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--- sav07_lecture_3_skeleton [2007/03/21 09:25]
vkuncak
+++ sav07_lecture_3_skeleton [2007/03/21 09:48]
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 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
+==== 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.
+==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ====
+First step: transform to disjunctive normal form.
+Next: reduce to integer linear programming:
+\begin{equation*}
+  Ax = b, x \geq 0
+\end{equation*}
+where $A \in {\cal Z}^{m,n}$ and $x \in {\cal Z}^n$.
+Then use small model theorem for integer linear programming (ILP).
+Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}.
+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).  Most 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.
 ===== Papers =====