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Web","\u002Fsnt-2nde\u002Fsb-web","0.snt-2nde\u002Fsb-web\u002F0.index",[41,42,46,50,54,58,62],{"title":37,"path":38,"stem":39},{"title":43,"path":44,"stem":45},"Histoire du Web","\u002Fsnt-2nde\u002Fsb-web\u002Fhistoire-web","0.snt-2nde\u002Fsb-web\u002F1.histoire-web",{"title":47,"path":48,"stem":49},"URL et hypertexte","\u002Fsnt-2nde\u002Fsb-web\u002Furl-hypertexte","0.snt-2nde\u002Fsb-web\u002F2.url-hypertexte",{"title":51,"path":52,"stem":53},"HTML et CSS — le contenu et la présentation","\u002Fsnt-2nde\u002Fsb-web\u002Fhtml-css","0.snt-2nde\u002Fsb-web\u002F3.html-css",{"title":55,"path":56,"stem":57},"Client-serveur et requêtes HTTP","\u002Fsnt-2nde\u002Fsb-web\u002Fclient-serveur-http","0.snt-2nde\u002Fsb-web\u002F4.client-serveur-http",{"title":59,"path":60,"stem":61},"Moteurs de recherche","\u002Fsnt-2nde\u002Fsb-web\u002Fmoteurs-recherche","0.snt-2nde\u002Fsb-web\u002F5.moteurs-recherche",{"title":63,"path":64,"stem":65},"Sécurité du navigateur et notions juridiques","\u002Fsnt-2nde\u002Fsb-web\u002Fsecurite-droits","0.snt-2nde\u002Fsb-web\u002F6.securite-droits",{"title":67,"path":68,"stem":69,"children":70},"Les réseaux sociaux","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F0.index",[71,72,76,80,84,88],{"title":67,"path":68,"stem":69},{"title":73,"path":74,"stem":75},"Panorama des réseaux sociaux","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fpanorama-rs","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F1.panorama-rs",{"title":77,"path":78,"stem":79},"Identité numérique et authentification","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fidentite-numerique","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F2.identite-numerique",{"title":81,"path":82,"stem":83},"Le modèle économique des réseaux sociaux","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fmodele-economique","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F3.modele-economique",{"title":85,"path":86,"stem":87},"Petit monde et graphes","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fpetit-monde-graphes","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F4.petit-monde-graphes",{"title":89,"path":90,"stem":91},"La cyberviolence","\u002Fsnt-2nde\u002Fsc-reseaux-sociaux\u002Fcyberviolence","0.snt-2nde\u002Fsc-reseaux-sociaux\u002F5.cyberviolence",{"title":93,"path":94,"stem":95,"children":96},"Les données structurées et leur traitement","\u002Fsnt-2nde\u002Fsd-donnees-structurees","0.snt-2nde\u002Fsd-donnees-structurees\u002F0.index",[97,98,102,106,110,114],{"title":93,"path":94,"stem":95},{"title":99,"path":100,"stem":101},"Données et formats","\u002Fsnt-2nde\u002Fsd-donnees-structurees\u002Fdonnees-formats","0.snt-2nde\u002Fsd-donnees-structurees\u002F1.donnees-formats",{"title":103,"path":104,"stem":105},"Données 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position","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fconfidentialite-position","0.snt-2nde\u002Fse-localisation-cartographie\u002F5.confidentialite-position",{"title":145,"path":146,"stem":147,"children":148},"Informatique embarquée et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee","0.snt-2nde\u002Fsf-info-embarquee\u002F0.index",[149,150,154,158],{"title":145,"path":146,"stem":147},{"title":151,"path":152,"stem":153},"Systèmes informatiques embarqués","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fsystemes-embarques","0.snt-2nde\u002Fsf-info-embarquee\u002F1.systemes-embarques",{"title":155,"path":156,"stem":157},"Capteurs et actionneurs","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fcapteurs-actionneurs","0.snt-2nde\u002Fsf-info-embarquee\u002F2.capteurs-actionneurs",{"title":159,"path":160,"stem":161},"IHM et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fihm-objet-connecte","0.snt-2nde\u002Fsf-info-embarquee\u002F3.ihm-objet-connecte",{"title":163,"path":164,"stem":165,"children":166},"Photographie numérique","\u002Fsnt-2nde\u002Fsg-photo-numerique","0.snt-2nde\u002Fsg-photo-numerique\u002F0.index",[167,168,172,176,180],{"title":163,"path":164,"stem":165},{"title":169,"path":170,"stem":171},"Photosites, pixels et résolution","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Fpixels-resolution","0.snt-2nde\u002Fsg-photo-numerique\u002F1.pixels-resolution",{"title":173,"path":174,"stem":175},"Les algorithmes de l'appareil photo","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Falgorithmes-photo","0.snt-2nde\u002Fsg-photo-numerique\u002F2.algorithmes-photo",{"title":177,"path":178,"stem":179},"Les métadonnées EXIF","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Fmetadonnees-exif","0.snt-2nde\u002Fsg-photo-numerique\u002F3.metadonnees-exif",{"title":181,"path":182,"stem":183},"Traiter une image en Python","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Ftraitement-rgb","0.snt-2nde\u002Fsg-photo-numerique\u002F4.traitement-rgb",{"title":185,"path":186,"stem":187,"children":188},"Notions transversales de programmation","\u002Fsnt-2nde\u002Fsh-prog-notions","0.snt-2nde\u002Fsh-prog-notions\u002F0.index",[189,190,194,198,202],{"title":185,"path":186,"stem":187},{"title":191,"path":192,"stem":193},"Séquences d'instructions et variables","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fsequences-affectations","0.snt-2nde\u002Fsh-prog-notions\u002F1.sequences-affectations",{"title":195,"path":196,"stem":197},"Instructions conditionnelles","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fconditionnelles","0.snt-2nde\u002Fsh-prog-notions\u002F2.conditionnelles",{"title":199,"path":200,"stem":201},"Boucles bornées et non bornées","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fboucles","0.snt-2nde\u002Fsh-prog-notions\u002F3.boucles",{"title":203,"path":204,"stem":205},"Définitions et appels de fonctions","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Ffonctions","0.snt-2nde\u002Fsh-prog-notions\u002F4.fonctions",{"title":207,"path":208,"stem":209,"children":210},"Démo des widgets MDC","\u002Fsnt-2nde\u002Fzz-demo-widgets","0.snt-2nde\u002Fzz-demo-widgets\u002F0.index",[211,212,216,220],{"title":207,"path":208,"stem":209},{"title":213,"path":214,"stem":215},"Démo statique — callouts, code, BO, illustration","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fstatique","0.snt-2nde\u002Fzz-demo-widgets\u002F1.statique",{"title":217,"path":218,"stem":219},"Démo interactive — self-eval et python-snippet","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Finteractif","0.snt-2nde\u002Fzz-demo-widgets\u002F2.interactif",{"title":221,"path":222,"stem":223},"Démo · 14 widgets de manipulation NSI","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fwidgets-manipulation","0.snt-2nde\u002Fzz-demo-widgets\u002F3.widgets-manipulation",{"title":225,"path":226,"stem":227,"children":228},"NSI — Première","\u002Fnsi-1ere","1.nsi-1ere\u002F0.index",[229,230,234,256,282,304,328,354,380,410],{"title":225,"path":226,"stem":227},{"title":231,"path":232,"stem":233},"Relecture pédagogique — 8 chapitres NSI 1ère (PA → PH)","\u002Fnsi-1ere\u002Freview","1.nsi-1ere\u002FREVIEW",{"title":235,"path":236,"stem":237,"children":238},"Histoire de l'informatique","\u002Fnsi-1ere\u002Fpa-histoire-info","1.nsi-1ere\u002Fpa-histoire-info\u002F0.index",[239,240,244,248,252],{"title":235,"path":236,"stem":237},{"title":241,"path":242,"stem":243},"De la préantiquité aux machines à calculer","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fpreantiquite-aux-machines","1.nsi-1ere\u002Fpa-histoire-info\u002F1.preantiquite-aux-machines",{"title":245,"path":246,"stem":247},"La machine universelle (1930-1950)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fmachine-universelle","1.nsi-1ere\u002Fpa-histoire-info\u002F2.machine-universelle",{"title":249,"path":250,"stem":251},"Les ordinateurs modernes (1947-1990)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Fordinateurs-modernes","1.nsi-1ere\u002Fpa-histoire-info\u002F3.ordinateurs-modernes",{"title":253,"path":254,"stem":255},"Réseaux, Web et IA (1969 à aujourd'hui)","\u002Fnsi-1ere\u002Fpa-histoire-info\u002Freseaux-et-web","1.nsi-1ere\u002Fpa-histoire-info\u002F4.reseaux-et-web",{"title":257,"path":258,"stem":259,"children":260},"Représentation des données — types et valeurs","\u002Fnsi-1ere\u002Fpb-types-valeurs","1.nsi-1ere\u002Fpb-types-valeurs\u002F0.index",[261,262,266,270,274,278],{"title":257,"path":258,"stem":259},{"title":263,"path":264,"stem":265},"Bases de numération et entiers positifs","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbases-entiers-positifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F1.bases-entiers-positifs",{"title":267,"path":268,"stem":269},"Entiers relatifs et complément à deux","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fentiers-relatifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs",{"title":271,"path":272,"stem":273},"Nombres flottants et virgule flottante","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fflottants","1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants",{"title":275,"path":276,"stem":277},"Textes et encodages — ASCII, ISO-8859-1, Unicode","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Ftextes-encodages","1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages",{"title":279,"path":280,"stem":281},"Valeurs booléennes et opérateurs logiques","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbooleens","1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens",{"title":283,"path":284,"stem":285,"children":286},"Représentation des données : types construits","\u002Fnsi-1ere\u002Fpc-types-construits","1.nsi-1ere\u002Fpc-types-construits\u002F0.index",[287,288,292,296,300],{"title":283,"path":284,"stem":285},{"title":289,"path":290,"stem":291},"Tableaux et listes","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftableaux-listes","1.nsi-1ere\u002Fpc-types-construits\u002F1.tableaux-listes",{"title":293,"path":294,"stem":295},"Listes en compréhension","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fcomprehension","1.nsi-1ere\u002Fpc-types-construits\u002F2.comprehension",{"title":297,"path":298,"stem":299},"Tuples et p-uplets nommés","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftuples","1.nsi-1ere\u002Fpc-types-construits\u002F3.tuples",{"title":301,"path":302,"stem":303},"Dictionnaires","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fdictionnaires","1.nsi-1ere\u002Fpc-types-construits\u002F4.dictionnaires",{"title":305,"path":306,"stem":307,"children":308},"Traitement de données en tables","\u002Fnsi-1ere\u002Fpd-tables-donnees","1.nsi-1ere\u002Fpd-tables-donnees\u002F0.index",[309,310,314,318,322],{"title":305,"path":306,"stem":307},{"title":311,"path":312,"stem":313},"Tables et CSV","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftables-csv","1.nsi-1ere\u002Fpd-tables-donnees\u002F1.tables-csv",{"title":315,"path":316,"stem":317},"Recherche et filtrage","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Frecherche","1.nsi-1ere\u002Fpd-tables-donnees\u002F2.recherche",{"title":319,"path":320,"stem":321},"Trier une table","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftri","1.nsi-1ere\u002Fpd-tables-donnees\u002F3.tri",{"title":323,"path":324,"stem":325,"children":326},"Fusion et indexation","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ffusion-indexation","1.nsi-1ere\u002Fpd-tables-donnees\u002F4.fusion-indexation",[327],{"title":323,"path":324,"stem":325},{"title":329,"path":330,"stem":331,"children":332},"IHM sur le Web","\u002Fnsi-1ere\u002Fpe-ihm-web","1.nsi-1ere\u002Fpe-ihm-web\u002F0.index",[333,334,338,342,346,350],{"title":329,"path":330,"stem":331},{"title":335,"path":336,"stem":337},"HTML et CSS — rappel de structure","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fhtml-css-bases","1.nsi-1ere\u002Fpe-ihm-web\u002F1.html-css-bases",{"title":339,"path":340,"stem":341},"Formulaires HTML","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fformulaires","1.nsi-1ere\u002Fpe-ihm-web\u002F2.formulaires",{"title":343,"path":344,"stem":345},"Événements JavaScript","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fevenements-js","1.nsi-1ere\u002Fpe-ihm-web\u002F3.evenements-js",{"title":347,"path":348,"stem":349},"Manipulation du DOM","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fdom-interaction","1.nsi-1ere\u002Fpe-ihm-web\u002F4.dom-interaction",{"title":351,"path":352,"stem":353},"Client, serveur et HTTP","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fclient-serveur-http","1.nsi-1ere\u002Fpe-ihm-web\u002F5.client-serveur-http",{"title":355,"path":356,"stem":357,"children":358},"Architectures matérielles et systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os","1.nsi-1ere\u002Fpf-archi-os\u002F0.index",[359,360,364,368,372,376],{"title":355,"path":356,"stem":357},{"title":361,"path":362,"stem":363},"Le modèle de von Neumann","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fvon-neumann","1.nsi-1ere\u002Fpf-archi-os\u002F1.von-neumann",{"title":365,"path":366,"stem":367},"Périphériques et interface homme-machine","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fperipheriques-ihm","1.nsi-1ere\u002Fpf-archi-os\u002F2.peripheriques-ihm",{"title":369,"path":370,"stem":371},"Les systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fsystemes-exploitation","1.nsi-1ere\u002Fpf-archi-os\u002F3.systemes-exploitation",{"title":373,"path":374,"stem":375},"Linux et le shell","\u002Fnsi-1ere\u002Fpf-archi-os\u002Flinux-shell","1.nsi-1ere\u002Fpf-archi-os\u002F4.linux-shell",{"title":377,"path":378,"stem":379},"Réseaux et protocoles","\u002Fnsi-1ere\u002Fpf-archi-os\u002Freseaux","1.nsi-1ere\u002Fpf-archi-os\u002F5.reseaux",{"title":381,"path":382,"stem":383,"children":384},"Langages et programmation","\u002Fnsi-1ere\u002Fpg-langages-prog","1.nsi-1ere\u002Fpg-langages-prog\u002F0.index",[385,386,390,394,398,402,406],{"title":381,"path":382,"stem":383},{"title":387,"path":388,"stem":389},"Diversité et unité des langages de programmation","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fdiversite-langages","1.nsi-1ere\u002Fpg-langages-prog\u002F1.diversite-langages",{"title":391,"path":392,"stem":393},"Constructions élémentaires","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fconstructions-elementaires","1.nsi-1ere\u002Fpg-langages-prog\u002F2.constructions-elementaires",{"title":395,"path":396,"stem":397},"Fonctions","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Ffonctions","1.nsi-1ere\u002Fpg-langages-prog\u002F3.fonctions",{"title":399,"path":400,"stem":401},"Spécification d'une fonction","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fspecification","1.nsi-1ere\u002Fpg-langages-prog\u002F4.specification",{"title":403,"path":404,"stem":405},"Mise au point des programmes","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fmise-au-point","1.nsi-1ere\u002Fpg-langages-prog\u002F5.mise-au-point",{"title":407,"path":408,"stem":409},"Utilisation de bibliothèques","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fbibliotheques","1.nsi-1ere\u002Fpg-langages-prog\u002F6.bibliotheques",{"title":411,"path":412,"stem":413,"children":414},"Algorithmique","\u002Fnsi-1ere\u002Fph-algorithmique","1.nsi-1ere\u002Fph-algorithmique\u002F0.index",[415,416,420,424,428,432,436],{"title":411,"path":412,"stem":413},{"title":417,"path":418,"stem":419},"Parcours séquentiel d'un tableau","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fparcours-sequentiel","1.nsi-1ere\u002Fph-algorithmique\u002F1.parcours-sequentiel",{"title":421,"path":422,"stem":423},"Bases de la complexité","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fcomplexite-bases","1.nsi-1ere\u002Fph-algorithmique\u002F2.complexite-bases",{"title":425,"path":426,"stem":427},"Tris par sélection et par insertion","\u002Fnsi-1ere\u002Fph-algorithmique\u002Ftris","1.nsi-1ere\u002Fph-algorithmique\u002F3.tris",{"title":429,"path":430,"stem":431},"Recherche dichotomique","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fdichotomie","1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie",{"title":433,"path":434,"stem":435},"Algorithmes gloutons","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fgloutons","1.nsi-1ere\u002Fph-algorithmique\u002F5.gloutons",{"title":437,"path":438,"stem":439},"Les k plus proches voisins (k-NN)","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fknn","1.nsi-1ere\u002Fph-algorithmique\u002F6.knn",{"title":441,"path":442,"stem":443,"children":444},"NSI — Terminale","\u002Fnsi-tale","2.nsi-tale\u002F0.index",[445,446,450,476,498,536,560],{"title":441,"path":442,"stem":443},{"title":447,"path":448,"stem":449},"Review pédagogique — chapitres NSI Terminale (TA, TB, TC, TD, TE)","\u002Fnsi-tale\u002Freview","2.nsi-tale\u002FREVIEW",{"title":451,"path":452,"stem":453,"children":454},"Structures de données","\u002Fnsi-tale\u002Fta-structures-donnees","2.nsi-tale\u002Fta-structures-donnees\u002F0.index",[455,456,460,464,468,472],{"title":451,"path":452,"stem":453},{"title":457,"path":458,"stem":459},"Objet, classe, interface","\u002Fnsi-tale\u002Fta-structures-donnees\u002Finterface-implementation","2.nsi-tale\u002Fta-structures-donnees\u002F1.interface-implementation",{"title":461,"path":462,"stem":463},"Implémenter une structure — l'exemple de la file FIFO","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fimplementer-une-structure","2.nsi-tale\u002Fta-structures-donnees\u002F2.implementer-une-structure",{"title":465,"path":466,"stem":467},"Listes, piles, files, dictionnaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Flistes-piles-files-dict","2.nsi-tale\u002Fta-structures-donnees\u002F3.listes-piles-files-dict",{"title":469,"path":470,"stem":471},"Arbres binaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Farbres-binaires","2.nsi-tale\u002Fta-structures-donnees\u002F4.arbres-binaires",{"title":473,"path":474,"stem":475},"Graphes","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fgraphes","2.nsi-tale\u002Fta-structures-donnees\u002F5.graphes",{"title":477,"path":478,"stem":479,"children":480},"Bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees","2.nsi-tale\u002Ftb-bases-de-donnees\u002F0.index",[481,482,486,490,494],{"title":477,"path":478,"stem":479},{"title":483,"path":484,"stem":485},"Le modèle relationnel","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fmodele-relationnel","2.nsi-tale\u002Ftb-bases-de-donnees\u002F1.modele-relationnel",{"title":487,"path":488,"stem":489},"Bases de données relationnelles et anomalies de schéma","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fbase-relationnelle","2.nsi-tale\u002Ftb-bases-de-donnees\u002F2.base-relationnelle",{"title":491,"path":492,"stem":493},"Le langage SQL","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsql","2.nsi-tale\u002Ftb-bases-de-donnees\u002F3.sql",{"title":495,"path":496,"stem":497},"Les systèmes de gestion de bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsgbd","2.nsi-tale\u002Ftb-bases-de-donnees\u002F4.sgbd",{"title":499,"path":500,"stem":501,"children":502},"Architectures matérielles, systèmes d'exploitation et réseaux","\u002Fnsi-tale\u002Ftc-archi-os-reseaux","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F0.index",[503,504,508,512,516,520,524,528,532],{"title":499,"path":500,"stem":501},{"title":505,"path":506,"stem":507},"Le système sur puce (SoC)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fsysteme-sur-puce","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F1.systeme-sur-puce",{"title":509,"path":510,"stem":511},"Processus et ordonnancement par l'OS","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprocessus-os","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F2.processus-os",{"title":513,"path":514,"stem":515},"L'interblocage (deadlock)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Finterblocage","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F3.interblocage",{"title":517,"path":518,"stem":519},"Principes du routage","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Froutage-principes","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F4.routage-principes",{"title":521,"path":522,"stem":523},"Protocoles de routage RIP et OSPF","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprotocoles-rip-ospf","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F5.protocoles-rip-ospf",{"title":525,"path":526,"stem":527},"Chiffrement symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-symetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F6.chiffrement-symetrique",{"title":529,"path":530,"stem":531},"Chiffrement asymétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-asymetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F7.chiffrement-asymetrique",{"title":533,"path":534,"stem":535},"HTTPS — l'échange de clé symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fhttps-tls","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F8.https-tls",{"title":381,"path":537,"stem":538,"children":539},"\u002Fnsi-tale\u002Ftd-langages-prog","2.nsi-tale\u002Ftd-langages-prog\u002F0.index",[540,541,544,548,552,556],{"title":381,"path":537,"stem":538},{"title":403,"path":542,"stem":543},"\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmise-au-point","2.nsi-tale\u002Ftd-langages-prog\u002F1.mise-au-point",{"title":545,"path":546,"stem":547},"Modularité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmodularite","2.nsi-tale\u002Ftd-langages-prog\u002F2.modularite",{"title":549,"path":550,"stem":551},"Récursivité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Frecursivite","2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite",{"title":553,"path":554,"stem":555},"Paradigmes de programmation","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fparadigmes","2.nsi-tale\u002Ftd-langages-prog\u002F4.paradigmes",{"title":557,"path":558,"stem":559},"Programme comme donnée — calculabilité et indécidabilité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fprogramme-comme-donnee","2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee",{"title":411,"path":561,"stem":562,"children":563},"\u002Fnsi-tale\u002Fte-algorithmique","2.nsi-tale\u002Fte-algorithmique\u002F0.index",[564,565,569,573,577,581],{"title":411,"path":561,"stem":562},{"title":566,"path":567,"stem":568},"Algorithmes sur les arbres binaires","\u002Fnsi-tale\u002Fte-algorithmique\u002Farbres-binaires-algos","2.nsi-tale\u002Fte-algorithmique\u002F1.arbres-binaires-algos",{"title":570,"path":571,"stem":572},"Parcours de graphes — DFS, BFS, cycles, chemins","\u002Fnsi-tale\u002Fte-algorithmique\u002Fgraphes-parcours","2.nsi-tale\u002Fte-algorithmique\u002F2.graphes-parcours",{"title":574,"path":575,"stem":576},"Diviser pour régner","\u002Fnsi-tale\u002Fte-algorithmique\u002Fdiviser-pour-regner","2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner",{"title":578,"path":579,"stem":580},"Programmation dynamique","\u002Fnsi-tale\u002Fte-algorithmique\u002Fprogrammation-dynamique","2.nsi-tale\u002Fte-algorithmique\u002F4.programmation-dynamique",{"title":582,"path":583,"stem":584},"Recherche textuelle — l'algorithme de Boyer-Moore","\u002Fnsi-tale\u002Fte-algorithmique\u002Frecherche-textuelle","2.nsi-tale\u002Fte-algorithmique\u002F5.recherche-textuelle",{"id":586,"title":557,"bo":587,"body":589,"description":1297,"eval":1298,"extension":1299,"labo":1298,"meta":1300,"navigation":1301,"path":558,"quizzes":1302,"readingTime":754,"seo":1309,"stem":559,"tp":1298,"__hash__":1310},"cours\u002F2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee.md",[588],"TD03",{"type":590,"value":591,"toc":1288},"minimark",[592,597,614,630,634,658,720,738,793,797,808,822,831,841,867,871,874,909,919,923,930,945,948,1017,1032,1041,1048,1106,1122,1194,1216,1220,1230,1244,1267,1271,1282,1284],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,601,605,606,609,610,613],"p",{},"Vous avez écrit du code toute l'année. Il est temps de ",[602,603,604],"strong",{},"regarder le code\nlui-même comme une donnée"," : un texte qu'un autre programme peut lire,\nanalyser, exécuter, transformer. Cette idée — anodine en apparence — fonde\ntoute l'informatique théorique, et débouche sur l'un de ses résultats les plus\nprofonds : ",[602,607,608],{},"certains problèmes parfaitement définis n'admettent aucun\nprogramme qui les résout",". C'est le contenu de l'item ",[611,612],"bo-ref",{"code":588},".",[598,615,616,617,620,621,624,625,629],{},"Le cœur de ce que vous allez voir est un cousin algorithmique du ",[602,618,619],{},"paradoxe\ndu menteur"," (« cette phrase est fausse » : si elle est vraie, elle est\nfausse ; si elle est fausse, elle est vraie). L'auteur de la démonstration,\n",[602,622,623],{},"Alan Turing",", est aussi celui dont le film ",[626,627,628],"em",{},"The Imitation Game"," (2014)\nretrace la contribution au déchiffrement d'Enigma pendant la Seconde Guerre\nmondiale — mais ses idées sur le calcul, formulées dès 1936, lui survivent\nencore plus largement.",[593,631,633],{"id":632},"un-programme-est-une-donnée","Un programme est une donnée",[598,635,636,637,641,642,645,646,649,650,653,654,657],{},"Quand vous tapez ",[638,639,640],"code",{},"python mon_script.py",", voici ce qui se passe : le programme\n",[602,643,644],{},"CPython"," (l'interpréteur Python, lui-même écrit en C) lit votre fichier\n",[638,647,648],{},"mon_script.py",", le ",[602,651,652],{},"parse"," en une structure de données interne (un arbre\nsyntaxique), puis l'",[602,655,656],{},"exécute",". Votre script n'est pas magique — c'est une\nchaîne de caractères, lue comme telle par un autre programme.",[659,660,661,677],"table",{},[662,663,664],"thead",{},[665,666,667,671,674],"tr",{},[668,669,670],"th",{},"Outil",[668,672,673],{},"Rôle",[668,675,676],{},"Exemple",[678,679,680,694,707],"tbody",{},[665,681,682,688,691],{},[683,684,685],"td",{},[602,686,687],{},"Interpréteur",[683,689,690],{},"Lit le code et l'exécute directement.",[683,692,693],{},"CPython, navigateur (JS).",[665,695,696,701,704],{},[683,697,698],{},[602,699,700],{},"Compilateur",[683,702,703],{},"Traduit le code en un autre langage (souvent du natif).",[683,705,706],{},"GCC pour le C, javac pour Java.",[665,708,709,714,717],{},[683,710,711],{},[602,712,713],{},"Transpileur",[683,715,716],{},"Traduit d'un langage source vers un autre langage source.",[683,718,719],{},"TypeScript → JavaScript.",[721,722,723],"note",{},[598,724,725,726,729,730,733,734,737],{},"CPython est lui-même un programme — écrit en C, ",[602,727,728],{},"compilé"," par GCC en un\nexécutable natif. Quand vous lancez Python, vous lancez un exécutable qui lit\nvotre ",[638,731,732],{},".py"," comme un humain lirait un livre. Le code que vous écrivez est la\n",[602,735,736],{},"donnée d'entrée"," de CPython.",[739,740,741],"python-snippet",{},[742,743,748],"pre",{"className":744,"code":745,"language":746,"meta":747,"style":747},"language-python shiki shiki-themes github-light github-light github-dark","# Un programme peut manipuler du code Python comme une chaîne :\ncode_source = \"x = 2 + 3\\nprint(x)\"\nexec(code_source)        # affiche 5 — Python exécute son propre code\n","python","",[638,749,750,759,781],{"__ignoreMap":747},[751,752,755],"span",{"class":753,"line":754},"line",1,[751,756,758],{"class":757},"sCsY4","# Un programme peut manipuler du code Python comme une chaîne :\n",[751,760,762,766,770,774,778],{"class":753,"line":761},2,[751,763,765],{"class":764},"sxrX7","code_source ",[751,767,769],{"class":768},"s8jYJ","=",[751,771,773],{"class":772},"sIIMD"," \"x = 2 + 3",[751,775,777],{"class":776},"sBjJW","\\n",[751,779,780],{"class":772},"print(x)\"\n",[751,782,784,787,790],{"class":753,"line":783},3,[751,785,786],{"class":776},"exec",[751,788,789],{"class":764},"(code_source)        ",[751,791,792],{"class":757},"# affiche 5 — Python exécute son propre code\n",[593,794,796],{"id":795},"calculabilité-ce-qui-peut-être-calculé","Calculabilité — ce qui peut être calculé",[598,798,799,800,803,804,807],{},"Une fonction est dite ",[602,801,802],{},"calculable"," s'il existe un ",[602,805,806],{},"algorithme"," (donc un\nprogramme, dans n'importe quel langage Turing-complet) qui la calcule en un\nnombre fini d'étapes pour toute entrée.",[598,809,810,813,814,817,818,821],{},[602,811,812],{},"Thèse de Church-Turing (énoncé informel)"," : la classe des fonctions\ncalculables est ",[602,815,816],{},"la même"," pour tous les modèles de calcul raisonnables —\nmachine de Turing, λ-calcul, Python, C, JavaScript. Le choix du langage ne\nchange pas ",[602,819,820],{},"ce qui est calculable",", seulement la commodité d'écriture.",[721,823,824],{},[598,825,826,827,830],{},"On parle bien d'une ",[602,828,829],{},"thèse"," et non d'un théorème : la notion intuitive de\n« fonction calculable » n'est pas un objet mathématique, on ne peut donc pas\ndémontrer formellement l'équivalence. Mais tous les modèles de calcul proposés\ndepuis 1936 se sont avérés équivalents — la thèse a force d'évidence\nexpérimentale.",[832,833,834],"warning",{},[598,835,836,837,840],{},"Il est tentant de croire qu'un langage plus puissant calcule plus de choses.\n",[602,838,839],{},"C'est faux",". Python ne sait pas calculer plus que ce qu'une machine de\nTuring peut calculer — il l'écrit juste plus vite.",[842,843,846,852],"self-eval-qcm",{"correct":844,"slug":845},"1","programme-comme-donnee-c1",[847,848,849],"template",{"v-slot:question":747},[598,850,851],{},"Calculabilité dépend-elle du langage utilisé ?",[847,853,854],{"v-slot:choices":747},[855,856,857,861,864],"ul",{},[858,859,860],"li",{},"Oui — un langage puissant calcule plus de fonctions",[858,862,863],{},"Non — tous les langages Turing-complets calculent les mêmes fonctions",[858,865,866],{},"Cela dépend du compilateur",[593,868,870],{"id":869},"le-problème-de-larrêt-lénoncé","Le problème de l'arrêt — l'énoncé",[598,872,873],{},"Voici une question simple en apparence :",[875,876,877],"blockquote",{},[598,878,879,880,883,884,886,887,890,891,894,895,898,899,901,902,904,905,908],{},"Existe-t-il un programme ",[638,881,882],{},"arret(p, e)"," qui, étant donné un programme ",[638,885,598],{}," et\nune entrée ",[638,888,889],{},"e",", ",[602,892,893],{},"répond toujours"," par ",[638,896,897],{},"True"," si l'exécution de ",[638,900,598],{}," sur ",[638,903,889],{},"\ntermine, et par ",[638,906,907],{},"False"," sinon ?",[598,910,911,912,915,916,613],{},"Un tel programme serait précieux : il détecterait toutes les boucles infinies\navant exécution. Hélas, ",[602,913,914],{},"un tel programme n'existe pas",". Le ",[602,917,918],{},"problème de\nl'arrêt est indécidable",[593,920,922],{"id":921},"la-preuve-par-auto-référence","La preuve par auto-référence",[598,924,925,926,929],{},"L'argument est dû à Alan Turing (1936). Il procède par ",[602,927,928],{},"l'absurde"," et\nexploite le fait qu'un programme est une donnée pour un autre programme.",[598,931,932,935,936,938,939,941,942,944],{},[602,933,934],{},"Supposons"," qu'une fonction ",[638,937,882],{}," existe, qui termine toujours et\nrépond correctement à la question : « le programme ",[638,940,598],{}," exécuté sur ",[638,943,889],{},"\ntermine-t-il ? ».",[598,946,947],{},"Construisons alors la fonction suivante :",[742,949,951],{"className":744,"code":950,"language":746,"meta":747,"style":747},"def paradox(p):\n    if arret(p, p):       # si p appliqué à p termine…\n        while True:        # …alors paradox boucle pour toujours.\n            pass\n    else:                  # sinon (p sur p ne termine pas)…\n        return            # …paradox termine immédiatement.\n",[638,952,953,965,976,990,996,1008],{"__ignoreMap":747},[751,954,955,958,962],{"class":753,"line":754},[751,956,957],{"class":768},"def",[751,959,961],{"class":960},"snPdu"," paradox",[751,963,964],{"class":764},"(p):\n",[751,966,967,970,973],{"class":753,"line":761},[751,968,969],{"class":768},"    if",[751,971,972],{"class":764}," arret(p, p):       ",[751,974,975],{"class":757},"# si p appliqué à p termine…\n",[751,977,978,981,984,987],{"class":753,"line":783},[751,979,980],{"class":768},"        while",[751,982,983],{"class":776}," True",[751,985,986],{"class":764},":        ",[751,988,989],{"class":757},"# …alors paradox boucle pour toujours.\n",[751,991,993],{"class":753,"line":992},4,[751,994,995],{"class":768},"            pass\n",[751,997,999,1002,1005],{"class":753,"line":998},5,[751,1000,1001],{"class":768},"    else",[751,1003,1004],{"class":764},":                  ",[751,1006,1007],{"class":757},"# sinon (p sur p ne termine pas)…\n",[751,1009,1011,1014],{"class":753,"line":1010},6,[751,1012,1013],{"class":768},"        return",[751,1015,1016],{"class":757},"            # …paradox termine immédiatement.\n",[598,1018,1019,1020,1023,1024,1031],{},"Le code de ",[638,1021,1022],{},"paradox"," est un programme Python parfaitement bien défini, ",[602,1025,1026,1027,1030],{},"à\ncondition que ",[638,1028,1029],{},"arret"," existe",". Posons-nous la question fatidique :",[598,1033,1034],{},[602,1035,1036,1037,1040],{},"Que se passe-t-il quand on exécute ",[638,1038,1039],{},"paradox(paradox)"," ?",[598,1042,1043,1044,1047],{},"Deux cas possibles, et ",[602,1045,1046],{},"les deux mènent à une contradiction"," :",[1049,1050,1051,1080],"ol",{},[858,1052,1053,1059,1060,1062,1063,1066,1067,1069,1070,1072,1073,1075,1076,1079],{},[602,1054,1055,1056,1058],{},"Si ",[638,1057,1039],{}," termine"," : alors par définition de ",[638,1061,1022],{},",\nc'est que ",[638,1064,1065],{},"arret(paradox, paradox)"," a retourné ",[638,1068,907],{},". Mais ",[638,1071,1029],{}," est\ncensé être correcte — donc ",[638,1074,1039],{}," ne termine ",[602,1077,1078],{},"pas",".\nContradiction.",[858,1081,1082,1087,1088,1090,1091,1066,1093,1095,1096,1098,1099,1101,1102,1105],{},[602,1083,1055,1084,1086],{},[638,1085,1039],{}," ne termine pas"," : alors par définition de\n",[638,1089,1022],{},", c'est que ",[638,1092,1065],{},[638,1094,897],{},". Mais\n",[638,1097,1029],{}," est censé être correcte — donc ",[638,1100,1039],{}," termine\n",[602,1103,1104],{},"bien",". Contradiction.",[598,1107,1108,1109,1111,1112,1115,1116,613],{},"L'hypothèse de départ (l'existence de ",[638,1110,1029],{},") conduit à une contradiction\ndans ",[602,1113,1114],{},"tous les cas",". Cette hypothèse est donc fausse. ",[602,1117,1118,1119,1121],{},"Aucun programme\n",[638,1120,1029],{}," ne peut exister",[1123,1124,1125],"tip",{},[598,1126,1127,1128,1131,1132,1134,1135,1138,1139,1186,1187,1190,1191,613],{},"L'argument exploite la ",[602,1129,1130],{},"diagonalisation"," : on construit un programme qui\ndemande à ",[638,1133,1029],{}," ce qui va lui arriver à lui-même, puis fait ",[602,1136,1137],{},"l'inverse"," —\nexactement le mécanisme du paradoxe du menteur, transposé en code. C'est la\nmême idée que dans la preuve de Cantor sur la non-dénombrabilité de\n",[751,1140,1143,1167],{"className":1141},[1142],"katex",[751,1144,1147],{"className":1145},[1146],"katex-mathml",[1148,1149,1151],"math",{"xmlns":1150},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1152,1153,1154,1162],"semantics",{},[1155,1156,1157],"mrow",{},[1158,1159,1161],"mi",{"mathvariant":1160},"double-struck","R",[1163,1164,1166],"annotation",{"encoding":1165},"application\u002Fx-tex","\\mathbb{R}",[751,1168,1172],{"className":1169,"ariaHidden":1171},[1170],"katex-html","true",[751,1173,1176,1181],{"className":1174},[1175],"base",[751,1177],{"className":1178,"style":1180},[1179],"strut","height:0.6889em;",[751,1182,1161],{"className":1183},[1184,1185],"mord","mathbb",". Le procédé est appelé ",[602,1188,1189],{},"auto-référence"," ou ",[602,1192,1193],{},"argument\ndiagonal",[842,1195,1198,1203],{"correct":1196,"slug":1197},"0","programme-comme-donnee-c2",[847,1199,1200],{"v-slot:question":747},[598,1201,1202],{},"Le problème de l'arrêt est indécidable signifie que…",[847,1204,1205],{"v-slot:choices":747},[855,1206,1207,1210,1213],{},[858,1208,1209],{},"aucun programme ne peut décider correctement pour TOUTE entrée",[858,1211,1212],{},"l ordinateur n est pas assez puissant",[858,1214,1215],{},"il faudrait un langage plus expressif",[593,1217,1219],{"id":1218},"conséquences","Conséquences",[598,1221,1222,1223,1225,1226,1229],{},"L'indécidabilité du problème de l'arrêt n'est ",[602,1224,1078],{}," une limite technique\nqu'on lèvera demain avec un meilleur compilateur. C'est une ",[602,1227,1228],{},"limite\nmathématique",", démontrée, universelle. Elle a des cousines :",[855,1231,1232,1238],{},[858,1233,1234,1237],{},[602,1235,1236],{},"Indécidabilité de l'équivalence"," : on ne peut pas écrire un programme\nqui décide si deux programmes calculent la même fonction.",[858,1239,1240,1243],{},[602,1241,1242],{},"Théorème de Rice (1953)"," : toute propriété non-triviale de la fonction\ncalculée par un programme est indécidable.",[598,1245,1246,1247,1250,1251,1254,1255,1258,1259,1262,1263,1266],{},"Cela dit, ",[602,1248,1249],{},"les outils approchés existent"," : analyseurs statiques (",[638,1252,1253],{},"mypy",",\n",[638,1256,1257],{},"pylint","), démonstrateurs (Coq, Lean), vérification de modèles. Ils ne\nrésolvent pas le problème de l'arrêt en général, mais répondent\n",[602,1260,1261],{},"correctement sur les cas qu'ils savent traiter"," — et ",[602,1264,1265],{},"refusent de\nrépondre"," ailleurs.",[593,1268,1270],{"id":1269},"pour-aller-plus-loin","Pour aller plus loin",[598,1272,1273,1274,1277,1278,1281],{},"Le résultat de Turing (1936) a précédé l'existence des ordinateurs. Il\nfonde l'informatique théorique : ce que vous lisez sur le problème P vs NP,\nsur les machines quantiques, sur la cryptographie, repose sur cette\ndistinction entre ",[602,1275,1276],{},"ce qui peut être calculé"," et ",[602,1279,1280],{},"ce qui ne le peut pas",".\nLe programme officiel s'arrête à l'argument informel — mais vous avez vu\nl'essentiel.",[611,1283],{"code":588},[1285,1286,1287],"style",{},"html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sIIMD, html code.shiki .sIIMD{--shiki-light:#032F62;--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}",{"title":747,"searchDepth":761,"depth":761,"links":1289},[1290,1291,1292,1293,1294,1295,1296],{"id":595,"depth":761,"text":596},{"id":632,"depth":761,"text":633},{"id":795,"depth":761,"text":796},{"id":869,"depth":761,"text":870},{"id":921,"depth":761,"text":922},{"id":1218,"depth":761,"text":1219},{"id":1269,"depth":761,"text":1270},"Un programme est une donnée pour un autre programme ; certaines questions sur les programmes n'admettent aucune réponse algorithmique — c'est le problème de l'arrêt.",null,"md",{},true,[1303,1307],{"slug":845,"kind":1304,"question":851,"choices":1305,"correct":844,"multi":1306,"explanation":747},"qcm",[860,863,866],false,{"slug":1197,"kind":1304,"question":1202,"choices":1308,"correct":1196,"multi":1306,"explanation":747},[1209,1212,1215],{"title":557,"description":1297},"_RCCzevtohoYq21_WhvTMub0VwRYY-VIdP848kO3KyU",{"id":1312,"title":1313,"bo":1314,"body":1315,"description":1517,"eval":1298,"extension":1299,"labo":1298,"meta":1518,"navigation":1306,"path":1519,"quizzes":1520,"readingTime":754,"seo":1521,"stem":1522,"tp":1298,"__hash__":1523},"coursMemo\u002F2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee.memo.md","Programme comme donnée — l'essentiel",[588],{"type":590,"value":1316,"toc":1510},[1317,1321,1326,1330,1383,1387,1425,1429,1444,1448,1507],[593,1318,1320],{"id":1319},"en-une-phrase","En une phrase",[598,1322,1323],{},[602,1324,1325],{},"Un programme est une donnée pour un autre programme (interpréteur, compilateur) ; la calculabilité ne dépend pas du langage ; le problème de l'arrêt est indécidable — aucun programme ne peut décider pour TOUTE entrée si un programme arbitraire termine.",[593,1327,1329],{"id":1328},"à-mémoriser","À mémoriser",[855,1331,1332,1337,1342,1348,1354,1377],{},[858,1333,1334,1336],{},[602,1335,687],{}," : lit et exécute (CPython, navigateur JS).",[858,1338,1339,1341],{},[602,1340,700],{}," : traduit le source vers une autre langue (GCC, javac).",[858,1343,1344,1347],{},[602,1345,1346],{},"Thèse de Church-Turing"," : tous les langages Turing-complets calculent les mêmes fonctions (thèse, pas théorème).",[858,1349,1350,1353],{},[602,1351,1352],{},"Problème de l'arrêt"," : indécidable — démontré par Turing (1936).",[858,1355,1356,1359,1360,1362,1363,1366,1367,1370,1371,1373,1374,1376],{},[602,1357,1358],{},"Argument"," : supposer ",[638,1361,882],{}," existe, construire ",[638,1364,1365],{},"paradox(p)"," qui boucle si ",[638,1368,1369],{},"arret(p, p)"," retourne ",[638,1372,897],{},", termine sinon. Examiner ",[638,1375,1039],{}," → contradiction dans les deux cas.",[858,1378,1379,1382],{},[602,1380,1381],{},"Théorème de Rice"," : toute propriété sémantique non-triviale est indécidable.",[593,1384,1386],{"id":1385},"à-savoir-reconnaître","À savoir reconnaître",[659,1388,1389,1399],{},[662,1390,1391],{},[665,1392,1393,1396],{},[668,1394,1395],{},"Question",[668,1397,1398],{},"Réponse",[678,1400,1401,1409,1417],{},[665,1402,1403,1406],{},[683,1404,1405],{},"Python peut-il calculer plus que C ?",[683,1407,1408],{},"Non, même classe de fonctions calculables.",[665,1410,1411,1414],{},[683,1412,1413],{},"Peut-on écrire un détecteur de boucles infinies parfait ?",[683,1415,1416],{},"Non — problème de l'arrêt.",[665,1418,1419,1422],{},[683,1420,1421],{},"Un compilateur traduit-il vers du natif ?",[683,1423,1424],{},"Souvent — GCC vers du C compilé en binaire.",[593,1426,1428],{"id":1427},"pièges-courants","Pièges courants",[855,1430,1431,1434,1441],{},[858,1432,1433],{},"Croire qu'un langage plus expressif calcule plus.",[858,1435,1436,1437,1440],{},"Confondre indécidable et difficile — c'est ",[602,1438,1439],{},"impossible",", pas juste lent.",[858,1442,1443],{},"Penser que les analyseurs statiques résolvent le problème de l'arrêt.",[593,1445,1447],{"id":1446},"argument-diagonal-schéma","Argument diagonal (schéma)",[742,1449,1451],{"className":744,"code":1450,"language":746,"meta":747,"style":747},"# Si arret(p, e) existait :\ndef paradox(p):\n    if arret(p, p):\n        while True: pass     # boucle\n    return                   # termine\n\n# paradox(paradox) : contradiction dans les deux cas.\n",[638,1452,1453,1458,1466,1473,1488,1496,1501],{"__ignoreMap":747},[751,1454,1455],{"class":753,"line":754},[751,1456,1457],{"class":757},"# Si arret(p, e) existait :\n",[751,1459,1460,1462,1464],{"class":753,"line":761},[751,1461,957],{"class":768},[751,1463,961],{"class":960},[751,1465,964],{"class":764},[751,1467,1468,1470],{"class":753,"line":783},[751,1469,969],{"class":768},[751,1471,1472],{"class":764}," arret(p, p):\n",[751,1474,1475,1477,1479,1482,1485],{"class":753,"line":992},[751,1476,980],{"class":768},[751,1478,983],{"class":776},[751,1480,1481],{"class":764},": ",[751,1483,1484],{"class":768},"pass",[751,1486,1487],{"class":757},"     # boucle\n",[751,1489,1490,1493],{"class":753,"line":998},[751,1491,1492],{"class":768},"    return",[751,1494,1495],{"class":757},"                   # termine\n",[751,1497,1498],{"class":753,"line":1010},[751,1499,1500],{"emptyLinePlaceholder":1301},"\n",[751,1502,1504],{"class":753,"line":1503},7,[751,1505,1506],{"class":757},"# paradox(paradox) : contradiction dans les deux cas.\n",[1285,1508,1509],{},"html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":747,"searchDepth":761,"depth":761,"links":1511},[1512,1513,1514,1515,1516],{"id":1319,"depth":761,"text":1320},{"id":1328,"depth":761,"text":1329},{"id":1385,"depth":761,"text":1386},{"id":1427,"depth":761,"text":1428},{"id":1446,"depth":761,"text":1447},"Mémo de révision sur calculabilité, interpréteur\u002Fcompilateur et indécidabilité du problème de l'arrêt.",{},"\u002Fnsi-tale\u002Ftd-langages-prog\u002Fprogramme-comme-donnee.memo",[],{"title":1313,"description":1517},"2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee.memo","FqqVIIz5p7mWJ3qu5cM-m4ZsGrSwiqxZyzBABH54jb8",{"id":1525,"title":1526,"body":1527,"description":1850,"extension":1299,"meta":1851,"navigation":1306,"path":1852,"seo":1853,"stem":1854,"__hash__":1855},"coursSlides\u002F2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee.slides.md","Programme comme donnée — diapositives",{"type":590,"value":1528,"toc":1848},[1529,1534,1540,1543,1547,1552,1562,1568,1570,1574,1619,1621,1625,1631,1640,1643,1645,1649,1654,1670,1682,1688,1690,1694,1700,1745,1750,1752,1756,1773,1787,1793,1800,1802,1806,1813,1815,1817,1833,1835,1839,1845],[1530,1531,1533],"h1",{"id":1532},"programme-comme-donnée","Programme comme donnée",[598,1535,1536,1537,1539],{},"Item ",[611,1538],{"code":588}," — Le code lui-même est une donnée.",[1541,1542],"hr",{},[1530,1544,1546],{"id":1545},"le-retournement","Le retournement",[598,1548,1549,1550,613],{},"Vous écrivez : ",[638,1551,640],{},[598,1553,1554,1555,1557,1558,1561],{},"CPython lit ",[638,1556,648],{}," comme ",[602,1559,1560],{},"texte",".\nLe parse en arbre syntaxique. L'exécute.",[598,1563,1564,1565,1567],{},"Votre code est la ",[602,1566,736],{}," d'un autre programme.",[1541,1569],{},[1530,1571,1573],{"id":1572},"interpréteur-vs-compilateur","Interpréteur vs compilateur",[659,1575,1576,1587],{},[662,1577,1578],{},[665,1579,1580,1582,1585],{},[668,1581,670],{},[668,1583,1584],{},"Action",[668,1586,676],{},[678,1588,1589,1599,1609],{},[665,1590,1591,1593,1596],{},[683,1592,687],{},[683,1594,1595],{},"Lit + exécute",[683,1597,1598],{},"CPython, JS",[665,1600,1601,1603,1606],{},[683,1602,700],{},[683,1604,1605],{},"Traduit en autre langage",[683,1607,1608],{},"GCC, javac",[665,1610,1611,1613,1616],{},[683,1612,713],{},[683,1614,1615],{},"Source → source",[683,1617,1618],{},"TypeScript → JS",[1541,1620],{},[1530,1622,1624],{"id":1623},"calculabilité","Calculabilité",[598,1626,1627,1628,1630],{},"Une fonction est ",[602,1629,802],{}," s'il existe un algorithme qui la calcule en un nombre fini d'étapes.",[598,1632,1633,1635,1636,1639],{},[602,1634,1346],{}," : tous les langages Turing-complets calculent les ",[602,1637,1638],{},"mêmes"," fonctions.",[598,1641,1642],{},"Python ≠ C en commodité — pas en puissance.",[1541,1644],{},[1530,1646,1648],{"id":1647},"le-problème-de-larrêt","Le problème de l'arrêt",[598,1650,879,1651,1653],{},[638,1652,882],{}," qui répond toujours :",[855,1655,1656,1665],{},[858,1657,1658,1660,1661,1664],{},[638,1659,897],{}," si ",[638,1662,1663],{},"p(e)"," termine ;",[858,1666,1667,1669],{},[638,1668,907],{}," sinon ;",[598,1671,1672,1673,1676,1677,1679,1680,1040],{},"…pour ",[602,1674,1675],{},"tout"," ",[638,1678,598],{}," et tout ",[638,1681,889],{},[598,1683,1684,1687],{},[602,1685,1686],{},"Non."," Démonstration de Turing (1936).",[1541,1689],{},[1530,1691,1693],{"id":1692},"construction-de-paradox","Construction de paradox",[598,1695,1696,1697,1699],{},"Supposons ",[638,1698,1029],{}," existe. Définissons :",[742,1701,1703],{"className":744,"code":1702,"language":746,"meta":747,"style":747},"def paradox(p):\n    if arret(p, p):\n        while True: pass     # boucle\n    else:\n        return                # termine\n",[638,1704,1705,1713,1719,1731,1738],{"__ignoreMap":747},[751,1706,1707,1709,1711],{"class":753,"line":754},[751,1708,957],{"class":768},[751,1710,961],{"class":960},[751,1712,964],{"class":764},[751,1714,1715,1717],{"class":753,"line":761},[751,1716,969],{"class":768},[751,1718,1472],{"class":764},[751,1720,1721,1723,1725,1727,1729],{"class":753,"line":783},[751,1722,980],{"class":768},[751,1724,983],{"class":776},[751,1726,1481],{"class":764},[751,1728,1484],{"class":768},[751,1730,1487],{"class":757},[751,1732,1733,1735],{"class":753,"line":992},[751,1734,1001],{"class":768},[751,1736,1737],{"class":764},":\n",[751,1739,1740,1742],{"class":753,"line":998},[751,1741,1013],{"class":768},[751,1743,1744],{"class":757},"                # termine\n",[598,1746,1747,1748,1040],{},"Question : que fait ",[638,1749,1039],{},[1541,1751],{},[1530,1753,1755],{"id":1754},"contradiction-par-auto-référence","Contradiction par auto-référence",[598,1757,1758,1761,1762,1764,1765,1768,1769,1075,1771,1105],{},[602,1759,1760],{},"Cas 1"," : ",[638,1763,1039],{}," termine.\n⇒ ",[638,1766,1767],{},"arret(paradox, paradox) == False","\n⇒ par hypothèse, ",[638,1770,1039],{},[602,1772,1078],{},[598,1774,1775,1761,1778,1780,1781,1768,1784,1786],{},[602,1776,1777],{},"Cas 2",[638,1779,1039],{}," ne termine pas.\n⇒ ",[638,1782,1783],{},"arret(paradox, paradox) == True",[638,1785,1039],{}," termine. Contradiction.",[598,1788,1789,1790,1792],{},"⇒ ",[638,1791,1029],{}," n'existe pas.",[598,1794,1795,1796,1799],{},"C'est le ",[602,1797,1798],{},"paradoxe du menteur"," transposé en code.",[1541,1801],{},[1530,1803,1805],{"id":1804},"schéma-de-largument","Schéma de l'argument",[1807,1808],"illustration",{"alt":1809,"kind":1810,"src":1811,"svg":1812},"Preuve de l'indécidabilité de l'arrêt : la fonction arret(p,e) répondrait True\u002FFalse, le programme paradox(p) boucle si arret dit True et termine sinon ; appliqué à lui-même, paradox(paradox) mène à une contradiction dans les deux cas, donc arret ne peut pas exister.","mermaid","\u002Fillustrations\u002F2.nsi-tale\u002Ftd-langages-prog\u002Fprobleme-arret-paradox.svg","\u003Csvg id=\"ill-2-nsi-tale-td-langages-prog-probleme-arret-paradox\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" xmlns:xlink=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink\" class=\"illustration-svg flowchart\" viewBox=\"0 0 531.921875 869.59375\" 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font-weight=\"normal\"> existe\u003C\u002Ftspan>\u003C\u002Ftspan>\u003C\u002Ftext>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"edgeLabel\" transform=\"translate(264.42578125, 376.5)\">\u003Cg class=\"label\" data-id=\"L_P_Q_0\" transform=\"translate(0, -8.500000953674316)\">\u003Cg>\u003Crect class=\"background\" style=\"\" x=\"-64.25\" y=\"-1.9999990463256836\" width=\"128.5\" height=\"21\"\u002F>\u003Ctext y=\"-10.1\" text-anchor=\"middle\" style=\"\">\u003Ctspan class=\"text-outer-tspan row\" x=\"0\" y=\"-0.1em\" dy=\"1.1em\" text-anchor=\"middle\">\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\">appel\u003C\u002Ftspan>\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\"> sur\u003C\u002Ftspan>\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\"> lui-même\u003C\u002Ftspan>\u003C\u002Ftspan>\u003C\u002Ftext>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"edgeLabel\" transform=\"translate(123.4453125, 635.09375)\">\u003Cg class=\"label\" data-id=\"L_Q_C1_0\" transform=\"translate(0, -8.500000953674316)\">\u003Cg>\u003Crect class=\"background\" style=\"\" x=\"-33.75\" y=\"-1.9999990463256836\" width=\"67.5\" height=\"21\"\u002F>\u003Ctext y=\"-10.1\" text-anchor=\"middle\" style=\"\">\u003Ctspan class=\"text-outer-tspan row\" x=\"0\" y=\"-0.1em\" dy=\"1.1em\" text-anchor=\"middle\">\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\">si\u003C\u002Ftspan>\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\"> termine\u003C\u002Ftspan>\u003C\u002Ftspan>\u003C\u002Ftext>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"edgeLabel\" transform=\"translate(405.40625, 635.09375)\">\u003Cg class=\"label\" data-id=\"L_Q_C2_0\" transform=\"translate(0, -8.500000953674316)\">\u003Cg>\u003Crect class=\"background\" style=\"\" x=\"-30.90625\" y=\"-1.9999990463256836\" width=\"61.8125\" height=\"21\"\u002F>\u003Ctext y=\"-10.1\" text-anchor=\"middle\" style=\"\">\u003Ctspan class=\"text-outer-tspan row\" x=\"0\" y=\"-0.1em\" dy=\"1.1em\" text-anchor=\"middle\">\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\">si\u003C\u002Ftspan>\u003Ctspan font-style=\"normal\" class=\"text-inner-tspan\" font-weight=\"normal\"> boucle\u003C\u002Ftspan>\u003C\u002Ftspan>\u003C\u002Ftext>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"edgeLabel\">\u003Cg class=\"label\" data-id=\"L_C1_X_0\" transform=\"translate(0, 0)\">\u003Ctext y=\"-10.1\" text-anchor=\"middle\">\u003Ctspan class=\"text-outer-tspan row\" x=\"0\" y=\"-0.1em\" dy=\"1.1em\" text-anchor=\"middle\"\u002F>\u003C\u002Ftext>\u003C\u002Fg>\u003C\u002Fg>\u003Cg>\u003Crect class=\"background\" style=\"stroke: none\"\u002F>\u003C\u002Fg>\u003Cg class=\"edgeLabel\">\u003Cg class=\"label\" data-id=\"L_C2_X_0\" transform=\"translate(0, 0)\">\u003Ctext y=\"-10.1\" text-anchor=\"middle\">\u003Ctspan class=\"text-outer-tspan row\" x=\"0\" y=\"-0.1em\" dy=\"1.1em\" text-anchor=\"middle\"\u002F>\u003C\u002Ftext>\u003C\u002Fg>\u003C\u002Fg>\u003Cg>\u003Crect class=\"background\" style=\"stroke: none\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"nodes\">\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-td-langages-prog-probleme-arret-paradox-flowchart-A-0\" data-look=\"classic\" transform=\"translate(264.42578125, 66)\">\u003Crect class=\"basic label-container\" style=\"\" x=\"-132\" y=\"-54\" width=\"264\" height=\"108\"\u002F>\u003Cg class=\"label\" style=\"\" transform=\"translate(-100, -42)\">\u003Crect\u002F>\u003CforeignObject width=\"200\" height=\"84\">\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" style=\"display: table; white-space: break-spaces; line-height: 1.5; max-width: 200px; text-align: center; width: 200px;\">\u003Cspan class=\"nodeLabel\">\u003Cp>arret(p, e)\u003Cbr \u002F>termine toujours et répond :\u003Cbr \u002F>True si p(e) termine, False sinon\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-td-langages-prog-probleme-arret-paradox-flowchart-P-1\" data-look=\"classic\" transform=\"translate(264.42578125, 269.5)\">\u003Crect class=\"basic label-container\" style=\"\" x=\"-132\" y=\"-64.5\" width=\"264\" height=\"129\"\u002F>\u003Cg class=\"label\" style=\"\" transform=\"translate(-100, -52.5)\">\u003Crect\u002F>\u003CforeignObject width=\"200\" height=\"105\">\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" style=\"display: table; white-space: break-spaces; line-height: 1.5; max-width: 200px; text-align: center; width: 200px;\">\u003Cspan class=\"nodeLabel\">\u003Cp>paradox(p) :\u003Cbr \u002F>si arret(p, p) == True → boucle (while True)\u003Cbr \u002F>si arret(p, p) == False → termine (return)\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-td-langages-prog-probleme-arret-paradox-flowchart-Q-2\" data-look=\"classic\" transform=\"translate(264.42578125, 505.796875)\">\u003Cpolygon points=\"90.796875,0 181.59375,-90.796875 90.796875,-181.59375 0,-90.796875\" class=\"label-container\" transform=\"translate(-90.296875, 90.796875)\"\u002F>\u003Cg class=\"label\" style=\"\" transform=\"translate(-64.296875, -10.5)\">\u003Crect\u002F>\u003CforeignObject width=\"128.59375\" height=\"21\">\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" style=\"display: table-cell; white-space: nowrap; line-height: 1.5; max-width: 200px; text-align: center;\">\u003Cspan class=\"nodeLabel\">\u003Cp>paradox(paradox) ?\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-td-langages-prog-probleme-arret-paradox-flowchart-C1-8\" data-look=\"classic\" transform=\"translate(123.4453125, 710.59375)\">\u003Crect class=\"basic label-container\" style=\"\" x=\"-115.4453125\" y=\"-33\" width=\"230.890625\" height=\"66\"\u002F>\u003Cg class=\"label\" style=\"\" transform=\"translate(-83.4453125, -21)\">\u003Crect\u002F>\u003CforeignObject width=\"166.890625\" height=\"42\">\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" style=\"display: table-cell; white-space: nowrap; line-height: 1.5; max-width: 200px; text-align: center;\">\u003Cspan class=\"nodeLabel\">\u003Cp>alors arret a dit False\u003Cbr \u002F>⇒ paradox devait boucler\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-td-langages-prog-probleme-arret-paradox-flowchart-C2-10\" data-look=\"classic\" transform=\"translate(405.40625, 710.59375)\">\u003Crect 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