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--- sav07_lecture_3_skeleton [2007/03/20 21:13]
vkuncak
+++ sav07_lecture_3_skeleton [2007/03/21 09:25]
vkuncak
@@ Line 90: / Line 90: @@
 We can apply these rules to reduce the size of formulas.
-==== Abstraction ====
-  * for proving properties
+==== Approximation ====
-  * for finding errors
+If (F -> G) is value, we say that F is stronger than F and we say G is weaker than F.
+When a formula would be too complicated, we can instead create a simpler approximate formula.  To be sound, if our goal is to prove a property, we need to generate a *larger* relation, which corresponds to a weaker formula describing a relation, and a stronger verification condition.  (If we were trying to identify counterexamples, we would do the opposite).
+We can replace "assume F" with "assume F1" where F1 is weaker.  Consequences:
+  * omtiting complex if conditionals (assuming both branches can happen - as in most type systems)
+  * replacing complex assignments with arbitrary change to variable: because x=t is havoc(x);assume(x=t) and we drop the assume
+This idea is important in static analysis.
 ==== Symbolic execution ====
 Symbolic execution converts programs into formulas by going forward.  It is therefore somewhat analogous to the way an [[interpreter]] for the language would work.  It is based on the notion of strongest postcondition.
@@ Line 103: / Line 114: @@
 While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward.
+==== Inferring Loop Invariants ====
+Suppose we compute strongest postcondition in a program where we unroll loop k times.
+  * What does it denote?
+  * What is its relationship to loop invariant?
+Weakening strategies
+  * maintain a conjunction
+  * drop conjuncts that do not remain true
+Alternative:
+  * decide that you will only loop for formulas of restricted form, as in abstract interpretation and data flow analysis (next week)
 ===== Proving quantifier-free linear arithmetic formulas =====