X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/blobdiff_plain/1b923f193392e3ce847882c24a128eff4bee9992..251b65007a909068d3344bd82b7c5ec0ffb0a21a:/annexePromelaProof.tex

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 \JFC{Voir section~\ref{sec:spin:proof}}
 
+Cette section donne les preuves des deux thÃ©orÃ¨mes de correction et complÃ©tude 
+du chapitre~\ref{chap:promela}.
+
+
+\begin{lemma}[Strategy Equivalence]\label{lemma:strategy}
+  Let $\phi$  be a  DDN with strategy  $(S^t)^{t \in  \Nats}$ and $\psi$  be its
+  translation.  There exists an execution of $\psi$ with weak fairness s.t.  the
+  scheduler makes \verb+update_elems+ update elements of $S^t$ at iteration $t$.
+\end{lemma}
+\begin{Proof}
+  The proof is direct  for $t=0$. Let us suppose it is  established until $t$ is
+  some $t_0$.  Let  us consider pseudo-periodic strategies.  Thanks  to the weak
+  fairness equity property, \verb+update_elems+ will modify elements of $S^t$ at
+  iteration $t$.
+\end{Proof}
+
+In     what     follows,     let     $Xd^t_{ji}$     be     the     value     of
+\verb+Xd[+$j$\verb+].v[+$i$\verb+]+  after  the   $t^{\text{th}}$  call  to  the
+function  \verb+fetch_values+.  Furthermore,  let $Y^k_{ij}$  be the  element at
+index  $k$  in  the  channel  \verb+channels[i].sent[j]+ of  size  $m$,  $m  \le
+\delta_0$; $Y^0_{ij}$ and $Y^{m-1}_{ij}$ are  respectively the head and the tail
+of the  channel.  Secondly, let $(M_{ij}^t)^{t \in  \{1, 1.5, 2, 2.5,\ldots\}}$ be a
+sequence such  that $M_{ij}^t$ is the  partial function that  associates to each
+$k$, $0 \le k \le  m-1$, the tuple $(Y^k_{ij},a^k_{ij},c^k_{ij})$ while entering
+into the \verb+update_elems+ at iteration $t$ where
+% \begin{itemize}
+% \item
+ $Y^k_{ij}$ is the value of the channel \verb+channels[i].sent[j]+
+  at index $k$,
+%\item 
+$a^k_{ij}$ is the date (previous to $t$) when $Y^k_{ij}$ has been added and
+%\item 
+$c^k_{ij}$ is the  first date at which the value is  available on $j$. So,
+  the value is removed from the channel $i\rightarrow j$ at date $c^k_{ij}+1$.
+%\end{itemize}
+$M_{ij}^t$ has the following signature:
+\begin{equation*}
+\begin{array}{rrcl}
+M_{ij}^t: & 
+\{0,\ldots, \textit{max}-1\} &\rightarrow & E_i\times \Nats \times \Nats \\
+& k  \in \{0,\ldots, m-1\} & \mapsto & M_{ij}(k)= (Y^k_{ij},a^k_{ij},c^k_{ij}).
+\end{array}  
+\end{equation*}
+
+Intuitively,  $M_{ij}^t$  is  the  memory  of  \verb+channels[i].sent[j]+  while
+starting the iteration $t$.  Notice that the domain of any $M_{ij}^1$ is $\{0\}$
+and   $M_{ij}^1(0)=(\verb+Xp[i]+,0,0)$:    indeed,   the   \verb+init+   process
+initializes \verb+channels[i].sent[j]+ with \verb+Xp[i]+.
+
+Let us  show how  to make the  indeterminism inside the  two functions\linebreak
+\verb+fetch_values+  and  \verb+diffuse_values+  compliant with  \Equ{eq:async}.
+The  function   $M_{ij}^{t+1}$  is  obtained   by  the  successive   updates  of
+$M_{ij}^{t}$  through   the  two  functions\linebreak   \verb+fetch_values+  and
+\verb+diffuse_values+.   Abusively,   let  $M_{ij}^{t+1/2}$  be   the  value  of
+$M_{ij}^{t}$ after the former function during iteration $t$.
+
+In  what follows, we  consider elements  $i$ and  $j$ both  in $\llbracket  1, n
+\rrbracket$   that  are   updated.   At   iteration   $t$,  $t   \geq  1$,   let
+$(Y^0_{ij},a^0_{ij},c^0_{ij})$ be the value of $M_{ij}^t(0)$ at the beginning of
+\verb+fetch_values+.   If $t$  is  equal  to $c^0_{ij}+1$  then  we execute  the
+instruction    that   assigns    $Y^0_{ij}$   (\textit{i.e.},     the   head    value   of
+\verb+channels[i].sent[j]+)  to   $Xd_{ji}^t$.   In  that   case,  the  function
+$M_{ij}^t$ is updated as follows: $M_{ij}^{t+1/2}(k) = M_{ij}^{t}(k+1)$ for each
+$k$, $0 \le k \le m-2$ and $m-1$ is removed from the domain of $M_{ij}^{t+1/2}$.
+Otherwise (\textit{i.e.}, when  $t < c^0_{ij}+1$ or when  the domain of $M_{ij}$
+is  empty)   the  \verb+skip+  statement  is  executed   and  $M_{ij}^{t+1/2}  =
+M_{ij}^{t}$.
+
+In the function \verb+diffuse_values+, if  there exists some $\tau$, $\tau\ge t$
+such that \mbox{$D^{\tau}_{ji} = t$}, let  $c_{ij}$ be defined by $ \min\{l \mid
+D^{l}_{ji} =  t \} $.  In  that case, we  execute the instruction that  adds the
+value   \verb+Xp[i]+   to  the   tail   of  \verb+channels[i].sent[j]+.    Then,
+$M_{ij}^{t+1}$ is defined  as an extension of $M_{ij}^{t+1/2}$  in $m$ such that
+$M_{ij}^{t+1}(m)$ is $(\verb+Xp[i]+,t,c_{ij})$.  Otherwise (\textit{i.e.}, when $\forall l
+\, .  \, l \ge t  \Rightarrow D^{l}_{ji} \neq t$ is established) the \verb+skip+
+statement is executed and $M_{ij}^{t+1} = M_{ij}^{t+1/2}$.
+
+
+\begin{lemma}[Existence of SPIN Execution]\label{lemma:execution}
+  For any sequences $(S^t)^{t  \in \Nats}$,\linebreak $(D^t)^{t \in \Nats}$, for
+  any map $F$ there exists a SPIN  execution such that for any iteration $t$, $t
+  \ge  1$, for  any $i$ and $j$ in  $\llbracket 1, n \rrbracket$  we  have the
+  following properties:
+   
+\noindent If the domain of $M_{ij}^t$ is not empty, then
+\begin{equation}
+  \left\{
+    \begin{array}{rcl}
+      M_{ij}^1(0) & = & \left(X_i^{D_{ji}^{0}}, 0,0 \right) \\
+      \textrm{if $t \geq 2$ then }M_{ij}^t(0) & = &
+      \left(X_i^{D_{ji}^{c}},D_{ji}^{c},c \right) \textrm{, }
+      c = \min\{l | D_{ji}^l > D_{ji}^{t-2} \}
+    \end{array}
+  \right.
+  \label{eq:Mij0}
+\end{equation}
+\noindent Secondly we have:
+\begin{equation}
+  \forall t'\, .\,   1 \le t' \le t  \Rightarrow   Xd^{t'}_{ji} = X^{D^{t'-1}_{ji}}_i
+  \label{eq:correct_retrieve}
+\end{equation}
+\noindent Thirdly, for any $k\in S^t$.  Then, the value of the computed variable
+\verb+Xp[k]+  at  the  end  of  the  \verb+update_elems+  process  is  equal  to
+$X_k^{t}$          \textit{i.e.},          $F_{k}\left(         X_1^{D_{k\,1}^{t-1}},\ldots,
+  X_{n}^{D_{k\,{n}}^{t-1}}\right)$ at the end of the $t^{\text{th}}$ iteration.
+\end{lemma}
+\begin{Proof}
+The proof is done by induction on the number of iterations.
+
+\paragraph{Initial case:}
+
+For the first  item, by definition of $M_{ij}^t$, we  have $M_{ij}^1(0) = \left(
+  \verb+Xp[i]+, 0,0 \right)$ that is obviously equal to $\left(X_i^{D_{ji}^{0}},
+  0,0 \right)$.
+
+Next, the first call to  the function \verb+fetch_value+ either assigns the head
+of   \verb+channels[i].sent[j]+  to   \verb+Xd[j].v[i]+  or   does   not  modify
+\verb+Xd[j].v[i]+.  Thanks to  the \verb+init+ process, both cases  are equal to
+\verb+Xp[i]+,  \textit{i.e.}, $X_i^0$.  The  equation (\ref{eq:correct_retrieve})  is then
+established.
+
+
+For  the last  item, let  $k$, $0  \le  k \le  n-1$.  At  the end  of the  first
+execution\linebreak   of   the  \verb+update_elems+   process,   the  value   of
+\verb+Xp[k]+       is\linebreak       $F(\verb+Xd[+k\verb+].v[0]+,       \ldots,
+\verb+Xd[+k\verb+].v[+n-1\verb+]+)$.  Thus,  by definition of $Xd$,  it is equal
+to $F(Xd^1_{k\,0}, \ldots,Xd^1_{k\,n-1})$.  Thanks to \Equ{eq:correct_retrieve},
+we can conclude the proof.
+
+
+
+\paragraph{Inductive case:}
+
+Suppose now that lemma~\ref{lemma:execution} is established until iteration $l$.
+
+First,  if domain  of definition  of the  function $M_{ij}^l$  is not  empty, by
+induction  hypothesis $M_{ij}^{l}(0)$  is  $\left(X_i^{D_{ji}^{c}}, D_{ji}^{c},c
+\right)$ where $c$ is $\min\{k | D_{ji}^k > D_{ji}^{l-2} \}$.
+
+At iteration $l$, if  $l < c + 1$ then the  \verb+skip+ statement is executed in
+the   \verb+fetch_values+  function.   Thus,   $M_{ij}^{l+1}(0)$  is   equal  to
+$M_{ij}^{l}(0)$.  Since $c > l-1$  then $D_{ji}^c > D_{ji}^{l-1}$ and hence, $c$
+is $\min\{k  | D_{ji}^k  > D_{ji}^{l-1} \}$.  Obviously, this implies  also that
+$D_{ji}^c > D_{ji}^{l-2}$ and $c=\min\{k | D_{ji}^k > D_{ji}^{l-2} \}$.
+
+We now consider that at iteration $l$, $l$ is $c + 1$.  In other words, $M_{ij}$
+is modified depending on the domain $\dom(M^l_{ij})$ of $M^l_{ij}$:
+\begin{itemize}
+\item  if $\dom(M_{ij}^{l})=\{0\}$  and $\forall  k\,  . \,  k\ge l  \Rightarrow
+  D^{k}_{ji} \neq l$  is established then $\dom(M_{ij}^{l+1})$ is  empty and the
+  first item of the lemma  is established; 
+\item if $\dom(M_{ij}^{l})=\{0\}$ and $\exists  k\, . \, k\ge l \land D^{k}_{ji}
+  = l$  is established then $M_{ij}^{l+1}(0)$  is $(\verb+Xp[i]+,l,c_{ij})$ that
+  is  added  in  the  \verb+diffuse_values+ function  s.t.\linebreak  $c_{ij}  =
+  \min\{k  \mid  D^{k}_{ji}  = l  \}  $.   Let  us  prove  that we  can  express
+  $M_{ij}^{l+1}(0)$  as  $\left(X_i^{D_{ji}^{c'}},D_{ji}^{c'},c' \right)$  where
+  $c'$ is  $\min\{k |  D_{ji}^k > D_{ji}^{l-1}  \}$.  First,  it is not  hard to
+  establish that  $D_{ji}^{c_{ij}}= l \geq  D_{ji}^{l} > D_{ji}^{l-1}$  and thus
+  $c_{ij}  \geq   c'$.   Next,  since   $\dom(M_{ij}^{l})=\{0\}$,  then  between
+  iterations $D_{ji}^{c}+1$ and $l-1$, the \texttt{diffuse\_values} function has
+  not updated $M_{ij}$.  Formally we have
+$$
+\forall t,k  \, .\, D_{ji}^c <  t < l \land  k \geq t  \Rightarrow D_{ji}^k \neq
+t.$$
+
+Particularly, $D_{ji}^{c'} \not  \in \{D_{ji}^{c}+1,\ldots,l-1\}$.  We can apply
+the     third    item     of    the     induction    hypothesis     to    deduce
+$\verb+Xp[i]+=X_i^{D_{ji}^{c'}}$ and we can conclude.
+
+\item  if   $\{0,1\}  \subseteq  \dom(M_{ij}^{l})$   then  $M_{ij}^{l+1}(0)$  is
+  $M_{ij}^{l}(1)$.   Let  $M_{ij}^{l}(1)=  \left(\verb+Xp[i]+, a_{ij}  ,  c_{ij}
+  \right)$.   By  construction $a_{ij}$  is  $\min\{t'  |  t' >  D_{ji}^c  \land
+  (\exists k \, .\, k \geq t' \land D_{ji}^k = t')\}$ and $c_{ij}$ is $\min\{k |
+  D_{ji}^k = a_{ij}\}$.  Let us show  $c_{ij}$ is equal to $\min\{k | D_{ji}^k >
+  D_{ji}^{l-1} \}$ further  referred as $c'$.  First we  have $D_{ji}^{c_{ij}} =
+  a_{ij} >  D_{ji}^c$. Since $c$  by definition is  greater or equal to  $l-1$ ,
+  then $D_{ji}^{c_{ij}}>  D_{ji}^{l-1}$ and then $c_{ij} \geq  c'$.  Next, since
+  $c$ is  $l-1$, $c'$ is $\min\{k |  D_{ji}^k > D_{ji}^{c} \}$  and then $a_{ij}
+  \leq  D_{ji}^{c'}$. Thus,  $c_{ij} \leq  c'$  and we  can conclude  as in  the
+  previous part.
+\end{itemize}
+
+
+The case where  the domain $\dom(M^l_{ij})$ is empty but  the formula $\exists k
+\, .\, k \geq  l \land D_{ji}^k = l$ is established  is equivalent to the second
+case given above and then is omitted.
+
+
+Secondly, let us focus on the formula~(\ref{eq:correct_retrieve}).  At iteration
+$l+1$, let $c'$ be defined as $\min\{k | D_{ji}^k > D_{ji}^{l-1} \}$.  Two cases
+have to be  considered depending on whether $D_{ji}^{l}$  and $D_{ji}^{l-1}$ are
+equal or not.
+\begin{itemize}
+\item If  $D_{ji}^{l} = D_{ji}^{l-1}$, since $D_{ji}^{c'}  > D_{ji}^{l-1}$, then
+  $D_{ji}^{c'} > D_{ji}^{l}$ and then $c'$  is distinct from $l$. Thus, the SPIN
+  execution detailed  above does not  modify $Xd_{ji}^{l+1}$.  It is  obvious to
+  establish   that   $Xd_{ji}^{l+1}  =   Xd_{ji}^{l}   =  X_i^{D_{ji}^{l-1}}   =
+  X_i^{D_{ji}^{l}}$.
+\item Otherwise $D_{ji}^{l}$ is greater than $D_{ji}^{l-1}$ and $c$ is thus $l$.
+  According     to     \Equ{eq:Mij0}     we     have    proved,     we     have
+  $M_{ij}^{l+1}(0)=(X_i^{D_{ji}^{l}},D_{ji}^{l},l)$.   Then  the SPIN  execution
+  detailed above  assigns $X_i^{D_{ji}^{l}}$ to $Xd_{ji}^{l+1}$,  which ends the
+  proof of (\ref{eq:correct_retrieve}).
+\end{itemize}
+
+We are left to prove the induction of  the third part of the lemma.  Let $k$, $k
+\in S^{l+1}$. % and $\verb+k'+ = k-1$.
+At the  end of the first  execution of the \verb+update_elems+  process, we have
+$\verb+Xp[+k\verb+]+=                                   F(\verb+Xd[+k\verb+][0]+,
+\ldots,\verb+Xd[+k\verb+][+n\verb+-1]+)+$.  By  definition of $Xd$,  it is equal
+to      $F(Xd^{l+1}_{k\,0},      \ldots,Xd^{l+1}_{k\,n-1})$.      Thanks      to
+\Equ{eq:correct_retrieve} we have proved, we can conclude the proof.
+\end{Proof}
+
+
+\begin{lemma}
+  Bounding the size of channels  to $\textit{max} = \delta_0$ is sufficient when
+  simulating a DDN where delays are bounded by $\delta_0$.
+\end{lemma}
+
+\begin{Proof}
+  For  any $i$,  $j$, at  each  iteration $t+1$,  thanks to  bounded delays  (by
+  $\delta_0$),  element $i$  has to  know at  worst $\delta_0$  values  that are
+  $X_j^{t}$, \ldots, $X_j^{t-\delta_0+1}$.  They  can be stored into any channel
+  of size $\delta_0$.
+\end{Proof}
+
+
+\begin{theorem}[Soundness wrt universal convergence property]\label{Theo:sound}
+  Let $\phi$ be  a DDN model and $\psi$ be its  translation.  If $\psi$ verifies
+  the  LTL  property  (\ref{eq:ltl:conv})  under weak  fairness  property,  then
+  iterations of $\phi$ are universally convergent.
+\end{theorem}
+\begin{Proof}
+%   For  the  case  where  the  strategy  is  finite,  one  notice  that  property
+%   verification  is achieved  under  weak fairness  property.  Instructions  that
+%   write or read into \verb+channels[j].sent[i]+ are continuously enabled leading
+%   to  convenient  available  dates  $D_{ji}$.   It is  then  easy  to  construct
+%   corresponding iterations of the DDN that are convergent.
+% \ANNOT{quel sens donnes-tu a \emph{convenient} ici ?}
+
+  Let us show the contraposition of the theorem.  The previous lemmas have shown
+  that for any  sequence of iterations of the DDN, there  exists an execution of
+  the PROMELA  model that  simulates them.   If some iterations  of the  DDN are
+  divergent, then  they prevent  the PROMELA model  from stabilizing,  \textit{i.e.},  not
+  verifying the LTL property (\ref{eq:ltl:conv}).
+\end{Proof}
+
+
+% \begin{Corol}[Soundness wrt universall convergence property]\label{Theo:sound}
+% Let $\phi$ be a DDN model where strategy, $X^(0)$ 
+% are only constrained to be pseudo-periodic and
+% in $E$ respectively.
+% Let $\psi$ be its translation.
+% If all the executions of $\psi$ converge, 
+% then  iterations of $\phi$ are universally convergent.
+% \end{Corol}
+
+
+
+\begin{theorem}[Completeness wrt universal convergence property]\label{Theo:completeness}
+  Let $\phi$ be a  DDN model and $\psi$ be its translation.   If $\psi$ does not
+  verify the LTL property  (\ref{eq:ltl:conv}) under weak fairness property then
+  the iterations of $\phi$ are divergent.
+\end{theorem}
+\begin{Proof}
+  For models $\psi$  that do not verify the  LTL property (\ref{eq:ltl:conv}) it
+  is easy  to construct corresponding iterations  of the DDN,  whose strategy is
+  pseudo-periodic since weak fairness property is taken into account.
+
+%   i.e. iterations that  are divergent.   Executions are
+%   performed under weak  fairness property; we then detail  what are continuously
+%   enabled:
+% \begin{itemize}
+% \item if the strategy is not  defined as periodic, elements $0$, \ldots, $n$ are
+%   infinitely often updated leading to pseudo-periodic strategy;
+% \item  instructions  that  write  or read  into  \verb+channels[j].sent[i]+  are
+%   continuously enabled leading to convenient available dates $D_{ji}$.
+% \end{itemize}
+% The simulated DDN does not stabilize and its iterations are divergent.
+ \end{Proof}
+
