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sav07_lecture_3_skeleton [2007/03/20 14:44] vkuncak |
sav07_lecture_3_skeleton [2007/03/20 14:58] vkuncak |
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====== Lecture 3 (Skeleton) ====== | ====== Lecture 3 (Skeleton) ====== | ||
+ | |||
+ | ===== Converting programs (with simple values) to formulas ===== | ||
+ | |||
+ | |||
==== 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 r has two state components, we can represent its meaning R(r) as | + | * we can represent relations using set comprehensions; if our program c has two state components, we can represent its meaning R( c ) as |
<latex> | <latex> | ||
\{((x_0,y_0),(x,y)) \mid F \} | \{((x_0,y_0),(x,y)) \mid F \} | ||
</latex> | </latex> | ||
- | where F is some formula that has x,y,x_0,y_0 as free variables. | + | where F is some formula that has x,y,x_0,y_0 as free variables. |
- | Our goal is to find rules for computing R(r) that are | + | * 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. |
+ | |||
+ | Our goal is to find rules for computing R( c ) that are | ||
* correct | * correct | ||
* efficient | * efficient | ||
- | * create formulas that we can prove later | + | * create formulas that we can effectively prove later |
+ | |||
+ | What exactly do we prove about the formula R( c ) ? | ||
+ | |||
+ | We prove that this formula is **valid** | ||
+ | |||
+ | R( c ) -> error=false | ||
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when c is a basic command. | when c is a basic command. | ||
- | ==== Accumulation of equalities ==== | + | |
+ | |||
+ | ==== Avoiding accumulation of equalities ==== | ||
This approach generates many variables and many frame conditions. | This approach generates many variables and many frame conditions. | ||
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But if a variable is equal to another, it can be substituted using the substitution rules | But if a variable is equal to another, it can be substituted using the substitution rules | ||
- | (exists x_1. x_1 = t & F(x_1)) <-> F(t) | + | (exists x_1. x_1=t & F(x_1)) <-> F(t) |
- | (forall x_1. x_1 = t -> F(x_1) <-> F(t) | + | (forall x_1. x_1=t -> F(x_1) <-> F(t) |
We can apply these rules to reduce the size of formulas. | We can apply these rules to reduce the size of formulas. |