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sav07_lecture_3_skeleton [2007/03/18 19:53] vkuncak |
sav07_lecture_3_skeleton [2007/03/20 14:35] vkuncak |
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+ | ====== Lecture 3 (Skeleton) ====== | ||
+ | |||
+ | ==== Context ==== | ||
+ | |||
+ | Recall that we can | ||
+ | * 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 | ||
+ | * we can represent relations using set comprehensions; if our program r has two state components, we can represent its meaning R(r) as | ||
+ | <latex> | ||
+ | \{((x_0,y_0),(x,y)) \mid F \} | ||
+ | </latex> | ||
+ | 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 ==== | ||
+ | |||
+ | 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) | ||
+ | |||
+ | In sequential composition we follow the rule for composition of relations. We want to get again formula with free variables x_0,y_0,x,y. So we need to do renaming. Let x_1,y_1,error_1 be fresh variables. | ||
+ | |||
+ | CR(c1 ; c2) = exists x_1,y_1,error_1. CR(c1)[x:=x_1,y:=y_1,error:=error_1] & CR(c2)[x:=x_1,y:=y_1,error:=error_1] | ||
+ | |||
==== Papers ==== | ==== Papers ==== | ||
- | * Compact verification conditions using weakest preconditions: http://research.microsoft.com/~leino/papers/krml157.pdf | + | * Verification condition generation in Spec#: http://research.microsoft.com/~leino/papers/krml157.pdf |
+ | * Loop invariant inference for set algebra formulas: {{hob-tcs.pdf}} | ||
+ | * Induction-iteration method for machine code checking: http://www.cs.wisc.edu/wpis/papers/pldi00.ps | ||
* 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 | ||