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--- sav07_lecture_3_skeleton [2007/03/21 09:51]
vkuncak
+++ sav07_lecture_3_skeleton [2007/03/21 10:39]
vkuncak
@@ Line 150: / Line 150: @@
 Proof: small model theorem.
@@ Line 159: / Line 163: @@
 Next: reduce to integer linear programming:
 \begin{equation*}
-  Ax = b, x \geq 0
+  A\vec x = \vec b, \vec x \geq \vec 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).
+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 $(2m+1)(\log n + \log m + \log a)$ bits
+  * we could encode the problem into SAT: use circuits for addition, comparison etc.
-Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}.
+Note: if small model theorem applies to conjunctions, it also applies to arbitrary QFPA formulas.
-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]].
+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).
+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