1. Midterm CSG270 Spring 2007
Karl Lieberherr

2. Question 1: SAT Rank 2 instance
14 : :       :            7 : 1    3            0  7 : 1        5      0  7 :       3   5      0  7 :   2    4         0  7 :   2         6   0  7 :         4   6   0
14: 1 2 = or(1 2)
7: 1 3 = or(!1 !3)
= F
adapted DIMACS format
find maximum assignment
and proof that it is maximum

3. Solution Strategy The MAX-CSP transition system gives many options:
Choose initial assignment. Has significant impact on length of proof. Best to start with a maximum assignment.
variable ordering. Irrelevant because start with maximum assignment.
value ordering: Also irrelevant.

4. rank 2
10: 1 = or(1)
5: 1 = or(!1)
14: 1 2 = or(1 2)
7: 1 3 = or(!1 !3)
SAT Rank 2 instance
N={1 !2 !3 4 5 !6}
unsat=1/9
{}|F|{}|N -> D UP*
{1* !3 !5 4 6}|F|{}|N -> SSR Restart
{}|F|5(1)|N -> UP*
{!1 2 !4 !6 5 3}F|5(1)|N -> SSR
{!1 2 !4 !6 5 3}F|5(1),0()|N -> Finale
end
14 : :       :            7 : 1    3            0  7 : 1        5      0  7 :       3   5      0  7 :   2    4         0  7 :   2         6   0  7 :         4   6   0
adapted DIMACS format

5. Rank 2 relations ba
10(1) = or(1) = or(*,1), don’t mention second argument
12(1) = or(1) = or(1,*), 10(2,1) = 12(1,2)
0() = empty clause

7. UP / D

8. Question 3a CSP = <comments> List(Comment)
Preamble <constraints> List(Constraint) EOF.
Comment = "c" <comment> Line .
Preamble =
"p" <numVariables> int
<numConstraints> int
["rank" <rank> Rank]. // default rank is 3
Constraint = Relation [Weight] ":" List(Variable) "0" .
Relation = <relationNumber> int .
Rank = <v> int Weight = <v> int .
If we omit “rank” we destroy the LL(1) property of the cd. Because what follows <rank> is also an int (a <relation>)
The cd becomes ambiguous: p : could mean: rank 3 relation3 but it could also mean: relation 3 weight 3.

9. Question 3c Introduce an auxiliary variable and express
x = (a or b)
x = (!a or b)
x = (!a or !b)
using relations of rank 3.
Iterate until all clauses are of rank 3 or less as in hw 1.
But what if we have constraints other relations than or and we express MAX-CSP. Is for MAX-CSP rank 3 sufficient?

10. Question 3c: DAJ public aspect hydrogen {
public void Molecule.print() {
Print v = new Print(); _s0(v); }
void Molecule._s0(Print v) {s0();}
void Molecule._s0() {
java.util.BitSet[] tokens = { new java.util.BitSet() };
tokens[0].set(0);
_s0(tokens); }
void Molecule._s0(java.util.BitSet[] tokens) {
{ java.util.BitSet[] newTokens = { new java.util.BitSet () };
if (tokens[0].get(0)) {
newTokens[0].set(0);
if (!newTokens[0].isEmpty())
if (atoms != null)
_s0_crossing_atoms(newTokens);
void Molecule._s0_crossing_atoms(java.util.BitSet[] token s) { this.atoms._s0(tokens); } …

11. Molecule = "<molecule" MoleculeId ">" Atoms Bonds "</molecule>".
Atoms = "<atomArray>" List(Atom) "</atomArray>".
Atom = "<atom" AtomId ElementType HydrogenCount ">" "</atom>".
MoleculeId = "id=" String.
AtomId = "id=" String.
ElementType = "elementType=" String.
HydrogenCount = "hydrogenCount=" String.
Bonds = "<bondArray>" List(Bond) "</bondArray>".
Bond = "<bond" BondId AtomRefs2 "/>".
BondId = "id=" String.
AtomRefs2 = "atomRefs2="
<first> String
<second> String.
List(S) ~ {S}.
aspect hydrogen {
declare traversal:
void print():
"from Molecule to HydrogenCount" (Print); }