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 | ||
|
sav07_lecture_3 [2007/03/22 13:35] yuanjianwz |
sav07_lecture_3 [2007/03/22 16:20] yuanjianwz |
||
|---|---|---|---|
| Line 96: | Line 96: | ||
| We can apply these rules to reduce the size of formulas. | We can apply these rules to reduce the size of formulas. | ||
| + | |||
| ==== Approximation ==== | ==== Approximation ==== | ||
| - | If (F -> G) is value, we say that F is stronger than F and we say G is weaker than F. | + | If (F -> G) is valid, 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). | 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). | ||
| Line 214: | Line 215: | ||
| Proof: small model theorem. | Proof: small model theorem. | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| Line 220: | Line 226: | ||
| The idea is to reduce the case, for example: | The idea is to reduce the case, for example: | ||
| - | ∃x,y,z.F | + | //∃x,y,z.F// |
| reduce to | reduce to | ||
| - | ∃x≤ M,y≤ M,z ≤ M.F | + | //∃x≤ M,y≤ M,z ≤ M.F// |
| - | We try to figure out M. | + | Then we try to figure out the boundary M. |
| Another example: | Another example: | ||
| - | ¬t1 < t2 | + | //¬t1 < t2// |
| reduce to | reduce to | ||
| - | t2+1≤t1 | + | //t2+1≤t1// |
| How about the not equal ? | How about the not equal ? | ||
| - | t1≠t2 | ||
| - | can be reduced to | ||
| - | (t1 < t2 ) ∨ (t2 < t1) => (t1 ≤ t2-1) ∨ (t2 ≤ t1-1) | ||
| - | |||
| + | //t1≠t2// | ||
| + | can be reduced to | ||
| + | //(t1 < t2 ) ∨ (t2 < t1) => (t1 ≤ t2-1) ∨ (t2 ≤ t1-1)// | ||
| First step: transform to disjunctive normal form. | First step: transform to disjunctive normal form. | ||