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sav08:normal_form_of_loop-free_programs [2009/03/04 18:57] vkuncak |
sav08:normal_form_of_loop-free_programs [2009/03/04 18:58] vkuncak |
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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: | ||
| + | \[ | ||
| + | (r_1 \cup r_2) \circ r_3 = (r_1 \circ r_3) \cup (r_2 \circ r_3) | ||
| + | \] | ||
| + | \[ | ||
| + | r_3 \circ (r_1 \cup r_2) = (r_3 \circ r_1) \cup (r_3 \circ r_2) | ||
| + | \] | ||
| + | Normal form: | ||
| + | \[ | ||
| + | \bigcup_{i=1}^n p_i | ||
| + | \] | ||
| + | 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: | ||