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sav08:normal_form_of_loop-free_programs [2009/03/04 18:57]
vkuncak
sav08:normal_form_of_loop-free_programs [2015/04/21 17:30] (current)
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 +====== Normal form for loop-free programs ======
 +
 +Example:
 +<​code>​
 +(if (x < 0) x=x+1 else x=x);
 +(if (y < 0) y=y+x else y=y);
 +</​code>​
 +
 +Without loops, after expressing conditionals using [] we obtain
 +  c ::=  x=T | assume(F) |  c [] c  |  c ; c 
 +
 +Laws:
 +\begin{equation*}
 +     (r_1 \cup r_2) \circ r_3 = (r_1 \circ r_3) \cup (r_2 \circ r_3)
 +\end{equation*}
 +\begin{equation*}
 +     r_3 \circ (r_1 \cup r_2) = (r_3 \circ r_1) \cup (r_3 \circ r_2)
 +\end{equation*}
 +Normal form:
 +\begin{equation*}
 +   ​\bigcup_{i=1}^n p_i
 +\end{equation*}
 +Each $p_i$ is of form $b_1 \circ \ldots \circ b_k$ for some $k$, where each $b_i$ is assignment or assume. ​ Each $p_i$ corresponds to one of the finitely paths from beginning to end of the acyclic control-flow graph for loop-free program.
 +
 +Length of normal form with sequences of if-then-else.
  
 We want to show: We want to show:
-\[+\begin{equation*}
     \{ P \} r \{ Q \}     \{ P \} r \{ Q \}
-\]+\end{equation*}
  
 ==== Verifying Each Path Separately ==== ==== Verifying Each Path Separately ====
  
 By normal form this is By normal form this is
-\[+\begin{equation*}
     \{ P \}  \bigcup_{i=1}^n p_i \{ Q \}     \{ P \}  \bigcup_{i=1}^n p_i \{ Q \}
-\]+\end{equation*}
 which is equivalent to which is equivalent to
-\[+\begin{equation*}
    ​\bigwedge_{i=1}^n (\{P\} p_i \{Q\})    ​\bigwedge_{i=1}^n (\{P\} p_i \{Q\})
-\]+\end{equation*}
  
 Note: the rule also applies to infinite union of paths (e.g. generated by loops). Note: the rule also applies to infinite union of paths (e.g. generated by loops).
 
sav08/normal_form_of_loop-free_programs.txt · Last modified: 2015/04/21 17:30 (external edit)
 
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