Differences

This shows you the differences between two versions of the page.

--- sav07_lecture_3_skeleton [2007/03/20 21:21]
vkuncak
+++ sav07_lecture_3_skeleton [2007/03/21 12:28]
vkuncak
@@ Line 2: / Line 2: @@
 ===== Converting programs (with simple values) to formulas =====
@@ Line 11: / Line 12: @@
   * 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.
-  * 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.
+  * simple values: variables are integers.  Later we will talk about modeling pointers and arrays, but what we say now applies
 Our goal is to find rules for computing R( c ) that are
@@ Line 24: / Line 25: @@
   R( c ) -> error=false
@@ Line 37: / Line 39: @@
   R(havoc x) = frame(x)
-  R(assume F) = F[x:=x_0, y:=y_0, error:=error_0]
+  R(assume F) = F[x:=x_0, y:=y_0, error:=error_0] & frame()
   R(assert F) = (F -> frame)
@@ Line 102: / Line 104: @@
 This idea is important in static analysis.
@@ Line 107: / Line 114: @@
 ==== 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 ====
 While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward.
+  wp(Q, x=t) =
+  wp(Q, assume F) =
+  wp(Q, assert F) =
+  wp(Q, c1 [] c2) =
+  wp(Q, c1 ; c2) =
+==== Inferring Loop Invariants ====
+Suppose we compute strongest postcondition in a program where we unroll loop k times.
+  * What does it denote?
+  * What is its relationship to loop invariant?
+Weakening strategies
+  * maintain a conjunction
+  * drop conjuncts that do not remain true
+Alternative:
+  * decide that you will only loop for formulas of restricted form, as in abstract interpretation and data flow analysis (next week)
 ===== Proving quantifier-free linear arithmetic formulas =====
+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
+Note: we can check satisfiability of $F\ \land\ (\mbox{error}=\mbox{true})$.
+==== 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 ::= A  |  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.
+==== 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
+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 $\log n + (2m+1)\log(ma)$ 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]].
+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
+==== 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 =====
@@ Line 124: / Line 243: @@
   * Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}}
   * Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract