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sav07_lecture_3_skeleton [2007/03/20 14:10] vkuncak |
sav07_lecture_3_skeleton [2007/03/20 14:32] vkuncak |
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====== Lecture 3 (Skeleton) ====== | ====== Lecture 3 (Skeleton) ====== | ||
+ | |||
+ | ==== Context ==== | ||
Recall that we can | Recall that we can | ||
* 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 has two state components, we can represent its meaning as | + | * we can represent relations using set comprehensions; if our program r has two state components, we can represent its meaning R(r) 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 mentions x,y,x_0,y_0. | + | 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 | ||
+ | * correct | ||
+ | * efficient | ||
+ | * create formulas that we can prove later | ||
+ | |||
+ | |||
+ | ==== Formulas for basic statements ==== | ||
+ | |||
+ | In our simple language, basic statements are assignment, havoc, assume, assert. | ||
+ | |||
+ | R(x=t) = (x=t & y=y_0 & error=error_0) | ||
+ | |||
+ | **Note**: all our statements will have the property that if error_0 = true, then error=true. That is, you can never recover from an error state. This is convenient: if we prove no errors at the end, then there were never errors in between. | ||
+ | |||
+ | **Note**: the condition y=y_0 & error=error_0 is called <b>frame condition</b>. There are as many conjuncts as there are components of the state. This can be annoying to write, so let us use shorthand frame(x) for it. The shorthand frame(x) denotes a conjunction of v=v_0 for all v that are distinct from x (in this case y and error). We can have zero or more variables as arguments of frame, so frame() means that nothing changes. | ||
+ | |||
+ | R(havoc x) = frame(x) | ||
+ | R(assume F) = F[x:=x_0, y:=y_0, error:=error_0] | ||
+ | R(assert F) = (F -> frame) | ||
+ | |||
+ | **Note**: | ||
+ | |||
+ | x=t is same as havoc(x);assume(x=t) | ||
+ | |||
+ | assert false = crash (stops with error) | ||
+ | |||
+ | assume true = skip (does nothing) | ||
+ | |||
+ | ==== Composing formulas using relation composition ==== | ||
- | Our goal is to compute this formula. | + | This is perhaps the most direct way of transforming programs to formulas. |
+ | It creates formulas that are linear in the size of the program. | ||
+ | Non-deterministic choice is union of relations, that is, disjunction of formulas: | ||
+ | CR(c1; c2) = CR(c1) | CR(c2) | ||
==== Papers ==== | ==== Papers ==== |