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sav07_lecture_3_skeleton [2007/03/21 09:25] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 10:59] vkuncak |
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This idea is important in static analysis. | This idea is important in static analysis. | ||
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==== Symbolic execution ==== | ==== Symbolic execution ==== | ||
- | Symbolic execution converts programs into formulas by going forward. It is therefore somewhat analogous to the way an [[interpreter]] for the language would work. It is based on the notion of strongest postcondition. | + | Symbolic execution converts programs into formulas by going forward. It is therefore somewhat analogous to the way an [[interpreter]] for the language would work. |
+ | Avoid renaming all the time. | ||
+ | SE(F,k, c1; c2) = SE(F & R(c1), k+1, c2) (update formula) | ||
+ | |||
+ | SE(F,k,(c1 [] c2); c2) = SE(F, k, c1) | SE(F,k,c2) (explore both branches) | ||
+ | |||
+ | Note: how many branches do we get? | ||
+ | |||
+ | Strongest postcondition: | ||
+ | \begin{equation*} | ||
+ | sp(P,r) = \{ s_2 \mid \exists s_1.\ s_1 \in P \land (s_1,s_2) \in r \} | ||
+ | \end{equation*} | ||
+ | Like composition of a set with a relation. It's called ''relational image'' of set $P$ under relation $r$. | ||
+ | |||
+ | Note: when proving our verification condition, instead of proving that semantics of relation implies error=false, it's same as proving that the formula for set sp(U,r) implies error=false, where U is the universal relation, or, in terms of formulas, computing the strongest postcondition of formula 'true'. | ||
==== Weakest preconditions ==== | ==== Weakest preconditions ==== | ||
While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward. | While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward. | ||
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+ | wp(Q, x=t) = | ||
+ | wp(Q, assume F) = | ||
+ | wp(Q, assert F) = | ||
+ | wp(Q, c1 [] c2) = | ||
+ | wp(Q, c1 ; c2) = | ||
==== Inferring Loop Invariants ==== | ==== Inferring Loop Invariants ==== | ||
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Alternative: | Alternative: | ||
* decide that you will only loop for formulas of restricted form, as in abstract interpretation and data flow analysis (next week) | * decide that you will only loop for formulas of restricted form, as in abstract interpretation and data flow analysis (next week) | ||
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===== Proving quantifier-free linear arithmetic formulas ===== | ===== Proving quantifier-free linear arithmetic formulas ===== | ||
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+ | Suppose that we obtain (one or more) verification conditions of the form | ||
+ | \begin{equation*} | ||
+ | F\ \rightarrow\ (\mbox{error}=\mbox{false}) | ||
+ | \end{equation*} | ||
+ | |||
+ | whose validity we need to prove. We here assume that F contains only | ||
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+ | Note: we can check satisfiability of $F\ \land\ (\mbox{error}=\mbox{true})$. | ||
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+ | ==== Quantifier Presburger arithmetic ==== | ||
+ | |||
+ | Here is the grammar: | ||
+ | |||
+ | var = x | y | z | ... (variables) | ||
+ | K = ... | -2 | -1 | 0 | 1 | 2 | ... (integer constants) | ||
+ | T ::= var | T + T | K * T (terms) | ||
+ | A ::= T=T | T <= T (atomic formulas) | ||
+ | F ::= F & F | F|F | ~F (formulas) | ||
+ | |||
+ | To get full Presburger arithmetic, allow existential and universal quantifiers in formula as well. | ||
+ | |||
+ | Note: we can assume we have boolean variables (such as 'error') as well, because we can represent them as 0/1 integers. | ||
+ | |||
+ | Satisfiability of quantifier-free Presburger arithmetic is decidable. | ||
+ | |||
+ | Proof: small model theorem. | ||
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+ | ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ||
+ | |||
+ | First step: transform to disjunctive normal form. | ||
+ | |||
+ | Next: reduce to integer linear programming: | ||
+ | \begin{equation*} | ||
+ | A\vec x = \vec b, \qquad \vec x \geq \vec 0 | ||
+ | \end{equation*} | ||
+ | where $A \in {\cal Z}^{m,n}$ and $x \in {\cal Z}^n$. | ||
+ | |||
+ | Then solve integer linear programming (ILP) problem | ||
+ | * [[wk>Integer Linear Programming]] | ||
+ | * online book chapter on ILP | ||
+ | * [[http://www.gnu.org/software/glpk/|GLPK]] tool | ||
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+ | We can prove small model theorem for ILP - gives bound on search. | ||
+ | |||
+ | Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}: | ||
+ | * solution of Ax=b (A regular) has as components rationals of form p/q with bounded p,q | ||
+ | * duality of linear programming | ||
+ | * obtains bound $M = n(ma)^{2m+1}$, which needs $(2m+1)(\log n + \log m + \log a)$ bits | ||
+ | * we could encode the problem into SAT: use circuits for addition, comparison etc. | ||
+ | |||
+ | Note: if small model theorem applies to conjunctions, it also applies to arbitrary QFPA formulas. | ||
+ | |||
+ | Moreover, one can improve these bounds. One tool based on these ideas is [[http://www.cs.cmu.edu/~uclid/|UCLID]]. | ||
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+ | Alternative: enumerate disjuncts of DNF on demand, each disjunct is a conjunction, then use ILP techniques (often first solve the underlying linear programming problem over reals). Many SMT tools are based on this idea (along with Nelson-Oppen combination: next class). | ||
+ | * [[http://www.cs.nyu.edu/acsys/cvc3/download.html|CVC3]] (successor of CVC Lite) | ||
+ | * [[http://combination.cs.uiowa.edu/smtlib/|SMT-LIB]] Standard for formulas, competition | ||
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+ | ==== Full Presburger arithmetic ==== | ||
+ | |||
+ | Full Presburger arithmetic is also decidable. | ||
+ | |||
+ | Approaches: | ||
+ | * Quantifier-Elimination (Omega tool from Maryland) - see homework | ||
+ | * Automata Theoretic approaches: LASH, MONA (as a special case) | ||
===== Papers ===== | ===== Papers ===== |