017 |
| Publisher | Elsevier Ltd Elsevier |
| Publisher_xml | – name: Elsevier Ltd – name: Elsevier |
| References | Bauer, Schneider (b26) 2010 Carloni, Passerone, Pinto, Sangiovanni-Vincentelli (b1) 2006; 1 E.A. Lee, H. Zheng, Operational semantics of hybrid systems, in: Hybrid Systems: Computation and Control (HSCC), Zurich, Switzerland, 2005, pp. 25–53. Halbwachs, Raymond, Ratel (b5) 1991 Pottier (b20) 2001; 170 Eker, Janneck, Lee, Liu, Liu, Ludvig, Neuendorffer, Sachs, Xiong (b24) 2003; 91 The Mathworks. Natick, MA, U.S.A., Simulink 7—Reference, seventh ed., 2010. T. Bourke, J.-L. Colaço, B. Pagano, C. Pasteur, M. Pouzet, A Synchronous-based Code Generator For Explicit Hybrid Systems Languages, in: Compiler Construction (CC), in: LNCS, 2015, London, UK, pp. 69–88. A. Benveniste, T. Bourke, B. Caillaud, M. Pouzet, Divide and recycle: types and compilation for a hybrid synchronous language, in: Languages, Compilers, Tools and Theory for Embedded Systems (LCTES), 2011, pp. 61–70. Halbwachs, Caspi, Raymond, Pilaud (b4) 1991; 79 Maler, Manna, Pnueli (b11) 1992; vol. 600 Dahlquist, Björck (b9) 2008 Pierce (b14) 2002 G. Berry, The constructive semantics of pure Esterel, 1999. Unpublished, Draft version 3 Benveniste, Caspi, Lublinerman, Tripakis (b18) 2008 The Mathworks, Natick, MA, U.S.A., Simulink 7—User’s Guide, seventh ed., 2010. Kahn (b17) 1974 Pouzet (b19) 2006 Palmgren (b22) 2000; 11 Pouzet, Raymond (b21) 2009 Lindstrøm (b23) 1988 . Amagbegnon, Besnard, Le Guernic (b6) 1995 Benveniste, Bourke, Caillaud, Pouzet (b10) 2012; 78 Bourke, Pouzet (b15) 2013 Gérard, Guatto, Pasteur, Pouzet (b8) 2012 Lee, Zheng (b25) 2007 10.1016/j.nahs.2017.04.004_b16 10.1016/j.nahs.2017.04.004_b3 10.1016/j.nahs.2017.04.004_b2 10.1016/j.nahs.2017.04.004_b12 10.1016/j.nahs.2017.04.004_b13 Pierce (10.1016/j.nahs.2017.04.004_b14) 2002 Halbwachs (10.1016/j.nahs.2017.04.004_b5) 1991 Pottier (10.1016/j.nahs.2017.04.004_b20) 2001; 170 Gérard (10.1016/j.nahs.2017.04.004_b8) 2012 Pouzet (10.1016/j.nahs.2017.04.004_b21) 2009 Lee (10.1016/j.nahs.2017.04.004_b25) 2007 Benveniste (10.1016/j.nahs.2017.04.004_b18) 2008 Pouzet (10.1016/j.nahs.2017.04.004_b19) 2006 Halbwachs (10.1016/j.nahs.2017.04.004_b4) 1991; 79 10.1016/j.nahs.2017.04.004_b7 Palmgren (10.1016/j.nahs.2017.04.004_b22) 2000; 11 Benveniste (10.1016/j.nahs.2017.04.004_b10) 2012; 78 Kahn (10.1016/j.nahs.2017.04.004_b17) 1974 Amagbegnon (10.1016/j.nahs.2017.04.004_b6) 1995 Eker (10.1016/j.nahs.2017.04.004_b24) 2003; 91 Dahlquist (10.1016/j.nahs.2017.04.004_b9) 2008 Carloni (10.1016/j.nahs.2017.04.004_b1) 2006; 1 Maler (10.1016/j.nahs.2017.04.004_b11) 1992; vol. 600 Bourke (10.1016/j.nahs.2017.04.004_b15) 2013 Bauer (10.1016/j.nahs.2017.04.004_b26) 2010 Lindstrøm (10.1016/j.nahs.2017.04.004_b23) 1988 |
| References_xml | – start-page: 207 year: 1991 end-page: 218 ident: b5 article-title: Generating efficient code from data-flow programs publication-title: 3rd International Symposium on Programming Language Implementation and Logic Programming (PLILP) – volume: 1 year: 2006 ident: b1 article-title: Languages and tools for hybrid systems design publication-title: Foundations & Trends in EDA – reference: T. Bourke, J.-L. Colaço, B. Pagano, C. Pasteur, M. Pouzet, A Synchronous-based Code Generator For Explicit Hybrid Systems Languages, in: Compiler Construction (CC), in: LNCS, 2015, London, UK, pp. 69–88. – start-page: 471 year: 1974 end-page: 475 ident: b17 article-title: The semantics of a simple language for parallel programming publication-title: IFIP 74 Congress – volume: 91 start-page: 127 year: 2003 end-page: 144 ident: b24 article-title: Taming heterogeneity—the Ptolemy Approach publication-title: Proc. IEEE – start-page: 113 year: 2013 end-page: 118 ident: b15 article-title: Zélus: A Synchronous Language with ODEs publication-title: Hybrid Systems: Computation and Control (HSCC) – reference: The Mathworks. Natick, MA, U.S.A., Simulink 7—Reference, seventh ed., 2010. – start-page: 114 year: 2007 end-page: 123 ident: b25 article-title: Leveraging synchronous language principles for heterogeneous modeling and design of embedded systems publication-title: Embedded Software (EMSOFT) – reference: The Mathworks, Natick, MA, U.S.A., Simulink 7—User’s Guide, seventh ed., 2010. – start-page: 163 year: 1995 end-page: 173 ident: b6 article-title: Implementation of the data-flow synchronous language Signal publication-title: Programming Languages Design and Implementation (PLDI) – reference: G. Berry, The constructive semantics of pure Esterel, 1999. Unpublished, Draft version 3, – volume: 170 start-page: 153 year: 2001 end-page: 183 ident: b20 article-title: Simplifying subtyping constraints: a theory publication-title: Inform. and Comput. – reference: A. Benveniste, T. Bourke, B. Caillaud, M. Pouzet, Divide and recycle: types and compilation for a hybrid synchronous language, in: Languages, Compilers, Tools and Theory for Embedded Systems (LCTES), 2011, pp. 61–70. – reference: E.A. Lee, H. Zheng, Operational semantics of hybrid systems, in: Hybrid Systems: Computation and Control (HSCC), Zurich, Switzerland, 2005, pp. 25–53. – year: 2008 ident: b18 article-title: Actors without directors: a Kahnian view of heterogeneous systems, Tech. Rep. TR-2008-6 – volume: vol. 600 start-page: 447 year: 1992 end-page: 484 ident: b11 article-title: From timed to hybrid systems publication-title: Real-Time: Theory in Practice – start-page: 1 year: 1988 end-page: 105 ident: b23 article-title: An invitation to nonstandard analysis publication-title: Nonstandard Analysis and Its Applications – year: 2002 ident: b14 article-title: Types and Programming Languages – reference: . – year: 2006 ident: b19 article-title: Lucid Synchrone, version 3. Tutorial and reference manual – start-page: 51 year: 2012 end-page: 60 ident: b8 article-title: A modular memory optimization for synchronous data-flow languages: application to arrays in a Lustre Compiler publication-title: Languages, Compilers and Tools for Embedded Systems (LCTES) – volume: 78 start-page: 877 year: 2012 end-page: 910 ident: b10 article-title: Non-standard semantics of hybrid systems modelers publication-title: J. Comput. System. Sci. – start-page: 41 year: 2010 end-page: 50 ident: b26 article-title: From synchronous programs to symbolic representations of hybrid systems publication-title: Hybrid Systems: Computation and Control (HSCC) – start-page: 215 year: 2009 end-page: 224 ident: b21 article-title: Modular static scheduling of synchronous data-flow networks: an efficient symbolic representation publication-title: Embedded Software (EMSOFT) – year: 2008 ident: b9 article-title: Numerical Methods in Scientific Computing: Volume 1 – volume: 11 start-page: 129 year: 2000 end-page: 138 ident: b22 article-title: Constructive nonstandard representations of generalized functions publication-title: Indag. Math. (N.S.) – volume: 79 start-page: 1305 year: 1991 end-page: 1320 ident: b4 article-title: The synchronous dataflow programming language publication-title: Proc. IEEE – ident: 10.1016/j.nahs.2017.04.004_b3 – year: 2002 ident: 10.1016/j.nahs.2017.04.004_b14 – year: 2006 ident: 10.1016/j.nahs.2017.04.004_b19 – volume: 170 start-page: 153 issue: 2 year: 2001 ident: 10.1016/j.nahs.2017.04.004_b20 article-title: Simplifying subtyping constraints: a theory publication-title: Inform. and Comput. doi: 10.1006/inco.2001.2963 – start-page: 41 year: 2010 ident: 10.1016/j.nahs.2017.04.004_b26 article-title: From synchronous programs to symbolic representations of hybrid systems – start-page: 51 year: 2012 ident: 10.1016/j.nahs.2017.04.004_b8 article-title: A modular memory optimization for synchronous data-flow languages: application to arrays in a Lustre Compiler – start-page: 114 year: 2007 ident: 10.1016/j.nahs.2017.04.004_b25 article-title: Leveraging synchronous language principles for heterogeneous modeling and design of embedded systems – ident: 10.1016/j.nahs.2017.04.004_b7 – ident: 10.1016/j.nahs.2017.04.004_b12 – start-page: 471 year: 1974 ident: 10.1016/j.nahs.2017.04.004_b17 article-title: The semantics of a simple language for parallel programming – volume: 1 year: 2006 ident: 10.1016/j.nahs.2017.04.004_b1 article-title: Languages and tools for hybrid systems design publication-title: Foundations & Trends in EDA – volume: 11 start-page: 129 issue: 1 year: 2000 ident: 10.1016/j.nahs.2017.04.004_b22 article-title: Constructive nonstandard representations of generalized functions publication-title: Indag. Math. (N.S.) doi: 10.1016/S0019-3577(00)88579-1 – start-page: 207 year: 1991 ident: 10.1016/j.nahs.2017.04.004_b5 article-title: Generating efficient code from data-flow programs – ident: 10.1016/j.nahs.2017.04.004_b2 doi: 10.1007/978-3-540-31954-2_2 – start-page: 163 year: 1995 ident: 10.1016/j.nahs.2017.04.004_b6 article-title: Implementation of the data-flow synchronous language Signal – year: 2008 ident: 10.1016/j.nahs.2017.04.004_b9 – start-page: 113 year: 2013 ident: 10.1016/j.nahs.2017.04.004_b15 article-title: Zélus: A Synchronous Language with ODEs – volume: vol. 600 start-page: 447 year: 1992 ident: 10.1016/j.nahs.2017.04.004_b11 article-title: From timed to hybrid systems – start-page: 215 year: 2009 ident: 10.1016/j.nahs.2017.04.004_b21 article-title: Modular static scheduling of synchronous data-flow networks: an efficient symbolic representation – year: 2008 ident: 10.1016/j.nahs.2017.04.004_b18 article-title: Actors without directors: a Kahnian view of heterogeneous systems, Tech. Rep. TR-2008-6 – ident: 10.1016/j.nahs.2017.04.004_b13 doi: 10.1145/1967677.1967687 – ident: 10.1016/j.nahs.2017.04.004_b16 doi: 10.1007/978-3-662-46663-6_4 – start-page: 1 year: 1988 ident: 10.1016/j.nahs.2017.04.004_b23 article-title: An invitation to nonstandard analysis – volume: 91 start-page: 127 issue: 1 year: 2003 ident: 10.1016/j.nahs.2017.04.004_b24 article-title: Taming heterogeneity—the Ptolemy Approach publication-title: Proc. IEEE doi: 10.1109/JPROC.2002.805829 – volume: 79 start-page: 1305 issue: 9 year: 1991 ident: 10.1016/j.nahs.2017.04.004_b4 article-title: The synchronous dataflow programming language lustre publication-title: Proc. IEEE doi: 10.1109/5.97300 – volume: 78 start-page: 877 issue: 3 year: 2012 ident: 10.1016/j.nahs.2017.04.004_b10 article-title: Non-standard semantics of hybrid systems modelers publication-title: J. Comput. System. Sci. doi: 10.1016/j.jcss.2011.08.009 |
| SSID | ssj0059284 |
| Score | 2.1494846 |
| Snippet | Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between... Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete-and continuous-time behaviors with complex interactions between... |
| SourceID | hal crossref elsevier |
| SourceType | Open Access Repository Enrichment Source Index Database Publisher |
| StartPage | 168 |
| SubjectTerms | Automatic Control Engineering Computer Science Embedded Systems Hybrid systems Mathematical Software Modeling and Simulation Other Synchronous programming languages Type systems |
| Title | A type-based analysis of causality loops in hybrid systems modelers |
| URI | https://dx.doi.org/10.1016/j.nahs.2017.04.004 https://inria.hal.science/hal-01549183 |
| Volume | 26 |
| WOSCitedRecordID | wos000410903700011&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 | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 issn: 1751-570X databaseCode: AIEXJ dateStart: 20070301 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: false ssIdentifier: ssj0059284 providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1La9wwEBbNpof2UPqkaZogSm_BYeWHZB1NSNiGdCk0LXszlqVtNgR72UdI_31mJFnepDS0hV6MkS3JaD5rRtLMN4R8ZFwnPDUyMjqBBYrmVSRTJSKTaqa0UUrY8LHvZ2I8zicT-cUfFyxtOgHRNPnNjZz_V1FDGQgbQ2f_QtyhUSiAexA6XEHscP0jwRd2WzVC9YQ0rD3nSF2tl87qvmrbufWDvfiJAVuezrlLi-PPd7zFOnZcGtUitHV4MLpTLazoDTqwI2xc5Ay6bIdn8OXODciDoz_7wLRHa4cz07SzjXjsH5XNC474a9rN7QlQeezO9kSIm-mdlHCaFRmLMjGcbM7D8eZEylyyHa-TmUsz9Mt073YeLg-bylKvM2Fpa10-43s02l-xT-ySIUN_nPMtsh2LTOYDsl18Op6cdvo7k7HNUx2-0YdaOa_A-z39zpzZuug25q2hcv6cPPMrDFo4ZLwgj0zzkjz9HOh5l6_IUUF7jNBOrrSd0oARajFCZw11GKFe2LTDyGvy7eT4_GgU-WwaUZ2IfBVVTElu4qTWUjEzTKZVpmQyBQWjhVGiNio1YFxOGcdbWOnWNXqogwGjdaJgHfCGDJq2MW8JFUhZpGudZFylYOPJJNcsVryChlPBxQ5h3aCUtaeax4wnV2XnU3hZ4kCWOJDlMC1hIHfIQagzd0QrD76ddWNdelPRmYAlQOPBeh9AMKED5FYfFWclllmyQlBw1_G7f2x8lzzp_4D3ZLBarM0eeVxfr2bLxb6H2S26XZwU |
| linkProvider | Elsevier |
| 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%3Ajournal&rft.genre=article&rft.atitle=A+type-based+analysis+of+causality+loops+in+hybrid+systems+modelers&rft.jtitle=Nonlinear+analysis.+Hybrid+systems&rft.au=Benveniste%2C+Albert&rft.au=Bourke%2C+Timothy&rft.au=Caillaud%2C+Benoit&rft.au=Pagano%2C+Bruno&rft.date=2017-11-01&rft.pub=Elsevier+Ltd&rft.issn=1751-570X&rft.volume=26&rft.spage=168&rft.epage=189&rft_id=info:doi/10.1016%2Fj.nahs.2017.04.004&rft.externalDocID=S1751570X17300286 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1751-570X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1751-570X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1751-570X&client=summon |