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Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav07_lecture_4 [2007/03/27 16:35] leander.eyer |
sav07_lecture_4 [2007/03/28 09:49] iulian.dragos |
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assume(x ∉ alloc); | assume(x ∉ alloc); | ||
alloc = alloc ∪ {x} | alloc = alloc ∪ {x} | ||
+ | |||
Line 275: | Line 276: | ||
==== Dynamically allocated arrays ==== | ==== Dynamically allocated arrays ==== | ||
- | When we allow dynamically allocated arrays, we introduce an additional parameter to the array function which identifies the array in question. | + | When we allow dynamically allocated arrays, we introduce a new global function **array** which maps a pair (arrayID, index) to values. |
x[i] = v; | x[i] = v; | ||
Line 289: | Line 290: | ||
===== Proving formulas with uninterpreted functions ===== | ===== Proving formulas with uninterpreted functions ===== | ||
+ | ==== Congruence closure algorithm ==== | ||
+ | The congruence closure algorithm can be used to proove the correctness of quantifier free formulas by examining congruence closures of the statements in the formula. | ||
+ | Recall the following properties of the relation **equivalence**: | ||
+ | - x = x (everything is equal to itself) (reflexivity) | ||
+ | - x = y -> y = x (symmetry) | ||
+ | - x = y ∧ y = z -> x = z (transitivity) | ||
+ | A congruence is an equivalence relationship with the additional property | ||
+ | * (x1 = x2 ∧ y1 = y2) -> f(x1, y1) = f(x2, y2) | ||
+ | a ≡ b (mod n) is a congruence in respect to addition. Indeed: | ||
- | + | a ≡ b (mod n) ∧ c ≡ d (mod n) -> a + c ≡ b + d (mod n) | |
- | + | ||
- | + | ||
- | + | ||
- | ==== Congruence closure algorithm ==== | + | |
- | + | ||
- | The congruence closure algorithm can be used to proove the correctness of quantifier free formulas by examining congruence closures of the statements in the formula. | + | |
- | + | ||
- | Recall the following properties of relations: | + | |
- | - x = x (everything is equal to itself) | + | |
- | - x = y -> y = x (reflexivity) | + | |
- | - x = y ∧ y = z -> x = z (transitivity) | + | |
- | - (x1 = x2 ∧ y1 = y2) -> f(x1, y1) = f(x2, y2) (equivalence in functions) | + | |
Equality is a congruence with respect to all function symbols. | Equality is a congruence with respect to all function symbols. |