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sav07_lecture_3_skeleton [2007/03/20 18:06] vkuncak |
sav07_lecture_3_skeleton [2007/03/20 18:26] wikiadmin |
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| ===== Converting programs (with simple values) to formulas ===== | ===== Converting programs (with simple values) to formulas ===== | ||
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| ==== Context ==== | ==== Context ==== | ||
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| * represent programs using guarded command language, e.g. desugaring of 'if' into non-deterministic choice and assume | * represent programs using guarded command language, e.g. desugaring of 'if' into non-deterministic choice and assume | ||
| * give meaning to guarded command language statements as relations | * give meaning to guarded command language statements as relations | ||
| - | * we can represent relations using set comprehensions; if our program c has two state components, we can represent its meaning R( c ) as | + | * we can represent relations using set comprehensions; if our program c has two state components, we can represent its meaning R( c ) as $\{((x_0,y_0),(x,y)) \mid F \}$, where F is some formula that has x,y,x_0,y_0 as free variables. |
| - | <latex> | + | |
| - | \{((x_0,y_0),(x,y)) \mid F \} | + | |
| - | </latex> | + | |
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| - | where F is some formula that has x,y,x_0,y_0 as free variables. | + | |
| * this is what I mean by ''simple values'': later we will talk about modeling pointers and arrays, but we will still use this as a starting point. | * this is what I mean by ''simple values'': later we will talk about modeling pointers and arrays, but we will still use this as a starting point. | ||
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| * efficient | * efficient | ||
| * create formulas that we can effectively prove later | * create formulas that we can effectively prove later | ||
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| What exactly do we prove about the formula R( c ) ? | What exactly do we prove about the formula R( c ) ? | ||
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| * 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 | ||
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