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CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207

CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207 paul.gibson@it-sudparis.eu http://www-public. it-sudparis.eu /~gibson/Teaching/Event-B/. Famille http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/Famille.pdf. wizards. ASCII. hommes <: P.

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CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207

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  1. CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207 paul.gibson@it-sudparis.eu http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/ Famille http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/Famille.pdf T&MSP-CSC 4504 : Langages formels et applications

  2. T&MSP-CSC 4504 : Langages formels et applications

  3. T&MSP-CSC 4504 : Langages formels et applications

  4. T&MSP-CSC 4504 : Langages formels et applications

  5. wizards T&MSP-CSC 4504 : Langages formels et applications

  6. ASCII hommes <: P T&MSP-CSC 4504 : Langages formels et applications

  7. ASCII Representations of the Mathematical Symbols http://www.b-core.com/ONLINEDOC/ASCIISymbols.html Atomic Symbols T&MSP-CSC 4504 : Langages formels et applications

  8. ASCII Representations of the Mathematical Symbols Unary Operators T&MSP-CSC 4504 : Langages formels et applications

  9. ASCII Representations of the Mathematical Symbols Unary Operators Assignment Operators T&MSP-CSC 4504 : Langages formels et applications

  10. ASCII Representations of the Mathematical Symbols Binary Operators T&MSP-CSC 4504 : Langages formels et applications

  11. ASCII Representations of the Mathematical Symbols Quantifiers T&MSP-CSC 4504 : Langages formels et applications

  12. Exemples T&MSP-CSC 4504 : Langages formels et applications

  13. Les relations: T&MSP-CSC 4504 : Langages formels et applications

  14. Exemples T&MSP-CSC 4504 : Langages formels et applications

  15. Les relations: T&MSP-CSC 4504 : Langages formels et applications

  16. Exemples T&MSP-CSC 4504 : Langages formels et applications

  17. Les fonctions: T&MSP-CSC 4504 : Langages formels et applications

  18. Bijection, injection and surjection From Wikipedia, the free encyclopedia T&MSP-CSC 4504 : Langages formels et applications

  19. P hommes femmes Preuve d'un théorème T&MSP-CSC 4504 : Langages formels et applications

  20. Use the Event-B Keyboard (view) for editing complex expressions ! per. not ( per : P & not(per : hommes) & not (per : femmes)) T&MSP-CSC 4504 : Langages formels et applications

  21. Comment? thm1 Proven Automatically √ Proving T&MSP-CSC 4504 : Langages formels et applications

  22. Open perspective • Event-B • Proving • …. • Choose theorem T&MSP-CSC 4504 : Langages formels et applications

  23. Specification: Epoux P hommes femmes Partial injection ASCII - epoux : femmes >+> hommes T&MSP-CSC 4504 : Langages formels et applications

  24. Specification: Epouse P hommes femmes Partial injection ASCII - epouse: hommes >+> femmes T&MSP-CSC 4504 : Langages formels et applications

  25. Epoux-Epouse: Attention! P hommes femmes Partial injection Partial injection ASCII - epoux : femmes >+> hommes epouse: hommes >+> femmes T&MSP-CSC 4504 : Langages formels et applications

  26. Problème: Theorem -epouse = epoux~ - n’est pas vrai -1 ASCII:epouse = epoux~ SYMBOL: epouse = epoux T&MSP-CSC 4504 : Langages formels et applications

  27. Problème: Theorem -epouse = epoux~ - n’est pas vrai Le prouveur n’est pas content T&MSP-CSC 4504 : Langages formels et applications

  28. Specification: Epoux-Epouse - Corrigé P hommes femmes Partial injection ASCII - AXIOM epoux : femmes >+> hommes AXIOM epouse = epoux~ T&MSP-CSC 4504 : Langages formels et applications

  29. Specification: Epoux/Epouse - Verification/Test T&MSP-CSC 4504 : Langages formels et applications

  30. Specification: Famille_ctx0 - Version finale Pourquoi pas la transformer en pdf (en utilisant plugin B2latex)? T&MSP-CSC 4504 : Langages formels et applications

  31. Specification: Famille_ctx0 - Version finale B2Latex T&MSP-CSC 4504 : Langages formels et applications

  32. \documentclass[10pt,a4paper]{report} \usepackage[top=3cm, bottom=2.5cm, left=3cm, right=2.5cm] {geometry} \usepackage {bsymb,b2latex} \usepackage{fancyhdr,lastpage} \lhead{\rm An Event-B Specification of Famille\_ctx0} \rhead {\rm Page \thepage~of \pageref{LastPage}} \lfoot{}\cfoot{}\rfoot{} \pagestyle{fancy} %--------------------------------------------------------- \begin{document} \thispagestyle{empty} \begin{description} \BTitle{Famille\_ctx0}{3 Mar 2009}{02:00:30 PM} \CONTEXT{Famille\_ctx0} \SETS \begin{description} \Item{ P } \end{description} \CONSTANTS \begin{description} \Item{ hommes } \Item{ femmes } \Item{ epoux } \Item{ epouse } \Item{ f } \Item{ ff } \Item{ h } \Item{ hh } \end{description} \AXIOMS \begin{description} \nItem{ axm\_hommes }{ hommes \subseteq P } \nItem{ axm\_femmes }{ femmes = P\setminus hommes } \nItem{ axm\_epoux }{ epoux \in femmes \pinj hommes } \nItem{ axm\_epouse }{ epouse = epoux^{-1} } \nItem{ axm\_fhffhh }{ f\in femmes \land h\in hommes \land ff\in femmes \land hh\in hommes } \end{description} \THEOREMS \begin{description} \nItem{ thm\_test\_axm\_femmes }{ \forall per\qdot \lnot ( per\in P \land \lnot (per \in hommes) \land \lnot (per \in femmes)) } \nItem{ thm\_epouxepouse }{ epoux = epoux; epouse; epoux } \nItem{ thm\_test\_epouse1 }{ \lnot (hh=h) \land \lnot (ff=f) \land epoux = \{ f\mapsto h, ff\mapsto hh\} \limp epouse(hh) = ff } \nItem{ thm\_test\_epouse2 }{ \lnot (hh=h) \land \lnot (ff=f) \land epoux = \{ f\mapsto h, ff\mapsto hh\} \limp epouse(h) = f } \end{description} \END \end{description} \end{document} Latex generated as <contextnameDate>.tex Convert to .pdf (using, eg, MiKTeX) T&MSP-CSC 4504 : Langages formels et applications

  33. Specification: Famille_ctx0.pdf T&MSP-CSC 4504 : Langages formels et applications

  34. Specification: Married - Famille_ctx1 extends Famille_ctx0 thm3 « married is a partial injection » Prover cannot prove it automatically Should we add it as an axiom? T&MSP-CSC 4504 : Langages formels et applications

  35. Specification: Married - Famille_ctx1 extends Famille_ctx0 T&MSP-CSC 4504 : Langages formels et applications

  36. Travaux Pratique (Context)– a completer (avant le 13) • Define: • mere et pere • fille et fils • frere et soeur • cousin • oncle et tante • Validate Theorems, eg: • oncle = cousin; pere • cousin = cousin~ • pere; enfants = mere; enfants • pere; mere~ = {} • pere; frere; fille <: cousin Before we look at the prover we first look (quickly) at Machines … T&MSP-CSC 4504 : Langages formels et applications

  37. Machine Famile_M0 T&MSP-CSC 4504 : Langages formels et applications

  38. Machine Famile_M0 T&MSP-CSC 4504 : Langages formels et applications

  39. Machine Famile_M0 event marry respects the invariant T&MSP-CSC 4504 : Langages formels et applications

  40. Machine Famile_M0 hommes … h1 h2 h3 h4 h5 h6 h7 h8 h9 epoux … … f2 f3 f4 f5 f6 f7 f8 f1 femmes State1 INITIALISATION: State0 hms = {h2} fms = {} ep = {} hms = {} fms = {} ep = {} add_h (h2) T&MSP-CSC 4504 : Langages formels et applications

  41. Machine Famile_M0 hommes … h1 h2 h3 h4 h5 h6 h7 h8 h9 epoux … … f2 f3 f4 f5 f6 f7 f8 f1 femmes State2 State1 hms = {h2} fms = {f3} ep = {} hms = {h2} fms = {} ep = {} add_f (f3) T&MSP-CSC 4504 : Langages formels et applications

  42. Machine Famile_M0 hommes … h1 h2 h3 h4 h5 h6 h7 h8 h9 epoux … … f2 f3 f4 f5 f6 f7 f8 f1 femmes State3 State2 hms = {h2} fms = {f3, f2} ep = {} hms = {h2} fms = {f3} ep = {} add_f (f2) T&MSP-CSC 4504 : Langages formels et applications

  43. Machine Famile_M0 hommes … h1 h2 h3 h4 h5 h6 h7 h8 h9 epoux … … f2 f3 f4 f5 f6 f7 f8 f1 femmes hms = {h2} fms = {f3, f2} ep = { f3 |-> h2} X √ State3 marry(f3,h2) hms = {h2} fms = {f3, f2} ep = {} hms = {h2} fms = {f3, f2} ep = { f2 |-> h2} marry(f2,h2) T&MSP-CSC 4504 : Langages formels et applications

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