/// <summary>
/// Adds to "lines" the lines implied by initial-variable columns not in basis
/// (see section 3.4.2 of Cousot and Halbwachs), and adds to "constraints" the
/// constraints to exclude those lines (see step 4.2 of section 3.4.3 of
/// Cousot and Halbwachs).
/// </summary>
/// <param name="lines"></param>
/// <param name="constraints"></param>
public void ProduceLines(ArrayList /*FrameElement*//*!*/ lines, ArrayList /*LinearConstraint*//*!*/ constraints)
{
Contract.Requires(constraints != null);
Contract.Requires(lines != null);
// for every initial variable not in basis
for (int i0 = 0; i0 < numInitialVars; i0++)
{
if (inBasis[i0] == -1)
{
FrameElement fe = new FrameElement();
LinearConstraint lc = new LinearConstraint(LinearConstraint.ConstraintRelation.EQ);
for (int i = 0; i < numInitialVars; i++)
{
if (i == i0)
{
fe.AddCoordinate((IVariable)cce.NonNull(dims[i]), Rational.ONE);
lc.SetCoefficient((IVariable)cce.NonNull(dims[i]), Rational.ONE);
}
else if (inBasis[i] != -1)
{
// i is a basis column
Contract.Assert(m[inBasis[i], i].HasValue(1));
Rational val = -m[inBasis[i], i0];
fe.AddCoordinate((IVariable)cce.NonNull(dims[i]), val);
lc.SetCoefficient((IVariable)cce.NonNull(dims[i]), val);
}
}
lines.Add(fe);
constraints.Add(lc);
}
}
}
/// <summary>
/// Worker method of TraverseVertices.
/// This method has no net effect on the tableau.
/// </summary>
/// <param name="basesSeenSoFar"></param>
/// <param name="vertices"></param>
/// <param name="rays"></param>
void TraverseBases(ArrayList /*bool[]*//*!*/ basesSeenSoFar, ArrayList /*FrameElement*//*!*/ vertices, ArrayList /*FrameElement*//*!*/ rays) {
Contract.Requires(rays != null);
Contract.Requires(vertices != null);
Contract.Requires(basesSeenSoFar != null);
CheckInvariant();
bool[] thisBasis = new bool[numSlackVars];
for (int i = numInitialVars; i < rhsColumn; i++) {
if (inBasis[i] != -1) {
thisBasis[i - numInitialVars] = true;
}
}
foreach (bool[]/*!*/ basis in basesSeenSoFar) {
Contract.Assert(basis != null);
Contract.Assert(basis.Length == numSlackVars);
for (int i = 0; i < numSlackVars; i++) {
if (basis[i] != thisBasis[i]) {
goto COMPARE_WITH_NEXT_BASIS;
}
}
// thisBasis and basis are the same--that is, basisColumns has been visited before--so
// we don't traverse anything from here
return;
COMPARE_WITH_NEXT_BASIS: {
}
}
// basisColumns has not been seen before; record thisBasis and continue with the traversal here
basesSeenSoFar.Add(thisBasis);
#if DEBUG_PRINT
Console.Write("TraverseBases, new basis: ");
foreach (bool t in thisBasis) {
Console.Write("{0}", t ? "*" : ".");
}
Console.WriteLine();
Dump();
#endif
// Add vertex
FrameElement v = new FrameElement();
for (int i = 0; i < rows; i++) {
int j = basisColumns[i];
if (j < numInitialVars) {
v.AddCoordinate((IVariable)cce.NonNull(dims[j]), m[i, rhsColumn]);
}
}
#if DEBUG_PRINT
Console.WriteLine(" Adding vertex: {0}", v);
#endif
vertices.Add(v);
// Add rays. Traverse all columns corresponding to slack variables that
// are not in basis (see second bullet of section 3.4.2 of Cousot and Halbwachs).
for (int i0 = numInitialVars; i0 < rhsColumn; i0++) {
if (inBasis[i0] != -1) {
// skip those slack-variable columns that are in basis
continue;
}
// check if slack-variable, non-basis column i corresponds to an extreme ray
for (int row = 0; row < rows; row++) {
if (m[row, i0].IsPositive) {
for (int k = numInitialVars; k < rhsColumn; k++) {
if (inBasis[k] != -1 && m[row, k].IsNonZero) {
// does not correspond to an extreme ray
goto CHECK_NEXT_SLACK_VAR;
}
}
}
}
// corresponds to an extreme ray
FrameElement ray = new FrameElement();
for (int i = 0; i < numInitialVars; i++) {
int j0 = inBasis[i];
Rational val = -m[j0, i0];
ray.AddCoordinate((IVariable)cce.NonNull(dims[i]), val);
}
#if DEBUG_PRINT
Console.WriteLine(" Adding ray: {0}", ray);
#endif
rays.Add(ray);
CHECK_NEXT_SLACK_VAR: {
}
}
// Continue traversal
for (int i = numInitialVars; i < rhsColumn; i++) {
int j = inBasis[i];
if (j != -1) {
// try moving i out of basis and some other slack-variable column into basis
for (int k = numInitialVars; k < rhsColumn; k++) {
if (inBasis[k] == -1 && m[j, k].IsPositive) {
Rational[] undo = Pivot(j, k);
// check if the new basis is feasible
for (int p = 0; p < rows; p++) {
int c = basisColumns[p];
if (numInitialVars <= c && c < rhsColumn && m[p, rhsColumn].IsNegative) {
// not feasible
goto AFTER_TRAVERSE;
}
}
TraverseBases(basesSeenSoFar, vertices, rays);
AFTER_TRAVERSE:
UnPivot(j, k, undo);
}
}
}
}
}
/// <summary>
/// Adds to "lines" the lines implied by initial-variable columns not in basis
/// (see section 3.4.2 of Cousot and Halbwachs), and adds to "constraints" the
/// constraints to exclude those lines (see step 4.2 of section 3.4.3 of
/// Cousot and Halbwachs).
/// </summary>
/// <param name="lines"></param>
/// <param name="constraints"></param>
public void ProduceLines(ArrayList /*FrameElement*//*!*/ lines, ArrayList /*LinearConstraint*//*!*/ constraints) {
Contract.Requires(constraints != null);
Contract.Requires(lines != null);
// for every initial variable not in basis
for (int i0 = 0; i0 < numInitialVars; i0++) {
if (inBasis[i0] == -1) {
FrameElement fe = new FrameElement();
LinearConstraint lc = new LinearConstraint(LinearConstraint.ConstraintRelation.EQ);
for (int i = 0; i < numInitialVars; i++) {
if (i == i0) {
fe.AddCoordinate((IVariable)cce.NonNull(dims[i]), Rational.ONE);
lc.SetCoefficient((IVariable)cce.NonNull(dims[i]), Rational.ONE);
} else if (inBasis[i] != -1) {
// i is a basis column
Contract.Assert(m[inBasis[i], i].HasValue(1));
Rational val = -m[inBasis[i], i0];
fe.AddCoordinate((IVariable)cce.NonNull(dims[i]), val);
lc.SetCoefficient((IVariable)cce.NonNull(dims[i]), val);
}
}
lines.Add(fe);
constraints.Add(lc);
}
}
}
/// <summary>
/// Worker method of TraverseVertices.
/// This method has no net effect on the tableau.
/// </summary>
/// <param name="basesSeenSoFar"></param>
/// <param name="vertices"></param>
/// <param name="rays"></param>
void TraverseBases(ArrayList /*bool[]*//*!*/ basesSeenSoFar, ArrayList /*FrameElement*//*!*/ vertices, ArrayList /*FrameElement*//*!*/ rays)
{
Contract.Requires(rays != null);
Contract.Requires(vertices != null);
Contract.Requires(basesSeenSoFar != null);
CheckInvariant();
bool[] thisBasis = new bool[numSlackVars];
for (int i = numInitialVars; i < rhsColumn; i++)
{
if (inBasis[i] != -1)
{
thisBasis[i - numInitialVars] = true;
}
}
foreach (bool[] /*!*/ basis in basesSeenSoFar)
{
Contract.Assert(basis != null);
Contract.Assert(basis.Length == numSlackVars);
for (int i = 0; i < numSlackVars; i++)
{
if (basis[i] != thisBasis[i])
{
goto COMPARE_WITH_NEXT_BASIS;
}
}
// thisBasis and basis are the same--that is, basisColumns has been visited before--so
// we don't traverse anything from here
return;
COMPARE_WITH_NEXT_BASIS : {
}
}
// basisColumns has not been seen before; record thisBasis and continue with the traversal here
basesSeenSoFar.Add(thisBasis);
#if DEBUG_PRINT
Console.Write("TraverseBases, new basis: ");
foreach (bool t in thisBasis)
{
Console.Write("{0}", t ? "*" : ".");
}
Console.WriteLine();
Dump();
#endif
// Add vertex
FrameElement v = new FrameElement();
for (int i = 0; i < rows; i++)
{
int j = basisColumns[i];
if (j < numInitialVars)
{
v.AddCoordinate((IVariable)cce.NonNull(dims[j]), m[i, rhsColumn]);
}
}
#if DEBUG_PRINT
Console.WriteLine(" Adding vertex: {0}", v);
#endif
vertices.Add(v);
// Add rays. Traverse all columns corresponding to slack variables that
// are not in basis (see second bullet of section 3.4.2 of Cousot and Halbwachs).
for (int i0 = numInitialVars; i0 < rhsColumn; i0++)
{
if (inBasis[i0] != -1)
{
// skip those slack-variable columns that are in basis
continue;
}
// check if slack-variable, non-basis column i corresponds to an extreme ray
for (int row = 0; row < rows; row++)
{
if (m[row, i0].IsPositive)
{
for (int k = numInitialVars; k < rhsColumn; k++)
{
if (inBasis[k] != -1 && m[row, k].IsNonZero)
{
// does not correspond to an extreme ray
goto CHECK_NEXT_SLACK_VAR;
}
}
}
}
// corresponds to an extreme ray
FrameElement ray = new FrameElement();
for (int i = 0; i < numInitialVars; i++)
{
int j0 = inBasis[i];
Rational val = -m[j0, i0];
ray.AddCoordinate((IVariable)cce.NonNull(dims[i]), val);
}
#if DEBUG_PRINT
Console.WriteLine(" Adding ray: {0}", ray);
#endif
rays.Add(ray);
CHECK_NEXT_SLACK_VAR : {
}
}
// Continue traversal
for (int i = numInitialVars; i < rhsColumn; i++)
{
int j = inBasis[i];
if (j != -1)
{
// try moving i out of basis and some other slack-variable column into basis
for (int k = numInitialVars; k < rhsColumn; k++)
{
if (inBasis[k] == -1 && m[j, k].IsPositive)
{
Rational[] undo = Pivot(j, k);
// check if the new basis is feasible
for (int p = 0; p < rows; p++)
{
int c = basisColumns[p];
if (numInitialVars <= c && c < rhsColumn && m[p, rhsColumn].IsNegative)
{
// not feasible
goto AFTER_TRAVERSE;
}
}
TraverseBases(basesSeenSoFar, vertices, rays);
AFTER_TRAVERSE:
UnPivot(j, k, undo);
}
}
}
}
}
public LinearConstraintSystem/*!*/ Join(LinearConstraintSystem/*!*/ lcs) {
Contract.Requires(lcs != null);
Contract.Ensures(Contract.Result<LinearConstraintSystem>() != null);
#endif
if (this.IsBottom()) {
return cce.NonNull(lcs.Clone());
} else if (lcs.IsBottom()) {
return cce.NonNull(this.Clone());
} else if (this.IsTop() || lcs.IsTop()) {
return new LinearConstraintSystem(new ArrayList /*LinearConstraint*/ ());
} else {
LinearConstraintSystem/*!*/ z;
// Start from the "larger" of the two frames (this is just a heuristic measure intended
// to save work).
Contract.Assume(this.FrameVertices != null);
Contract.Assume(this.FrameRays != null);
Contract.Assume(this.FrameLines != null);
Contract.Assume(lcs.FrameVertices != null);
Contract.Assume(lcs.FrameRays != null);
Contract.Assume(lcs.FrameLines != null);
if (this.FrameVertices.Count + this.FrameRays.Count + this.FrameLines.Count - this.FrameDimensions.Count <
lcs.FrameVertices.Count + lcs.FrameRays.Count + lcs.FrameLines.Count - lcs.FrameDimensions.Count) {
z = cce.NonNull(lcs.Clone());
lcs = this;
} else {
z = cce.NonNull(this.Clone());
}
#if DEBUG_PRINT
Console.WriteLine("DEBUG: LinearConstraintSystem.Join ---------------");
Console.WriteLine("z:");
z.Dump();
Console.WriteLine("lcs:");
lcs.Dump();
#endif
// Start by explicating the implicit lines of z for the dimensions dims(lcs)-dims(z).
foreach (IVariable/*!*/ dim in lcs.FrameDimensions) {
Contract.Assert(dim != null);
if (!z.FrameDimensions.Contains(dim)) {
z.FrameDimensions.Add(dim);
FrameElement line = new FrameElement();
line.AddCoordinate(dim, Rational.ONE);
// Note: AddLine is not called (because the line already exists in z--it's just that
// it was represented implicitly). Instead, just tack the explicit representation onto
// FrameLines.
Contract.Assume(z.FrameLines != null);
z.FrameLines.Add(line);
#if DEBUG_PRINT
Console.WriteLine("Join: After explicating line: {0}", line);
z.Dump();
#endif
}
}
// Now, the vertices, rays, and lines can be added.
foreach (FrameElement/*!*/ v in lcs.FrameVertices) {
Contract.Assert(v != null);
z.AddVertex(v);
#if DEBUG_PRINT
Console.WriteLine("Join: After adding vertex: {0}", v);
z.Dump();
#endif
}
foreach (FrameElement/*!*/ r in lcs.FrameRays) {
Contract.Assert(r != null);
z.AddRay(r);
#if DEBUG_PRINT
Console.WriteLine("Join: After adding ray: {0}", r);
z.Dump();
#endif
}
foreach (FrameElement/*!*/ l in lcs.FrameLines) {
Contract.Assert(l != null);
z.AddLine(l);
#if DEBUG_PRINT
Console.WriteLine("Join: After adding line: {0}", l);
z.Dump();
#endif
}
// also add to z the implicit lines of lcs
foreach (IVariable/*!*/ dim in z.FrameDimensions) {
Contract.Assert(dim != null);
if (!lcs.FrameDimensions.Contains(dim)) {
// "dim" is a dimension that's explicit in "z" but implicit in "lcs"
FrameElement line = new FrameElement();
line.AddCoordinate(dim, Rational.ONE);
z.AddLine(line);
#if DEBUG_PRINT
Console.WriteLine("Join: After adding lcs's implicit line: {0}", line);
z.Dump();
#endif
}
}
z.SimplifyFrame();
z.SimplifyConstraints();
z.CheckInvariant();
#if DEBUG_PRINT
Console.WriteLine("Join: Returning z:");
z.Dump();
Console.WriteLine("----------------------------------------");
#endif
return z;
}
}
public static void RunValidationC() {
// Run the example in section 3.4.3 of Cousot and Halbwachs backwards, that is, from
// from to constraints.
IVariable/*!*/ dim1 = new TestVariable("X");
IVariable/*!*/ dim2 = new TestVariable("Y");
IVariable/*!*/ dim3 = new TestVariable("Z");
Contract.Assert(dim1 != null);
Contract.Assert(dim2 != null);
Contract.Assert(dim3 != null);
FrameElement s0 = new FrameElement();
s0.AddCoordinate(dim1, Rational.ONE);
s0.AddCoordinate(dim2, Rational.FromInts(1, 2));
s0.AddCoordinate(dim3, Rational.FromInts(-1, 2));
FrameElement s1 = new FrameElement();
s1.AddCoordinate(dim1, Rational.ONE);
s1.AddCoordinate(dim2, Rational.FromInts(-1, 2));
s1.AddCoordinate(dim3, Rational.FromInts(1, 2));
FrameElement s2 = new FrameElement();
s2.AddCoordinate(dim1, Rational.FromInt(3));
s2.AddCoordinate(dim2, Rational.FromInts(-3, 2));
s2.AddCoordinate(dim3, Rational.FromInts(3, 2));
FrameElement r0 = new FrameElement();
r0.AddCoordinate(dim1, Rational.ONE);
r0.AddCoordinate(dim2, Rational.FromInts(1, 2));
r0.AddCoordinate(dim3, Rational.FromInts(-1, 2));
FrameElement r1 = new FrameElement();
r1.AddCoordinate(dim1, Rational.ONE);
r1.AddCoordinate(dim2, Rational.ZERO);
r1.AddCoordinate(dim3, Rational.ZERO);
FrameElement d0 = new FrameElement();
d0.AddCoordinate(dim1, Rational.ZERO);
d0.AddCoordinate(dim2, Rational.ONE);
d0.AddCoordinate(dim3, Rational.ONE);
LinearConstraintSystem lcs = new LinearConstraintSystem(s0);
lcs.Dump();
lcs.AddVertex(s1);
lcs.Dump();
lcs.AddVertex(s2);
lcs.Dump();
lcs.AddRay(r0);
lcs.Dump();
lcs.AddRay(r1);
lcs.Dump();
lcs.AddLine(d0);
lcs.Dump();
lcs.SimplifyConstraints();
lcs.Dump();
#if LATER
lcs.GenerateFrameFromConstraints(); // should give us back the original frame...
#endif
}
public static void RunValidationA() {
IVariable/*!*/ dim1 = new TestVariable("X");
IVariable/*!*/ dim2 = new TestVariable("Y");
IVariable/*!*/ dim3 = new TestVariable("Z");
Contract.Assert(dim1 != null);
Contract.Assert(dim2 != null);
Contract.Assert(dim3 != null);
FrameElement s1 = new FrameElement();
s1.AddCoordinate(dim1, Rational.ONE);
s1.AddCoordinate(dim2, Rational.MINUS_ONE);
s1.AddCoordinate(dim3, Rational.ZERO);
FrameElement s2 = new FrameElement();
s2.AddCoordinate(dim1, Rational.MINUS_ONE);
s2.AddCoordinate(dim2, Rational.ONE);
s2.AddCoordinate(dim3, Rational.ZERO);
FrameElement r1 = new FrameElement();
r1.AddCoordinate(dim1, Rational.ZERO);
r1.AddCoordinate(dim2, Rational.ZERO);
r1.AddCoordinate(dim3, Rational.ONE);
FrameElement d1 = new FrameElement();
d1.AddCoordinate(dim1, Rational.ONE);
d1.AddCoordinate(dim2, Rational.ONE);
d1.AddCoordinate(dim3, Rational.ZERO);
// create lcs from frame -- cf. Cousot/Halbwachs 1978, section 3.3.1.1
LinearConstraintSystem lcs = new LinearConstraintSystem(s1);
lcs.Dump();
lcs.AddVertex(s2);
lcs.Dump();
lcs.AddRay(r1);
lcs.Dump();
lcs.AddLine(d1);
lcs.Dump();
lcs.SimplifyConstraints();
lcs.Dump();
#if LATER
lcs.GenerateFrameFromConstraints(); // should give us back the original frame...
#endif
Console.WriteLine("IsSubset? {0}", lcs.IsSubset(lcs.Clone()));
lcs.Dump();
}
