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# Lab for Automated Reasoning and Analysis LARA

# Differences

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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). |