Noname manuscript No will be inserted by the editor

Marc Pouzet

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Noname manuscript No. (will be inserted by the editor) Modular Static Scheduling of Synchronous Data-flow Networks An efficient symbolic representation Marc Pouzet · Pascal Raymond Received: January 2010 / Accepted: date Abstract This paper addresses the question of producing modular sequential impera- tive code from synchronous data-flow networks. Precisely, given a system with several input and output flows, how to decompose it into a minimal number of classes executed atomically and statically scheduled without restricting possible feedback loops between input and output? Though this question has been identified by Raymond in the early years of Lustre, it has almost been left aside until the recent work of Lublinerman, Szegedy and Tripakis. The problem is proven to be intractable, in the sense that it belongs to the family of optimization problems where the corresponding decision problem — there exists a solution with size c — is NP-complete. Then, the authors derive an iterative algorithm looking for solutions for c = 1, 2, ... where each step is encoded as a satisfiability (SAT) problem. Despite the apparent intractability of the problem, our experience is that real pro- grams do not exhibit such a complexity. Based on earlier work by Raymond, the current paper presents a new encoding of the problem in terms of input/output relations. This encoding simplifies the problem, in the sense that it rejects some solutions, while keep- ing all the optimal ones.

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• problem — there


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• code

Sujets

• Laboratoire
• Raymond
• Output
• Sequence

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

NonamemanuscriptNo. (willbeinsertedbytheeditor) ModularStaticSchedulingofSynchronousData-flow Networks Anefficientsymbolicrepresentation MarcPouzet ∙ PascalRaymond Received:January2010/Accepted:date AbstractThispaperaddressesthequestionofproducingmodularsequentialimpera- tivecodefromsynchronousdata-flownetworks.Precisely,givenasystemwithseveral inputandoutputflows,howtodecomposeitintoaminimalnumberofclassesexecuted atomicallyandstaticallyscheduledwithoutrestrictingpossiblefeedbackloopsbetween inputandoutput? ThoughthisquestionhasbeenidentifiedbyRaymondintheearlyyearsof Lustre , ithasalmostbeenleftasideuntiltherecentworkofLublinerman,SzegedyandTripakis. Theproblemisproventobeintractable,inthesensethatitbelongstothefamilyof optimizationproblemswherethecorrespondingdecisionproblem—thereexistsa solutionwithsizec—isNP-complete.Then,theauthorsderiveaniterativealgorithm lookingforsolutionsforc=1,2,...whereeachstepisencodedasasatisfiability(SAT) problem. Despitetheapparentintractabilityoftheproblem,ourexperienceisthatrealpro- gramsdonotexhibitsuchacomplexity.BasedonearlierworkbyRaymond,thecurrent paperpresentsanewencodingoftheproblemintermsofinput/outputrelations.This encodingsimplifiestheproblem,inthesensethatitrejectssomesolutions,whilekeep- ingalltheoptimalones.Itallows,inpolynomialtime,(1)toidentifynodesforwhich severalschedulesarefeasibleandthusarepossiblesourcesofcombinatorialexplosion; (2)toobtainsolutionswhichinsomecasesarealreadyoptimal;(3)otherwise,toget anontriviallowerboundforctostartaniterativecombinatorialsearch.Themethod hasbeenvalidatedonseveralindustrialexamples. Revisedandextendedversionof[14].ThisworkissupportedbytheSYNCHRONICSlarge scaleinitiativeofINRIA. MarcPouzet E ´ coleNormaleSupe´rieureandUniversite´PierreetMarieCurie. Physicaladdress:E ´ coleNormaleSupe´rieure,45rued’Ulm,75230Pariscedex05,France. E-mail:[e-mail]. PascalRaymond LaboratoireVERIMAG,2avenuedeVignate,38610Gie`res,France. E-mail:[e-mail].

2 Fig.1A Scade (v5)block-diagram Thesolutionappliestoalargeclassofblock-diagramformalismsbasedonatomic computationsandadelayoperator,rangingfromsynchronouslanguagessuchas Lus- tre or Scade tomodelingtoolssuchas Simulink . KeywordsReal-timesystems,Synchronouslanguages,Block-diagrams,Compilation, NP-completeness,Partialorders,Preorders 1Introduction Thesynchronousblock-diagramordata-flowformalismisnowpreeminentinavariety ofdesigntoolsforembeddedsystems.Sequentialcodegenerationofsynchronousblock- diagramshavebeenconsideredintheearlyyearsof Lustre [7]and Signal [1]andis 21 providedbyindustrialtoolssuchas Scade and RtBuilder foralmostfifteenyears. Thoughithasbeenconsideredmorerecently,modelingandsimulationtoolssuchas Simulink 3 and Modelica 4 arenowequippedwithautomaticcodegenerators. Wefocushereontheproblemofgeneratingimperative,sequentialcode,imple- mentingthefunctionalbehaviorofaparalleldata-flownetwork.Wekeepabstracted the(somehoworthogonal)problemofdatamanagement,thatis,howvaluesareac- tuallypassedfromonenodetoanotherandeventheinterpretationofeachoperator. Inparticular,weaddressbothdata-flownetworkswithdiscrete-time(e.g., Scade )or continuous-time(e.g., Simulink )semantics.Figure1givesanexampleofa Scade block-diagramandFigure2,anexampleofa Simulink one. Whateverbethesemanticsofnodesinanetwork,therearebasicallytwotypes ofatomicnodes.Instantaneousnodesneedtoevaluatealltheirargumentsinorderto producetheiroutputs(e.g.,combinatorialfunctions).Onthecontrary,delaynodesare abletoproducetheiroutputsbeforereadingtheirinputs.Theycorrespondtounitary registersinsynchronousdesigns(theso-called pre operatorof Lustre ),initialized buffersinKahnprocessnetworks[9,10]orcontinuousintegratorsin Simulink .Delays 1 http://www.esterel-technologies.com/scade/ 2 http://www.geensoft.com/en/article/rtbuilder 3 http://www.mathworks.com/product/simulink 4 http://www.modelica.org