Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
sav07_lecture_3_skeleton [2007/03/21 11:04] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 14:27] vkuncak |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== Lecture 3 (Skeleton) ====== | ====== Lecture 3 (Skeleton) ====== | ||
| - | ===== Converting programs (with simple values) to formulas ===== | + | Summary of what we are doing in today's class: |
| + | {{vcg-big-picture.png}} | ||
| + | |||
| + | ===== Verification condition generation: converting programs into formulas ===== | ||
| ==== Context ==== | ==== Context ==== | ||
| Line 158: | Line 161: | ||
| - | ===== Proving quantifier-free linear arithmetic formulas ===== | + | |
| + | ===== One useful decision procedure: Proving quantifier-free linear arithmetic formulas ===== | ||
| Suppose that we obtain (one or more) verification conditions of the form | Suppose that we obtain (one or more) verification conditions of the form | ||
| Line 165: | Line 169: | ||
| \end{equation*} | \end{equation*} | ||
| - | whose validity we need to prove. We here assume that F contains only | + | whose validity we need to prove. We here assume that F contains only linear arithmetic. Note: we can check satisfiability of $F\ \land\ (\mbox{error}=\mbox{true})$. We show an algorithm to check this satisfiability. |
| - | + | ||
| - | Note: we can check satisfiability of $F\ \land\ (\mbox{error}=\mbox{true})$. | + | |
| ==== Quantifier Presburger arithmetic ==== | ==== Quantifier Presburger arithmetic ==== | ||
| Line 177: | Line 179: | ||
| T ::= var | T + T | K * T (terms) | T ::= var | T + T | K * T (terms) | ||
| A ::= T=T | T <= T (atomic formulas) | A ::= T=T | T <= T (atomic formulas) | ||
| - | F ::= F & F | F|F | ~F (formulas) | + | F ::= A | F & F | F|F | ~F (formulas) |
| To get full Presburger arithmetic, allow existential and universal quantifiers in formula as well. | To get full Presburger arithmetic, allow existential and universal quantifiers in formula as well. | ||
| Line 186: | Line 188: | ||
| Proof: small model theorem. | Proof: small model theorem. | ||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ||
| Line 243: | Line 237: | ||
| * Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}} | * Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}} | ||
| * Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract | * Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract | ||
| - | |||
| - | |||
| - | |||
| - | |||