diff options
| author |  tabarbe <unknown> | 2010-08-26 23:37:01 +0000 |
|---|---|---|
| committer | tabarbe <unknown> | 2010-08-26 23:37:01 +0000 |
| commit | 47171ab9f9d31dab0d5e0a4c3c95c763452e9295 (patch) | |
| tree | 402d453ee1c63dff1a04d03eabfc2bef32eed4ed /Source/AIFramework/Polyhedra/LinearConstraintSystem.cs | |
| parent | 8b0392fe672ce820ba07af673fe9177babdee00b (diff) | |
Boogie: Renaming the AIFramework sources in preparation for committal of my port of the project
Diffstat (limited to 'Source/AIFramework/Polyhedra/LinearConstraintSystem.cs')
| -rw-r--r-- | Source/AIFramework/Polyhedra/LinearConstraintSystem.cs | 1856 |
1 files changed, 1856 insertions, 0 deletions
diff --git a/Source/AIFramework/Polyhedra/LinearConstraintSystem.cs b/Source/AIFramework/Polyhedra/LinearConstraintSystem.cs new file mode 100644 index 00000000..e444b0ca --- /dev/null +++ b/Source/AIFramework/Polyhedra/LinearConstraintSystem.cs @@ -0,0 +1,1856 @@ +//-----------------------------------------------------------------------------
+//
+// Copyright (C) Microsoft Corporation. All Rights Reserved.
+//
+//-----------------------------------------------------------------------------
+namespace Microsoft.AbstractInterpretationFramework
+{
+ using System.Collections;
+ using System.Collections.Generic;
+ using System.Diagnostics;
+ using System;
+ using Microsoft.SpecSharp.Collections;
+ using Microsoft.Contracts;
+ using Microsoft.Basetypes;
+ using IMutableSet = Microsoft.Boogie.Set;
+ using HashSet = Microsoft.Boogie.Set;
+ using ISet = Microsoft.Boogie.Set;
+
+ /// <summary>
+ /// Represents a system of linear constraints (constraint/frame representations).
+ /// </summary>
+ public class LinearConstraintSystem
+ {
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ Data structure ----------------------------------------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ public /*maybe null*/ ArrayList /*LinearConstraint!*/ Constraints;
+ /*maybe null*/ ArrayList /*FrameElement!*/ FrameVertices;
+ /*maybe null*/ ArrayList /*FrameElement!*/ FrameRays;
+ IMutableSet/*IVariable!*/! FrameDimensions;
+ /*maybe null*/ ArrayList /*FrameElement!*/ FrameLines;
+ // Invariant: Either all of Constraints, FrameVertices, FrameRays, and FrameLines are
+ // null, or all are non-null.
+ // Invariant: Any dimension mentioned in Constraints, FrameVertices, FrameRays, or
+ // FrameLines is mentioned in FrameDimensions.
+ // The meaning of FrameDimensions is that for any dimension x not in FrameDimensions,
+ // there is an implicit line along dimension x (that is, (<x,1>)).
+
+ void CheckInvariant()
+ {
+ if (Constraints == null)
+ {
+ System.Diagnostics.Debug.Assert(FrameVertices == null);
+ System.Diagnostics.Debug.Assert(FrameRays == null);
+ System.Diagnostics.Debug.Assert(FrameLines == null);
+ System.Diagnostics.Debug.Assert(FrameDimensions.Count == 0);
+ }
+ else
+ {
+ System.Diagnostics.Debug.Assert(FrameVertices != null);
+ System.Diagnostics.Debug.Assert(FrameRays != null);
+ System.Diagnostics.Debug.Assert(FrameLines != null);
+
+ foreach (LinearConstraint! cc in Constraints)
+ {
+#if FIXED_DESERIALIZER
+ assert Forall{IVariable! var in cc.GetDefinedDimensions(); FrameDimensions.Contains(var)};
+#endif
+ assert cc.coefficients.Count != 0;
+ }
+ foreach (ArrayList /*FrameElement*/! FrameComponent in new ArrayList /*FrameElement*/ [] {FrameVertices, FrameRays, FrameLines})
+ {
+ foreach (FrameElement fe in FrameComponent)
+ {
+ if (fe == null) continue;
+#if FIXED_DESERIALIZER
+ assert Forall{IVariable! var in fe.GetDefinedDimensions(); FrameDimensions.Contains(var)};
+#endif
+ }
+ }
+ }
+ }
+
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ Constructors ------------------------------------------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ /// <summary>
+ /// Creates a LinearConstraintSystem representing the bottom element, that is, representing
+ /// an unsatisfiable system of constraints.
+ /// </summary>
+ [NotDelayed]
+ public LinearConstraintSystem()
+ {
+ FrameDimensions = new HashSet /*IVariable!*/ ();
+ base();
+ CheckInvariant();
+ }
+
+ /// <summary>
+ /// Constructs a linear constraint system with constraints "cs".
+ /// The constructor captures all constraints in "cs".
+ /// </summary>
+ /// <param name="cs"></param>
+ [NotDelayed]
+ public LinearConstraintSystem(ArrayList /*LinearConstraint!*/! cs)
+#if BUG_159_HAS_BEEN_FIXED
+ requires Forall{LinearConstraint! cc in cs; cc.coefficients.Count != 0};
+#endif
+ {
+ ArrayList constraints = new ArrayList /*LinearConstraint!*/ (cs.Count);
+ foreach (LinearConstraint! cc in cs)
+ {
+ constraints.Add(cc);
+ }
+ Constraints = constraints;
+ FrameDimensions = new HashSet /*IVariable!*/ (); // to please compiler; this value will be overridden in the call to GenerateFrameConstraints below
+ base();
+
+ GenerateFrameFromConstraints();
+ SimplifyConstraints();
+ CheckInvariant();
+#if DEBUG_PRINT
+ Console.WriteLine("LinearConstraintSystem: constructor produced:");
+ Dump();
+#endif
+ }
+
+ /// <summary>
+ /// Constructs a linear constraint system corresponding to given vertex. This constructor
+ /// is only used in the test harness--it is not needed for abstract interpretation.
+ /// </summary>
+ /// <param name="v"></param>
+ [NotDelayed]
+ LinearConstraintSystem(FrameElement! v)
+ {
+ IMutableSet! frameDims = v.GetDefinedDimensions();
+ ArrayList /*LinearConstraint!*/ constraints = new ArrayList /*LinearConstraint!*/ ();
+ foreach (IVariable! dim in frameDims)
+ {
+ LinearConstraint lc = new LinearConstraint(LinearConstraint.ConstraintRelation.EQ);
+ lc.SetCoefficient(dim, Rational.ONE);
+ lc.rhs = v[dim];
+ constraints.Add(lc);
+ }
+ FrameDimensions = frameDims;
+ Constraints = constraints;
+
+ ArrayList /*FrameElement*/ frameVertices = new ArrayList /*FrameElement*/ ();
+ frameVertices.Add(v);
+ FrameVertices = frameVertices;
+
+ FrameRays = new ArrayList /*FrameElement*/ ();
+ FrameLines = new ArrayList /*FrameElement*/ ();
+
+ base();
+ CheckInvariant();
+ }
+
+ void ChangeIntoBottom()
+ {
+ Constraints = null;
+ FrameVertices = null;
+ FrameRays = null;
+ FrameLines = null;
+ IMutableSet ss;
+ FrameDimensions.Clear(); // no implicit lines
+ }
+
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ Public operations and their support routines ----------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ public bool IsBottom()
+ {
+ return Constraints == null;
+ }
+
+ public bool IsTop()
+ {
+ return Constraints != null && Constraints.Count == 0;
+ }
+
+ [Pure]
+ public override string! ToString()
+ {
+ if (Constraints == null)
+ {
+ return "<bottom>";
+ }
+ else if (Constraints.Count == 0)
+ {
+ return "<top>";
+ }
+ else
+ {
+ string z = null;
+ foreach (LinearConstraint! lc in Constraints)
+ {
+ string s = lc.ToString();
+ if (z == null)
+ {
+ z = s;
+ }
+ else
+ {
+ z += " AND " + s;
+ }
+ }
+ assert z != null;
+ return z;
+ }
+ }
+
+
+ public ICollection<IVariable!>! FreeVariables()
+ ensures result.IsReadOnly;
+ {
+ List<IVariable!> list = new List<IVariable!>();
+ foreach (IVariable! v in FrameDimensions) { list.Add(v); }
+ return (!)list.AsReadOnly();
+ }
+
+ /// <summary>
+ /// Note: This method requires that all dimensions are of type Variable, something that's
+ /// not required elsewhere in this class.
+ /// </summary>
+ /// <returns></returns>
+ public IExpr! ConvertToExpression(ILinearExprFactory! factory)
+ {
+ if (this.Constraints == null) {
+ return factory.False;
+ }
+ if (this.Constraints.Count == 0) {
+ return factory.True;
+ }
+
+ IExpr result = null;
+ foreach (LinearConstraint! lc in Constraints)
+ {
+ IExpr conjunct = lc.ConvertToExpression(factory);
+ result = (result == null) ? conjunct : (IExpr)factory.And(conjunct, result);
+ }
+ assert result != null;
+ return result;
+ }
+
+
+ /* IsSubset(): determines if 'lcs' is a subset of 'this'
+ * -- See Cousot/Halbwachs 1978, section
+ */
+ public bool IsSubset(LinearConstraintSystem! lcs)
+ {
+ if (lcs.IsBottom())
+ {
+ return true;
+ }
+ else if (this.IsBottom())
+ {
+ return false;
+#if DEBUG
+#else
+ } else if (this.IsTop()) { // optimization -- this case not needed for correctness
+ return true;
+ } else if (lcs.IsTop()) { // optimization -- this case not needed for correctness
+ return false;
+#endif
+ }
+ else
+ {
+ // phase 0: check if frame dimensions are a superset of the constraint dimensions
+ ISet /*IVariable!*/! frameDims = lcs.GetDefinedDimensions();
+#if DEBUG_PRINT
+ Console.WriteLine("DEBUG: IsSubset:");
+ Console.WriteLine(" --- this:");
+ this.Dump();
+ Console.WriteLine(" --- lcs:");
+ lcs.Dump();
+ Console.WriteLine(" ---");
+#endif
+ foreach (LinearConstraint! cc in (!)this.Constraints)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" cc: {0}", cc);
+ Console.WriteLine(" cc.GetDefinedDimensions(): {0}", cc.GetDefinedDimensions());
+#endif
+ if (!forall{IVariable! var in cc.GetDefinedDimensions(); frameDims.Contains(var)})
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> phase 0 subset violated, return false from IsSubset");
+#endif
+ return false;
+ }
+ }
+ }
+
+ // phase 1: check frame vertices against each constraint...
+ foreach (FrameElement! v in (!)lcs.FrameVertices)
+ {
+ foreach (LinearConstraint! cc in this.Constraints)
+ {
+ Rational q = cc.EvaluateLhs(v);
+ if (cc.Relation == LinearConstraint.ConstraintRelation.LE)
+ {
+ if (!(q <= cc.rhs))
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> phase 1a subset violated, return false from IsSubset");
+#endif
+ return false;
+ }
+ }
+ else
+ {
+ if (!(q == cc.rhs))
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> phase 1b subset violated, return false from IsSubset");
+#endif
+ return false;
+ }
+ }
+ }
+ }
+
+ // phase 2: check frame rays against each constraint...
+ // To check if a ray "r" falls within a constraint "cc", we add the vector "r" to
+ // any point "p" on the side of the half-space or plane described by constraint, and
+ // then check if the resulting point satisfies the constraint. That is, we check (for
+ // an inequality constraint with coefficients a1,a2,...,an and right-hand side
+ // constant C):
+ // a1*(r1+p1) + a2*(r2+p2) + ... + an*(rn+pn) <= C
+ // Equivalently:
+ // a1*r1 + a2*r2 + ... + an*rn + a1*p1 + a2*p2 + ... + an*pn <= C
+ // To find a point "p", we can pick out a coordinate, call it 1, with a non-zero
+ // coefficient in the constraint, and then choose "p" as the point that has the
+ // value C/a1 in coordinate 1 and has 0 in all other coordinates. We then check:
+ // a1*r1 + a2*r2 + ... + an*rn + a1*(C/a1) + a2*0 + ... + an*0 <= C
+ // which simplifies to:
+ // a1*r1 + a2*r2 + ... + an*rn + C <= C
+ // which in turn simplifies to:
+ // a1*r1 + a2*r2 + ... + an*rn <= 0
+ // If the constraint is an equality constraint, we simply replace "<=" with "=="
+ // above.
+ foreach (FrameElement! r in (!)lcs.FrameRays)
+ {
+ System.Diagnostics.Debug.Assert(r != null, "encountered a null ray...");
+ foreach (LinearConstraint! cc in this.Constraints)
+ {
+ System.Diagnostics.Debug.Assert(cc != null, "encountered an null constraint...");
+ Rational q = cc.EvaluateLhs(r);
+ if (cc.Relation == LinearConstraint.ConstraintRelation.LE)
+ {
+ if (q.IsPositive)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> phase 2a subset violated, return false from IsSubset");
+#endif
+ return false;
+ }
+ }
+ else
+ {
+ if (q.IsNonZero)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> phase 2b subset violated, return false from IsSubset");
+#endif
+ return false;
+ }
+ }
+ }
+ }
+
+ // phase 3: check frame lines against each constraint...
+ // To check if a line "L" falls within a constraint "cc", we check if both the
+ // vector "L" and "-L", interpreted as rays, fall within the constraint. From
+ // the discussion above, this means we check the following two properties:
+ // a1*L1 + a2*L2 + ... + an*Ln <= 0 (*)
+ // a1*(-L1) + a2*(-L2) + ... + an*(-Ln) <= 0
+ // The second of these lines can be rewritten as:
+ // - a1*L1 - a2*L2 - ... - an*Ln <= 0
+ // which is equivalent to:
+ // -1 * (a1*L1 + a2*L2 + ... + an*Ln) <= 0
+ // Multiplying both sides by -1 and flipping the direction of the inequality,
+ // we have:
+ // a1*L1 + a2*L2 + ... + an*Ln >= 0 (**)
+ // Putting (*) and (**) together, we conclude that we need to check:
+ // a1*L1 + a2*L2 + ... + an*Ln == 0
+ // If the constraint is an equality constraint, we end up with the same equation.
+ foreach (FrameElement! line in (!)lcs.FrameLines)
+ {
+ System.Diagnostics.Debug.Assert(line != null, "encountered a null line...");
+ foreach (LinearConstraint! cc in this.Constraints)
+ {
+ System.Diagnostics.Debug.Assert(cc != null, "encountered an null constraint...");
+ Rational q = cc.EvaluateLhs(line);
+ if (q.IsNonZero)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> phase 3 subset violated, return false from IsSubset");
+#endif
+ return false;
+ }
+ }
+ }
+
+#if DEBUG_PRINT
+ Console.WriteLine(" ---> IsSubset returns true");
+#endif
+ return true;
+ }
+
+ public LinearConstraintSystem! Meet(LinearConstraintSystem! lcs)
+ requires this.Constraints != null;
+ requires lcs.Constraints != null;
+ {
+ ArrayList /*LinearConstraint*/ clist = new ArrayList(this.Constraints.Count + lcs.Constraints.Count);
+ clist.AddRange(this.Constraints);
+ clist.AddRange(lcs.Constraints);
+ return new LinearConstraintSystem(clist);
+ }
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem Join(LinearConstraintSystem lcs)
+ {
+ Console.WriteLine("===================================================================================");
+ Console.WriteLine("DEBUG: Join");
+ Console.WriteLine("Join: this=");
+ Dump();
+ Console.WriteLine("Join: lcs=");
+ lcs.Dump();
+ LinearConstraintSystem z = JoinX(lcs);
+ Console.WriteLine("----------Join------------------------------>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>");
+ Console.WriteLine("Join: result=");
+ z.Dump();
+ Console.WriteLine("===================================================================================");
+ return z;
+ }
+#endif
+
+ /// <summary>
+ /// The join is computed as described in section 4.4 in Cousot and Halbwachs.
+ /// </summary>
+ /// <param name="lcs"></param>
+ /// <returns></returns>
+#if DEBUG_PRINT
+ public LinearConstraintSystem JoinX(LinearConstraintSystem lcs)
+#else
+ public LinearConstraintSystem! Join(LinearConstraintSystem! lcs)
+#endif
+ {
+ if (this.IsBottom())
+ {
+ return (!) lcs.Clone();
+ }
+ else if (lcs.IsBottom())
+ {
+ return (!) 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).
+ assume this.FrameVertices != null;
+ assume this.FrameRays != null;
+ assume this.FrameLines != null;
+ assume lcs.FrameVertices != null;
+ assume lcs.FrameRays != null;
+ 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 = (!) lcs.Clone();
+ lcs = this;
+ }
+ else
+ {
+ z = (!) 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)
+ {
+ 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.
+ 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)
+ {
+ z.AddVertex(v);
+#if DEBUG_PRINT
+ Console.WriteLine("Join: After adding vertex: {0}", v);
+ z.Dump();
+#endif
+ }
+ foreach (FrameElement! r in lcs.FrameRays)
+ {
+ z.AddRay(r);
+#if DEBUG_PRINT
+ Console.WriteLine("Join: After adding ray: {0}", r);
+ z.Dump();
+#endif
+ }
+ foreach (FrameElement! l in lcs.FrameLines)
+ {
+ 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)
+ {
+ 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;
+ }
+ }
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem Widen(LinearConstraintSystem lcs)
+ {
+ Console.WriteLine("===================================================================================");
+ Console.WriteLine("DEBUG: Widen");
+ Console.WriteLine("Widen: this=");
+ Dump();
+ Console.WriteLine("Widen: lcs=");
+ lcs.Dump();
+ LinearConstraintSystem z = WidenX(lcs);
+ Console.WriteLine("----------Widen------------------------------>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>");
+ Console.WriteLine("Widen: result=");
+ z.Dump();
+ Console.WriteLine("===================================================================================");
+ return z;
+ }
+#endif
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem WidenX(LinearConstraintSystem lcs)
+#else
+ public LinearConstraintSystem! Widen(LinearConstraintSystem! lcs)
+#endif
+ {
+ if (this.IsBottom())
+ {
+ return (!) lcs.Clone();
+ }
+ else if (lcs.IsBottom())
+ {
+ return (!) this.Clone();
+ }
+ else if (this.IsTop() || lcs.IsTop())
+ {
+ return new LinearConstraintSystem(new ArrayList /*LinearConstraint*/ ());
+ }
+
+ // create new LCS, we will add only verified constraints to this...
+ ArrayList /*LinearConstraint*/ newConstraints = new ArrayList /*LinearConstraint*/ ();
+ assume this.Constraints != null;
+ foreach (LinearConstraint! ccX in this.Constraints)
+ {
+ LinearConstraint cc = ccX;
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: Starting to check constraint: {0}", cc);
+#endif
+ if (cc.IsConstant())
+ {
+ // (Can this ever occur in the stable state of a LinearConstraintSystem? --KRML)
+ // constraint is unaffected by the frame components
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: --Adding it!");
+#endif
+ newConstraints.Add(cc);
+ continue;
+ }
+
+ // PHASE I: verify constraints against all frame vertices...
+
+ foreach (FrameElement! vertex in (!)lcs.FrameVertices)
+ {
+ Rational lhs = cc.EvaluateLhs(vertex);
+ if (lhs > cc.rhs)
+ {
+ // the vertex does not satisfy the inequality <=
+ if (cc.Relation == LinearConstraint.ConstraintRelation.LE)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out because of vertex: {0}", vertex);
+#endif
+ goto CHECK_NEXT_CONSTRAINT;
+ }
+ else
+ {
+ // ... but it does satisfy the inequality >=
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out <= because of vertex: {0}", vertex);
+#endif
+ cc = cc.ChangeRelationToAtLeast();
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: left with constraint: {0}", cc);
+#endif
+ }
+ }
+ else if (cc.Relation == LinearConstraint.ConstraintRelation.EQ && lhs < cc.rhs)
+ {
+ // the vertex does not satisfy the inequality >=, and the constraint is an equality constraint
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out >= because of vertex: {0}", vertex);
+#endif
+ cc = cc.ChangeRelation(LinearConstraint.ConstraintRelation.LE);
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: left with contraint: {0}", cc);
+#endif
+ }
+ }
+
+ // PHASE II: verify constraints against all frame rays...
+
+ foreach (FrameElement! ray in (!)lcs.FrameRays)
+ {
+ // The following assumes the constraint to have some dimension with a non-zero coefficient
+ Rational lhs = cc.EvaluateLhs(ray);
+ if (lhs.IsPositive)
+ {
+ // the ray does not satisfy the inequality <=
+ if (cc.Relation == LinearConstraint.ConstraintRelation.LE)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out because of ray: {0}", ray);
+#endif
+ goto CHECK_NEXT_CONSTRAINT;
+ }
+ else
+ {
+ // ... but it does satisfy the inequality >=
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out <= because of ray: {0}", ray);
+#endif
+ cc = cc.ChangeRelationToAtLeast();
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: left with contraint: {0}", cc);
+#endif
+ }
+ }
+ else if (cc.Relation == LinearConstraint.ConstraintRelation.EQ && lhs.IsNegative)
+ {
+ // the ray does not satisfy the inequality >=, and the constraint is an equality constraint
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out >= because of ray: {0}", ray);
+#endif
+ cc = cc.ChangeRelation(LinearConstraint.ConstraintRelation.LE);
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: left with constraint: {0}", cc);
+#endif
+ }
+ }
+
+ // PHASE III: verify constraints against all frame lines...
+
+ foreach (FrameElement! line in (!)lcs.FrameLines)
+ {
+ // The following assumes the constraint to have some dimension with a non-zero coefficient
+ Rational lhs = cc.EvaluateLhs(line);
+ if (!lhs.IsZero)
+ {
+ // The line satisfies neither the inequality <= nor the equality ==
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: throwing out because of line: {0}", line);
+#endif
+ goto CHECK_NEXT_CONSTRAINT;
+ }
+ }
+
+ // constraint has been verified, so add to new constraint system
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: --Adding it!");
+#endif
+ newConstraints.Add(cc);
+
+ CHECK_NEXT_CONSTRAINT: {}
+#if DEBUG_PRINT
+ Console.WriteLine("Widen checking: done with that constraint"); +#endif
+ }
+
+ return new LinearConstraintSystem(newConstraints);
+ }
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem Project(IVariable! dim)
+ {
+ Console.WriteLine("===================================================================================");
+ Console.WriteLine("DEBUG: Project(dim={0})", dim);
+ Console.WriteLine("Project: this=");
+ Dump();
+ LinearConstraintSystem z = ProjectX(dim);
+ Console.WriteLine("----------Project------------------------------>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>");
+ Console.WriteLine("Project: result=");
+ z.Dump();
+ Console.WriteLine("===================================================================================");
+ return z;
+ }
+#endif
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem ProjectX(IVariable! dim)
+#else
+ public LinearConstraintSystem! Project(IVariable! dim)
+#endif
+ requires this.Constraints != null;
+ {
+ ArrayList /*LinearConstraint!*/! cc = Project(dim, Constraints);
+ return new LinearConstraintSystem(cc);
+ }
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem Rename(IVariable! oldName, IVariable! newName)
+ {
+ Console.WriteLine("===================================================================================");
+ Console.WriteLine("DEBUG: Rename(oldName={0}, newName={1})", oldName, newName);
+ Console.WriteLine("Rename: this=");
+ Dump();
+ LinearConstraintSystem z = RenameX(oldName, newName);
+ Console.WriteLine("----------Rename------------------------------>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>");
+ Console.WriteLine("Rename: result=");
+ z.Dump();
+ Console.WriteLine("===================================================================================");
+ return z;
+ }
+#endif
+
+#if DEBUG_PRINT
+ public LinearConstraintSystem RenameX(IVariable! oldName, IVariable! newName)
+#else
+ public LinearConstraintSystem! Rename(IVariable! oldName, IVariable! newName)
+#endif
+ {
+ if (this.Constraints == null)
+ {
+ System.Diagnostics.Debug.Assert(this.FrameVertices == null);
+ System.Diagnostics.Debug.Assert(this.FrameRays == null);
+ System.Diagnostics.Debug.Assert(this.FrameLines == null);
+ return this;
+ }
+ IMutableSet /*IVariable!*/! dims = this.FrameDimensions;
+ if (!dims.Contains(oldName))
+ {
+ return this;
+ }
+
+ LinearConstraintSystem z = new LinearConstraintSystem();
+ z.FrameDimensions = (HashSet! /*IVariable!*/)dims.Clone();
+ z.FrameDimensions.Remove(oldName);
+ z.FrameDimensions.Add(newName);
+
+ z.Constraints = new ArrayList /*LinearConstraint!*/ (this.Constraints.Count);
+ foreach (LinearConstraint! lc in (!)this.Constraints)
+ {
+ z.Constraints.Add(lc.Rename(oldName, newName));
+ }
+ z.FrameVertices = RenameInFE((!)this.FrameVertices, oldName, newName);
+ z.FrameRays = RenameInFE((!)this.FrameRays, oldName, newName);
+ z.FrameLines = RenameInFE((!)this.FrameLines, oldName, newName);
+ return z;
+ }
+
+ static ArrayList /*FrameElement*/ RenameInFE(ArrayList! /*FrameElement*/ list, IVariable! oldName, IVariable! newName)
+ {
+ ArrayList/*FrameElement!*/! z = new ArrayList/*FrameElement!*/ (list.Count);
+ foreach (FrameElement! fe in list)
+ {
+ z.Add(fe.Rename(oldName, newName));
+ }
+ System.Diagnostics.Debug.Assert(z.Count == list.Count);
+ return z;
+ }
+
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ support routines --------------------------------------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ /// <summary>
+ /// Returns a set of constraints that is the given set of constraints with dimension "dim"
+ /// projected out. See Cousot and Halbwachs, section 3.3.1.1.
+ /// </summary>
+ /// <param name="dim"></param>
+ /// <param name="constraints"></param>
+ /// <returns></returns>
+ static ArrayList /*LinearConstraint!*/! Project(IVariable! dim, ArrayList /*LinearConstraint!*/! constraints)
+ {
+ // Sort the inequality constaints into ones where dimension "dim" is 0, negative, and
+ // positive, respectively. Put equality constraints with a non-0 "dim" into "eq".
+ ArrayList /*LinearConstraint!*/! final = new ArrayList /*LinearConstraint!*/ ();
+ ArrayList /*LinearConstraint!*/! negative = new ArrayList /*LinearConstraint!*/ ();
+ ArrayList /*LinearConstraint!*/! positive = new ArrayList /*LinearConstraint!*/ ();
+ ArrayList /*LinearConstraint!*/! eq = new ArrayList /*LinearConstraint!*/ ();
+ foreach (LinearConstraint! cc in constraints)
+ {
+ Rational coeff = cc[dim];
+ if (coeff.IsZero)
+ {
+ LinearConstraint lc = (!) cc.Clone();
+ if (!lc.IsConstant())
+ {
+ lc.RemoveDimension(dim);
+ final.Add(lc);
+ }
+ }
+ else if (cc.Relation == LinearConstraint.ConstraintRelation.EQ)
+ {
+ eq.Add(cc);
+ }
+ else if (coeff.IsNegative)
+ {
+ negative.Add(cc);
+ }
+ else
+ {
+ System.Diagnostics.Debug.Assert(coeff.IsPositive);
+ positive.Add(cc);
+ }
+ }
+
+ if (eq.Count != 0)
+ {
+ LinearConstraint eqConstraint = (LinearConstraint!)eq[eq.Count-1];
+ eq.RemoveAt(eq.Count-1);
+ Rational eqC = -eqConstraint[dim];
+
+ foreach (ArrayList /*LinearConstraint!*/! list in new ArrayList[]{eq, negative, positive})
+ {
+ foreach (LinearConstraint! lcX in list)
+ {
+ LinearConstraint lc = (!) lcX.Clone();
+ lc.AddMultiple(lc[dim]/eqC, eqConstraint);
+ System.Diagnostics.Debug.Assert(lc[dim].IsZero);
+ if (!lc.IsConstant())
+ {
+ lc.RemoveDimension(dim);
+ final.Add(lc);
+ }
+ else
+ {
+ System.Diagnostics.Debug.Assert(lc.IsConstantSatisfiable());
+ }
+ }
+ }
+ }
+ else
+ {
+ // Consider all pairs of constraints with (negative,positive) coefficients of "dim".
+ foreach (LinearConstraint! cn in negative)
+ {
+ Rational dn = -cn[dim];
+ System.Diagnostics.Debug.Assert(dn.IsNonNegative);
+ foreach (LinearConstraint! cp in positive)
+ {
+ Rational dp = cp[dim];
+
+ LinearConstraint lc = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ lc.AddMultiple(dn, cp);
+ lc.AddMultiple(dp, cn);
+ System.Diagnostics.Debug.Assert(lc[dim].IsZero);
+ if (!lc.IsConstant())
+ {
+ lc.RemoveDimension(dim);
+ final.Add(lc);
+ }
+ else
+ {
+ System.Diagnostics.Debug.Assert(lc.IsConstantSatisfiable());
+ }
+ }
+ }
+ }
+
+ return final;
+ }
+
+ /// <summary>
+ /// Initializes FrameVertices, FrameRays, FrameLines, and FrameDimensions, see
+ /// Cousot and Halbwachs, section 3.4. Any previous values of these fields are
+ /// ignored and overwritten.
+ ///
+ /// If the set of Constraints is unsatisfiable, then "this" is changed into Bottom.
+ /// </summary>
+ void GenerateFrameFromConstraints()
+ {
+ if (Constraints == null)
+ {
+ FrameVertices = null;
+ FrameRays = null;
+ FrameLines = null;
+ FrameDimensions = new HashSet /*IVariable!*/ ();
+ return;
+ }
+
+ // Step 1 (see Cousot and Halbwachs, section 3.4.3): create a Simplex Tableau.
+#if DEBUG_PRINT
+ Console.WriteLine("DEBUG: --- GenerateFrameFromConstraint ---");
+ Console.WriteLine("Constraints:");
+ foreach (LinearConstraint cc in Constraints)
+ {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+ SimplexTableau tableau = new SimplexTableau(Constraints);
+#if DEBUG_PRINT
+ Console.WriteLine("Initial tableau:");
+ tableau.Dump();
+#endif
+ FrameDimensions = tableau.GetDimensions();
+#if DEBUG_PRINT
+ Console.WriteLine("Dimensions:");
+ foreach (object dim in FrameDimensions)
+ {
+ Console.Write(" {0}", dim);
+ }
+ Console.WriteLine();
+#endif
+
+ // Step 3 and 2: Put as many initial variables as possible into basis, then check if
+ // we reached a feasible basis
+ tableau.AddInitialVarsToBasis();
+#if DEBUG_PRINT
+ Console.WriteLine("Tableau after Step 3:");
+ tableau.Dump();
+#endif
+ if (!tableau.IsFeasibleBasis)
+ {
+ // The polyhedron is empty (according to Cousot and Halbwachs)
+ ChangeIntoBottom();
+ return;
+ }
+
+ FrameVertices = new ArrayList /*FrameElement*/ ();
+ FrameRays = new ArrayList /*FrameElement*/ ();
+ FrameLines = new ArrayList /*FrameElement*/ ();
+ if (FrameDimensions.Count == 0)
+ {
+ // top element
+ return;
+ }
+
+ if (tableau.AllInitialVarsInBasis)
+ {
+ // All initial variables are in basis; there are no lines.
+#if DEBUG_PRINT
+ Console.WriteLine("Tableau after Steps 2 and 3 (all initial variables in basis):");
+ tableau.Dump();
+#endif
+ }
+ else
+ {
+ // There are lines
+#if DEBUG_PRINT
+ Console.WriteLine("Tableau after Steps 2 and 3 (NOT all initial variables in basis--there are lines):");
+ tableau.Dump();
+#endif
+ // Step 4.2: Pick out the lines, then produce the tableau for a new polyhedron without those lines.
+ ArrayList /*LinearConstraint*/ moreConstraints = (ArrayList! /*LinearConstraint*/) Constraints.Clone();
+ tableau.ProduceLines(FrameLines, moreConstraints);
+ tableau = new SimplexTableau(moreConstraints);
+#if DEBUG_PRINT
+ Console.WriteLine("Lines produced:");
+ foreach (FrameElement line in FrameLines)
+ {
+ Console.WriteLine(" {0}", line);
+ }
+ Console.WriteLine("The new list of constraints is:");
+ foreach (LinearConstraint c in moreConstraints)
+ {
+ Console.WriteLine(" {0}", c);
+ }
+ Console.WriteLine("Tableau after producing lines in Step 4.2:");
+ tableau.Dump();
+#endif
+
+ // Repeat step 3 for the new tableau.
+ // Since the new tableau contains no lines, the following call should cause all initial
+ // variables to be in basis (see step 4.2 in section 3.4.3 of Cousot and Halbwachs).
+ tableau.AddInitialVarsToBasis();
+ System.Diagnostics.Debug.Assert(tableau.AllInitialVarsInBasis);
+ System.Diagnostics.Debug.Assert(tableau.IsFeasibleBasis); // the new tableau represents a set of feasible constraints, so this basis should be found to be feasible
+#if DEBUG_PRINT
+ Console.WriteLine("Tableau after all initial variables have been moved into basis:");
+ tableau.Dump();
+#endif
+ }
+
+ // Step 4.1: One vertex has been found. Find all others, too.
+ tableau.TraverseVertices(FrameVertices, FrameRays);
+#if DEBUG_PRINT
+ Console.WriteLine("Tableau after vertex traversal:");
+ tableau.Dump();
+#endif
+ }
+
+ class LambdaDimension : IVariable
+ {
+ readonly int id;
+ static int count = 0;
+
+ /// <summary>
+ /// Return the name of the variable
+ /// </summary>
+ public string! Name
+ {
+ get
+ {
+ return this.ToString();
+ }
+ }
+
+ public LambdaDimension()
+ {
+ id = count;
+ count++;
+ }
+ [Pure]
+ public override string! ToString()
+ {
+ return "lambda" + id;
+ }
+ [Pure]
+ public object DoVisit(ExprVisitor! visitor)
+ {
+ return visitor.VisitVariable(this);
+ }
+ }
+
+ /// <summary>
+ /// Adds a vertex to the frame of "this" and updates Constraints accordingly, see
+ /// Cousot and Halbwachs, section 3.3.1.1. However, this method does not simplify
+ /// Constraints after the operation; that remains the caller's responsibility (which
+ /// gives the caller the opportunity to make multiple calls to AddVertex, AddRay,
+ /// and AddLine before calling SimplifyConstraints).
+ /// Assumes Constraints (and the frame fields) to be non-null.
+ /// </summary>
+ /// <param name="vertex"></param>
+ void AddVertex(FrameElement! vertex)
+ requires this.FrameVertices != null;
+ {
+#if DEBUG_PRINT
+ Console.WriteLine("DEBUG: AddVertex called on {0}", vertex);
+ Console.WriteLine(" Initial constraints:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+
+ FrameVertices.Add(vertex.Clone());
+#if FIXED_DESERIALIZER
+ assert Forall{IVariable! var in vertex.GetDefinedDimensions(); FrameDimensions.Contains(var)};
+#endif
+
+ // We use a new temporary dimension.
+ IVariable! lambda = new LambdaDimension();
+
+ // We change the constraints A*X <= B into
+ // A*X + (A*vector - B)*lambda <= A*vector.
+ // That means that each row k in A (which corresponds to one LinearConstraint
+ // in Constraints) is changed by adding
+ // (A*vector - B)[k] * lambda
+ // to row k and changing the right-hand side of row k to
+ // (A*vector)[k]
+ // Note:
+ // (A*vector - B)[k]
+ // = { vector subtraction is pointwise }
+ // (A*vector)[k] - B[k]
+ // = { A*vector is a row vector whose every row i is the dot-product of
+ // row i of A with the column vector "vector" }
+ // A[k]*vector - B[k]
+ foreach (LinearConstraint! cc in (!)Constraints)
+ {
+ Rational d = cc.EvaluateLhs(vertex);
+ cc.SetCoefficient(lambda, d - cc.rhs);
+ cc.rhs = d;
+ }
+
+ // We also add the constraints that lambda lies between 0 ...
+ LinearConstraint la = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ la.SetCoefficient(lambda, Rational.MINUS_ONE);
+ la.rhs = Rational.ZERO;
+ Constraints.Add(la);
+ // ... and 1.
+ la = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ la.SetCoefficient(lambda, Rational.ONE);
+ la.rhs = Rational.ONE;
+ Constraints.Add(la);
+#if DEBUG_PRINT
+ Console.WriteLine(" Constraints after addition:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+
+ // Finally, project out the dummy dimension.
+ Constraints = Project(lambda, Constraints);
+
+#if DEBUG_PRINT
+ Console.WriteLine(" Resulting constraints:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+ }
+
+ /// <summary>
+ /// Adds a ray to the frame of "this" and updates Constraints accordingly, see
+ /// Cousot and Halbwachs, section 3.3.1.1. However, this method does not simplify
+ /// Constraints after the operation; that remains the caller's responsibility (which
+ /// gives the caller the opportunity to make multiple calls to AddVertex, AddRay,
+ /// and AddLine before calling SimplifyConstraints).
+ /// Assumes Constraints (and the frame fields) to be non-null.
+ /// </summary>
+ /// <param name="ray"></param>
+ void AddRay(FrameElement! ray)
+ requires this.FrameRays != null;
+ {
+#if DEBUG_PRINT
+ Console.WriteLine("DEBUG: AddRay called on {0}", ray);
+ Console.WriteLine(" Initial constraints:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+
+ FrameRays.Add(ray.Clone());
+#if FIXED_DESERIALIZER
+ assert Forall{IVariable! var in ray.GetDefinedDimensions(); FrameDimensions.Contains(var)};
+#endif
+
+ // We use a new temporary dimension.
+ IVariable! lambda = new LambdaDimension();
+
+ // We change the constraints A*X <= B into
+ // A*X - (A*ray)*lambda <= B.
+ // That means that each row k in A (which corresponds to one LinearConstraint
+ // in Constraints) is changed by subtracting
+ // (A*ray)[k] * lambda
+ // from row k.
+ // Note:
+ // (A*ray)[k]
+ // = { A*ray is a row vector whose every row i is the dot-product of
+ // row i of A with the column vector "ray" }
+ // A[k]*ray
+ foreach (LinearConstraint! cc in (!)Constraints)
+ {
+ Rational d = cc.EvaluateLhs(ray);
+ cc.SetCoefficient(lambda, -d);
+ }
+
+ // We also add the constraints that lambda is at least 0.
+ LinearConstraint la = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ la.SetCoefficient(lambda, Rational.MINUS_ONE);
+ la.rhs = Rational.ZERO;
+ Constraints.Add(la);
+#if DEBUG_PRINT
+ Console.WriteLine(" Constraints after addition:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+
+ // Finally, project out the dummy dimension.
+ Constraints = Project(lambda, Constraints);
+
+#if DEBUG_PRINT
+ Console.WriteLine(" Resulting constraints:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+ }
+
+ /// <summary>
+ /// Adds a line to the frame of "this" and updates Constraints accordingly, see
+ /// Cousot and Halbwachs, section 3.3.1.1. However, this method does not simplify
+ /// Constraints after the operation; that remains the caller's responsibility (which
+ /// gives the caller the opportunity to make multiple calls to AddVertex, AddRay,
+ /// and AddLine before calling SimplifyConstraints).
+ /// Assumes Constraints (and the frame fields) to be non-null.
+ /// </summary>
+ /// <param name="line"></param>
+ void AddLine(FrameElement! line)
+ requires this.FrameLines != null;
+ {
+ // Note: The code for AddLine is identical to that of AddRay, except the AddLine
+ // does not introduce the constraint 0 <= lambda. (One could imagine sharing the
+ // code between AddRay and AddLine.)
+#if DEBUG_PRINT
+ Console.WriteLine("DEBUG: AddLine called on {0}", line);
+ Console.WriteLine(" Initial constraints:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+
+ FrameLines.Add(line.Clone());
+#if FIXED_DESERIALIZER
+ assert Forall{IVariable! var in line.GetDefinedDimensions(); FrameDimensions.Contains(var)};
+#endif
+
+ // We use a new temporary dimension.
+ IVariable! lambda = new LambdaDimension();
+
+ // We change the constraints A*X <= B into
+ // A*X - (A*line)*lambda <= B.
+ // That means that each row k in A (which corresponds to one LinearConstraint
+ // in Constraints) is changed by subtracting
+ // (A*line)[k] * lambda
+ // from row k.
+ // Note:
+ // (A*line)[k]
+ // = { A*line is a row vector whose every row i is the dot-product of
+ // row i of A with the column vector "line" }
+ // A[k]*line
+ foreach (LinearConstraint! cc in (!)Constraints)
+ {
+ Rational d = cc.EvaluateLhs(line);
+ cc.SetCoefficient(lambda, -d);
+ }
+
+#if DEBUG_PRINT
+ Console.WriteLine(" Constraints after addition:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+
+ // Finally, project out the dummy dimension.
+ Constraints = Project(lambda, Constraints);
+
+#if DEBUG_PRINT
+ Console.WriteLine(" Resulting constraints:");
+ foreach (LinearConstraint cc in Constraints) {
+ Console.WriteLine(" {0}", cc);
+ }
+#endif
+ }
+
+ ISet /*IVariable!*/! GetDefinedDimensions()
+ {
+ HashSet /*IVariable!*/! dims = new HashSet /*IVariable!*/ ();
+ foreach (ArrayList p in new ArrayList[]{FrameVertices, FrameRays, FrameLines})
+ {
+ if (p != null)
+ {
+ foreach (FrameElement! element in p)
+ {
+ foreach (IVariable! dim in element.GetDefinedDimensions())
+ {
+ dims.Add(dim);
+ }
+ }
+ }
+ }
+ return dims;
+ }
+
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ Simplification routines -------------------------------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ /// <summary>
+ /// Uses the Constraints to simplify the frame. See section 3.4.4 of Cousot and Halbwachs.
+ /// </summary>
+ void SimplifyFrame()
+ requires this.Constraints != null;
+ {
+ SimplificationStatus[]! status;
+
+ SimplifyFrameElements((!)FrameVertices, true, Constraints, out status);
+ RemoveIrrelevantFrameElements(FrameVertices, status, null);
+
+ SimplifyFrameElements((!)FrameRays, false, Constraints, out status);
+ RemoveIrrelevantFrameElements(FrameRays, status, FrameLines);
+ }
+
+ enum SimplificationStatus { Irrelevant, Relevant, More };
+
+ /// <summary>
+ /// For each i, sets status[i] to:
+ /// <ul>
+ /// <li>Irrelevant if ff[i] is irrelevant</li>
+ /// <li>Relevant if ff[i] is irrelevant</li>
+ /// <li>More if vertices is true and ray ff[i] can be replaced by a line ff[i]</li>
+ /// </ul>
+ /// </summary>
+ /// <param name="ff"></param>
+ /// <param name="vertices">true if "ff" contains vertices; false if "ff" contains rays</param>
+ /// <param name="constraints"></param>
+ /// <param name="status"></param>
+ static void SimplifyFrameElements(ArrayList! /*FrameElement*/ ff, bool vertices,
+ ArrayList! /*LinearConstraint*/ constraints,
+ out SimplificationStatus[]! status)
+ {
+ status = new SimplificationStatus[ff.Count];
+ bool[,] sat = new bool[ff.Count, constraints.Count];
+ for (int i = 0; i < ff.Count; i++)
+ {
+ FrameElement f = (FrameElement!)ff[i];
+ int cnt = 0;
+ for (int c = 0; c < constraints.Count; c++)
+ {
+ LinearConstraint lc = (LinearConstraint!)constraints[c];
+ bool s = lc.IsSaturatedBy(f, vertices);
+ if (s)
+ {
+ sat[i,c] = true;
+ cnt++;
+ }
+ }
+ if (!vertices && cnt == constraints.Count)
+ {
+ status[i] = SimplificationStatus.More;
+ }
+ else
+ {
+ status[i] = SimplificationStatus.Relevant;
+ }
+ }
+
+ CheckPairSimplifications(sat, status);
+ }
+
+ /// <summary>
+ /// Requires sat.GetLength(0) == status.Length.
+ /// </summary>
+ /// <param name="sat"></param>
+ /// <param name="status"></param>
+ static void CheckPairSimplifications(bool[,]! sat, SimplificationStatus[]! status)
+ requires sat.GetLength(0) == status.Length;
+ {
+ int M = sat.GetLength(0);
+ int N = sat.GetLength(1);
+
+ for (int i = 0; i < M-1; i++)
+ {
+ if (status[i] != SimplificationStatus.Relevant)
+ {
+ continue;
+ }
+ for (int j = i+1; j < M; j++)
+ {
+ if (status[j] != SimplificationStatus.Relevant)
+ {
+ continue;
+ }
+ // check (sat[i,*] <= sat[j,*]) and (sat[i,*] >= sat[j,*])
+ int cmp = 0; // -1: (sat[i,*] <= sat[j,*]), 0: equal, 1: (sat[i,*] >= sat[j,*])
+ for (int c = 0; c < N; c++)
+ {
+ if (cmp < 0)
+ {
+ if (sat[i,c] && !sat[j,c])
+ {
+ // incomparable
+ goto NEXT_PAIR;
+ }
+ }
+ else if (0 < cmp)
+ {
+ if (!sat[i,c] && sat[j,c])
+ {
+ // incomparable
+ goto NEXT_PAIR;
+ }
+ }
+ else if (sat[i,c] != sat[j,c])
+ {
+ if (!sat[i,c])
+ {
+ cmp = -1;
+ }
+ else
+ {
+ cmp = 1;
+ }
+ }
+ }
+ if (cmp <= 0)
+ {
+ // sat[i,*] <= sat[j,*] holds, so mark i as irrelevant
+ status[i] = SimplificationStatus.Irrelevant;
+ goto NEXT_OUTER;
+ }
+ else
+ {
+ // sat[i,*] >= sat[j,*] holds, so mark j as irrelevant
+ status[j] = SimplificationStatus.Irrelevant;
+ }
+ NEXT_PAIR: {}
+ }
+ NEXT_OUTER: {}
+ }
+ }
+
+ static void RemoveIrrelevantFrameElements(ArrayList! /*FrameElement*/ ff, SimplificationStatus[]! status,
+ /*maybe null*/ ArrayList /*FrameElement*/ lines)
+ requires ff.Count == status.Length;
+ {
+ for (int j = ff.Count - 1; 0 <= j; j--)
+ {
+ switch (status[j])
+ {
+ case SimplificationStatus.Relevant:
+ break;
+ case SimplificationStatus.Irrelevant:
+#if DEBUG_PRINT
+ Console.WriteLine("Removing irrelevant {0}: {1}", lines == null ? "vertex" : "ray", ff[j]);
+#endif
+ ff.RemoveAt(j);
+ break;
+ case SimplificationStatus.More:
+ System.Diagnostics.Debug.Assert(lines != null);
+ FrameElement f = (FrameElement)ff[j];
+#if DEBUG_PRINT
+ Console.WriteLine("Changing ray into line: {0}", f);
+#endif
+ ff.RemoveAt(j);
+ assert lines != null;
+ lines.Add(f);
+ break;
+ }
+ }
+ }
+
+ /// <summary>
+ /// Uses the frame to simplify Constraints. See section 3.3.1.2 of Cousot and Halbwachs.
+ ///
+ /// Note: This code does not necessarily eliminate all irrelevant equalities; Cousot and
+ /// Halbwachs only claim that the technique eliminates all irrelevant inequalities.
+ /// </summary>
+ void SimplifyConstraints()
+ {
+ if (Constraints == null)
+ {
+ return;
+ }
+ assume this.FrameVertices != null;
+ assume this.FrameRays != null;
+
+ SimplificationStatus[] status = new SimplificationStatus[Constraints.Count];
+ /*readonly*/ int feCount = FrameVertices.Count + FrameRays.Count;
+
+ // Create a table that keeps track of which constraints are satisfied by which vertices and rays
+ bool[,] sat = new bool[Constraints.Count, FrameVertices.Count + FrameRays.Count];
+ for (int i = 0; i < Constraints.Count; i++)
+ {
+ status[i] = SimplificationStatus.Relevant;
+ LinearConstraint lc = (LinearConstraint!)Constraints[i];
+ int cnt = 0; // number of vertices and rays that saturate lc
+ for (int j = 0; j < FrameVertices.Count; j++)
+ {
+ FrameElement vertex = (FrameElement!)FrameVertices[j];
+ if (lc.IsSaturatedBy(vertex, true))
+ {
+ sat[i,j] = true;
+ cnt++;
+ }
+ }
+ if (cnt == 0)
+ {
+ // no vertex saturates the constraint, so the constraint is irrelevant
+ status[i] = SimplificationStatus.Irrelevant;
+ continue;
+ }
+ for (int j = 0; j < FrameRays.Count; j++)
+ {
+ FrameElement ray = (FrameElement!)FrameRays[j];
+ if (lc.IsSaturatedBy(ray, false))
+ {
+ sat[i, FrameVertices.Count + j] = true;
+ cnt++;
+ }
+ }
+ if (cnt == feCount)
+ {
+ status[i] = SimplificationStatus.More;
+ }
+ else
+ {
+ // Cousot and Halbwachs says that all equalities are found in the way we just tested.
+ // If I understand that right, then we should not get here if the constraint is an
+ // equality constraint. The following assertion tests my understanding. --KRML
+ System.Diagnostics.Debug.Assert(lc.Relation == LinearConstraint.ConstraintRelation.LE);
+ }
+ }
+
+ CheckPairSimplifications(sat, status);
+
+ // Finally, make the changes to the list of constraints
+ for (int i = Constraints.Count - 1; 0 <= i; i--)
+ {
+ switch (status[i])
+ {
+ case SimplificationStatus.Relevant:
+ break;
+ case SimplificationStatus.Irrelevant:
+#if DEBUG_PRINT
+ Console.WriteLine("Removing irrelevant constraint: {0}", Constraints[i]);
+#endif
+ Constraints.RemoveAt(i);
+ break;
+ case SimplificationStatus.More:
+ LinearConstraint lc = (LinearConstraint!)Constraints[i];
+ if (lc.Relation == LinearConstraint.ConstraintRelation.LE)
+ {
+#if DEBUG_PRINT
+ Console.WriteLine("Converting the following constraint into an equality: {0}", lc);
+#endif
+ LinearConstraint lcEq = lc.ChangeRelation(LinearConstraint.ConstraintRelation.EQ);
+ Constraints[i] = lcEq;
+ }
+ break;
+ }
+ }
+
+ foreach (LinearConstraint! lc in Constraints) {
+ lc.Normalize();
+ }
+ }
+
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ Cloning routines --------------------------------------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ public LinearConstraintSystem! Clone()
+ {
+ LinearConstraintSystem z = new LinearConstraintSystem();
+ z.FrameDimensions = (IMutableSet /*IVariable!*/!)this.FrameDimensions.Clone();
+ if (this.Constraints != null)
+ {
+ z.Constraints = DeeperListCopy_LC(this.Constraints);
+ z.FrameVertices = DeeperListCopy_FE((!)this.FrameVertices);
+ z.FrameRays = DeeperListCopy_FE((!)this.FrameRays);
+ z.FrameLines = DeeperListCopy_FE((!)this.FrameLines);
+ }
+ else
+ {
+ System.Diagnostics.Debug.Assert(this.FrameVertices == null);
+ System.Diagnostics.Debug.Assert(this.FrameRays == null);
+ System.Diagnostics.Debug.Assert(this.FrameLines == null);
+ // the constructor should already have set these fields of z to null
+ System.Diagnostics.Debug.Assert(z.Constraints == null);
+ System.Diagnostics.Debug.Assert(z.FrameVertices == null);
+ System.Diagnostics.Debug.Assert(z.FrameRays == null);
+ System.Diagnostics.Debug.Assert(z.FrameLines == null);
+ }
+ return z;
+ }
+
+ /// <summary>
+ /// Clones "list" and the elements of "list".
+ /// </summary>
+ /// <param name="list"></param>
+ /// <returns></returns>
+ ArrayList /*LinearConstraint*/ DeeperListCopy_LC(ArrayList! /*LinearConstraint*/ list)
+ {
+ ArrayList /*LinearConstraint*/ z = new ArrayList /*LinearConstraint*/ (list.Count);
+ foreach (LinearConstraint! lc in list)
+ {
+ z.Add(lc.Clone());
+ }
+ System.Diagnostics.Debug.Assert(z.Count == list.Count);
+ return z;
+ }
+
+ /// <summary>
+ /// Clones "list" and the elements of "list".
+ /// </summary>
+ /// <param name="list"></param>
+ /// <returns></returns>
+ ArrayList /*FrameElement*/ DeeperListCopy_FE(ArrayList! /*FrameElement*/ list)
+ {
+ ArrayList /*FrameElement*/ z = new ArrayList /*FrameElement*/ (list.Count);
+ foreach (FrameElement! fe in list)
+ {
+ z.Add(fe.Clone());
+ }
+ System.Diagnostics.Debug.Assert(z.Count == list.Count);
+ return z;
+ }
+
+ // --------------------------------------------------------------------------------------------------------
+ // ------------------ Debugging and unit test routines ----------------------------------------------------
+ // --------------------------------------------------------------------------------------------------------
+
+ public void Dump()
+ {
+ Console.WriteLine(" Constraints:");
+ if (Constraints == null)
+ {
+ Console.WriteLine(" <bottom>");
+ }
+ else
+ {
+ foreach (LinearConstraint cc in Constraints)
+ {
+ Console.WriteLine(" {0}", cc);
+ }
+ }
+
+ Console.WriteLine(" FrameDimensions: {0}", FrameDimensions);
+
+ Console.WriteLine(" FrameVerticies:");
+ if (FrameVertices == null)
+ {
+ Console.WriteLine(" <null>");
+ }
+ else
+ {
+ foreach (FrameElement fe in FrameVertices)
+ {
+ Console.WriteLine(" {0}", fe);
+ }
+ }
+
+ Console.WriteLine(" FrameRays:");
+ if (FrameRays == null)
+ {
+ Console.WriteLine(" <null>");
+ }
+ else
+ {
+ foreach (FrameElement fe in FrameRays)
+ {
+ Console.WriteLine(" {0}", fe);
+ }
+ }
+
+ Console.WriteLine(" FrameLines:");
+ if (FrameLines == null)
+ {
+ Console.WriteLine(" <null>");
+ }
+ else
+ {
+ foreach (FrameElement fe in FrameLines)
+ {
+ Console.WriteLine(" {0}", fe);
+ }
+ }
+ }
+
+ class TestVariable : IVariable {
+ readonly string! name;
+
+ public string! Name
+ {
+ get
+ {
+ return name;
+ }
+ }
+
+ public TestVariable(string! name) {
+ this.name = name;
+ }
+ [Pure]
+ public object DoVisit(ExprVisitor! visitor) {
+ return visitor.VisitVariable(this);
+ }
+ }
+
+ public static void RunValidationA()
+ {
+ IVariable! dim1 = new TestVariable("X");
+ IVariable! dim2 = new TestVariable("Y");
+ IVariable! dim3 = new TestVariable("Z");
+
+ 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();
+ }
+
+ /// <summary>
+ /// Tests the example in section 3.4.3 of Cousot and Halbwachs.
+ /// </summary>
+ public static void RunValidationB()
+ {
+ IVariable! X = new TestVariable("X");
+ IVariable! Y = new TestVariable("Y");
+ IVariable! Z = new TestVariable("Z");
+ ArrayList /*LinearConstraint*/ cs = new ArrayList /*LinearConstraint*/ ();
+
+ LinearConstraint c = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ c.SetCoefficient(X, Rational.MINUS_ONE);
+ c.SetCoefficient(Y, Rational.ONE);
+ c.SetCoefficient(Z, Rational.MINUS_ONE);
+ c.rhs = Rational.ZERO;
+ cs.Add(c);
+
+ c = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ c.SetCoefficient(X, Rational.MINUS_ONE);
+ c.rhs = Rational.MINUS_ONE;
+ cs.Add(c);
+
+ c = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ c.SetCoefficient(X, Rational.MINUS_ONE);
+ c.SetCoefficient(Y, Rational.MINUS_ONE);
+ c.SetCoefficient(Z, Rational.ONE);
+ c.rhs = Rational.ZERO;
+ cs.Add(c);
+
+ c = new LinearConstraint(LinearConstraint.ConstraintRelation.LE);
+ c.SetCoefficient(Y, Rational.MINUS_ONE);
+ c.SetCoefficient(Z, Rational.ONE);
+ c.rhs = Rational.FromInt(3);
+ cs.Add(c);
+
+ LinearConstraintSystem lcs = new LinearConstraintSystem(cs);
+ Console.WriteLine("==================== The final linear constraint system ====================");
+ lcs.Dump();
+ }
+
+ 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");
+
+ 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
+ }
+ }
+}
\ No newline at end of file |