Theory Conclusion

(*<*)
―‹ ********************************************************************
 * Project         : HOL-CSPM - Architectural operators for HOL-CSP
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 * Author          : Benoît Ballenghien, Safouan Taha, Burkhart Wolff.
 *
 * This file       : Conclusion
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chapter‹ Conclusion ›

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theory Conclusion                                               
  imports "HOL-CSPM" "HOL-Library.LaTeXsugar"
begin 
  (*>*)



text ‹In this session, we defined three architectural operators: const‹GlobalDet›,
      const‹MultiSync›, and const‹MultiSeq› as respective generalizations 
      of term‹P □ Q›, term‹P ⟦S⟧ Q›, and term‹P ; Q›.
      The generalization of term‹P ⊓ Q›, term‹GlobalNdet›, is already in
      session‹HOL-CSP› since it is required for some algebraic laws.›



text ‹We did this in a fully-abstract way, that is:
      ▪ const‹Det› is commutative, idempotent and admits const‹STOP› as a neutral element so
        we defined const‹GlobalDet› on a typ‹'a set› term‹A› by making it equal
        to const‹STOP› when term‹A = {}›. Continuity only holds for finite cases,
        while the operator is always defined.

      ▪ const‹Ndet› is also commutative and idempotent so in session‹HOL-CSP›
        const‹GlobalNdet› has been defined
        on a typ‹'a set› term‹A› by making it equal to const‹STOP› when term‹A = {}›.
        Beware of the fact that const‹STOP› is not the
        neutral element for const‹Ndet› (this operator does not admit a neutral element) so 
        we ❙‹do not have› the equality
        @{term [display, eta_contract = false] ‹(⊓p ∈ {a}. P p) = P a ⊓ (⊓p ∈ {}. P p)›}
        while this holds for const‹Det› and const‹GlobalDet›).
        Again, continuity only holds for finite cases.

      ▪ const‹Sync› is commutative but is not idempotent so we defined const‹MultiSync›
        on a typ‹'a multiset› term‹M› to keep the multiplicity of the processes.
        We made it equal to const‹STOP› when term‹M = {#}› but like const‹Ndet›,
        const‹Sync› does not admit a neutral element so beware of the fact that in general
        @{term [display, eta_contract = false] ‹(⟦S⟧p ∈# {#a#}. P p) ≠ P a ⟦S⟧ (⟦S⟧p ∈# {#}. P p)›}.
        By construction, multiset are finite and therefore continuity holds.

      ▪ const‹Seq› is neither commutative nor idempotent, and term‹SKIP r› is neutral only on
        the left hand side (note if the second type typ‹'r› of typ‹('a, 'r) processptick› is actually
        typ‹unit›, that is to say we go back to the old version without parameterized termination,
        it is neutral element on both sides, see @{thm Seq_SKIP SKIP_Seq}).
        Therefore we defined const‹MultiSeq› on a typ‹'a list› term‹L› to keep the multiplicity
        and the order of the processes, and the folding is done on the reversed list in order to
        enjoy the neutrality of term‹SKIP r› on the left hand side.
        For example, proving
        @{term [eta_contract = false] ‹SEQ p ∈@ L1. P p ; SEQ p ∈@ L2. P p = SEQ p ∈@ (L1 @ L2). P p›}
        in general requires term‹L2 ≠ []›.›



text ‹We presented two examples: Dining philosophers, and and the 'Plain old telephone system' POTS.

      In both, we underlined the usefulness of the architectural operators
      for modeling complex systems.›

text ‹Finally we provided powerful results on const‹events_of› and const‹deadlock_free›
      among which the most important is undoubtedly:
      
         @{thm [display, eta_contract = false] MultiInter_deadlock_free[no_vars]}
      
      This theorem allows, for example, to establish:

         @{term [display, eta_contract = false]
             ‹0 < n ⟹ deadlock_free (||| m ∈# mset [0..<n]. P m)›}

      under the assumption that a family of processes parameterized by
      term‹m› @{text "::"} typ‹nat› verifies term‹∀m < n::nat. deadlock_free (P m)›.
      ›

text ‹More recently, two operators const‹Throw› and const‹Interrupt› have been added.
      The corresponding continuities and algebraic laws can also be found
      in this session.›

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end
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