Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav08:sign_analysis_of_expressions_and_programs [2008/05/06 23:50] vkuncak |
sav08:sign_analysis_of_expressions_and_programs [2008/05/07 01:08] vkuncak |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== Sign Analysis of Expressions and Programs ====== | ====== Sign Analysis of Expressions and Programs ====== | ||
+ | |||
+ | ===== Sign Analysis of Expressions ===== | ||
Suppose we want to test quickly whether the result of an expression is positive, negative or zero. | Suppose we want to test quickly whether the result of an expression is positive, negative or zero. | ||
- | What is the sign of | + | What is (quickly) the sign of |
\[ | \[ | ||
(31283321 + 8629184) \times (-34234) \times (-4 + (123123 \times (-3))) | (31283321 + 8629184) \times (-34234) \times (-4 + (123123 \times (-3))) | ||
Line 13: | Line 15: | ||
\] | \] | ||
++++ | ++++ | ||
+ | |||
+ | What is (quickly) the sign of | ||
+ | \[ | ||
+ | (28166461706 + (723497 \times (- 38931))) \times 42 | ||
+ | \] | ||
+ | ++++| | ||
+ | \[ | ||
+ | (pos \oplus (pos \otimes neg)) \otimes pos = (pos \oplus neg) \otimes pos = \top \otimes pos = \top | ||
+ | \] | ||
+ | ++++ | ||
+ | |||
+ | Concrete domain: $\mathbb{Z} = \{ \ldots, -2, -1, 0, 1, 2, \ldots \}$ | ||
+ | |||
+ | Concrete operations: ${+}, {\times} : \mathbb{Z}^2 \to \mathbb{Z}$ | ||
+ | |||
+ | Abstract domain: $A = \{ neg, nul, pos, \top \}$ | ||
+ | |||
+ | Abstract operations: $\oplus, \otimes : A^2 \to A$ defined by tables: | ||
+ | |||
+ | ===== Sign Analysis of Programs ===== | ||
+ | |||
+ | Prove that this program never reports error: | ||
+ | |||
+ | <code> | ||
+ | i = 20; | ||
+ | x = 2; | ||
+ | while (i > 0) { | ||
+ | x = x + 4; | ||
+ | i = i - 1; | ||
+ | } | ||
+ | if (x==0) { | ||
+ | error; | ||
+ | } else { | ||
+ | y = 1000/x; | ||
+ | } | ||
+ | </code> | ||
+ | |||
+ | Abstract state: map each variable to element of $A$. | ||
+ | |||
+ | * computation over control-flow graph | ||
+ | |||
+ | What does it mean for such computation to be correct? | ||