Theory Conclusion

(*<*)
―‹ ******************************************************************** 
 * Project         : HOL-CSP_OpSem - Operational semantics for HOL-CSP
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 * Author          : Benoît Ballenghien, Burkhart Wolff.
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 * This file       : Conclusion
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chapter ‹ Conclusion›

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theory  Conclusion
  imports Recovered_Laws CSP_New_Laws OpSem_Deadlock_Results
begin
  (*>*)

text ‹We started by defining the operators const‹Sliding›, const‹Throw› and const‹Interrupt›
      and provided on them several new laws, especially monotony, "step-law"
      (behaviour with @{term [source] ‹□a ∈ A → P a›}) and continuity.›

text ‹We defined the const‹initials› notion, and described its behaviour with the reference
      processes and the operators of session‹HOL-CSP› and session‹HOL-CSPM›
      (which is already a minor contribution).›

text ‹As main contribution, we defined the @{const [source] ‹After.After›} operator which represents
      a bridge between the denotational and the versions of operational semantics for CSP.
      We made the construction as generic as possible, by exploiting the locale mechanism.
      Therefore we derive the correspondence between denotational and operational 
      semantics by construction. Based on failure divergence or trace divergence refinements, 
      the two operational semantics correspond to the versions described in
      cite‹"roscoe:csp:1998" and "DBLP:journals/entcs/Roscoe15"›.

      We have slight variations that can open up for discussion.

      Thus, we provided a formal theory of operational behaviour for CSP, which is, to our
      knowledge, done for the first time for the entire language and the complete FD-Semantics
      model. Some of the proofs turned out to be extremely complex and out of reach of
      paper-and-pencil reasoning.›

text ‹A notable point is that the experimental order term‹(⊑DT)› behaves surprisingly well:
      initially pushed in session‹HOL-CSP› for pure curiosity, it looks promising for future
      applications, since it gives a direct handle for an operational trace semantics for
      non-diverging processes which is executable.›

text ‹Another take-away is the development of alternatives with term‹(⊑F)› and
      term‹(⊑T)› orders but this remains a bit disappointing because their monotony w.r.t.
      to some operators does not allow to recover all the laws of 
      cite‹"roscoe:csp:1998" and "DBLP:journals/entcs/Roscoe15"›.›

text ‹As a bonus we provided in theory‹HOL-CSP_OpSem.CSP_New_Laws› some powerful laws for CSP.
      Here, we recall only the most important ones:
      
      \begin{center}  
      @{thm [mode = Rule, eta_contract = false] bij_Renaming_Hiding}

      @{thm [mode = Rule, eta_contract = false] bij_Renaming_Sync}

      @{thm [mode = Rule, eta_contract = false] Hiding_Mprefix_non_disjoint}
      \end{center}›

text ‹Finally, we discovered that the @{const [source] After.After} operator and its extensions
      @{const [source] AfterExt.Aftertick} and @{const [source] AfterExt.Aftertrace} have a real
      interest even without the construction of operational semantics.
      
      With induction rules based on @{const [source] AfterExt.Aftertrace}, we could for
      example prove the following theorem:

      \begin{center}  
      @{thm [mode = Rule] AfterExt.data_independence_deadlock_free_Sync_bis}
      \end{center}›

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end
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