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|
//
// IntervalInference.cs
//
// Authors:
// Alexander Chebaturkin (chebaturkin@gmail.com)
//
// Copyright (C) 2012 Alexander Chebaturkin
//
// Permission is hereby granted, free of charge, to any person obtaining
// a copy of this software and associated documentation files (the
// "Software"), to deal in the Software without restriction, including
// without limitation the rights to use, copy, modify, merge, publish,
// distribute, sublicense, and/or sell copies of the Software, and to
// permit persons to whom the Software is furnished to do so, subject to
// the following conditions:
//
// The above copyright notice and this permission notice shall be
// included in all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
// LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
// OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
// WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
//
using System;
using System.Collections.Generic;
using Mono.CodeContracts.Static.DataStructures;
using Mono.CodeContracts.Static.Lattices;
namespace Mono.CodeContracts.Static.Analysis.Numerical {
static class IntervalInference {
public static class ConstraintsFor {
public static IDictionary<TVar, Sequence<TInterval>> GreaterEqualThanZero<TEnv, TVar, TExpr, TInterval>
(TExpr expr, IExpressionDecoder<TVar, TExpr> decoder, TEnv env)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
var result = new Dictionary<TVar, Sequence<TInterval>> ();
var variable = decoder.UnderlyingVariable (expr);
AddToResult (result, variable, env.Eval (expr).Meet (env.Context.Positive));
if (!decoder.IsVariable (expr)) {
Polynomial<TVar, TExpr> zeroPoly; // poly(0)
if (!Polynomial<TVar, TExpr>.TryToPolynomial (new[] {Monomial<TVar>.From (Rational.Zero)}, out zeroPoly))
throw new AbstractInterpretationException (
"It can never be the case that the conversion of a list of monomials into a polynomial fails.");
Polynomial<TVar, TExpr> exprPoly; // poly(expr)
Polynomial<TVar, TExpr> fullPoly; // '0 <= poly(expr)' polynome
if (Polynomial<TVar, TExpr>.TryBuildFrom (expr, decoder, out exprPoly) &&
Polynomial<TVar, TExpr>.TryToPolynomial (ExpressionOperator.LessEqualThan, zeroPoly, exprPoly, out fullPoly) &&
fullPoly.IsIntervalForm) {
var k = fullPoly.Left[0].Coeff; // k != 0
TVar x;
fullPoly.Left[0].IsSingleVariable (out x);
Rational constraint;
if (Rational.TryDivide (fullPoly.Right[0].Coeff, k, out constraint)) {
TInterval interval;
if (k > 0L) // +x <= constraint
interval = env.Eval (x).Meet (env.Context.For (Rational.MinusInfinity, constraint));
else // -x <= -constraint ==> x >= constraint
interval = env.Eval (x).Meet (env.Context.For (constraint, Rational.PlusInfinity));
AddToResult (result, x, interval);
}
}
}
return result;
}
public static IDictionary<TVar, Sequence<TInterval>> LessEqualThan<TEnv, TVar, TExpr, TInterval>
(TExpr left, TExpr right, IExpressionDecoder<TVar, TExpr> decoder, TEnv env, out bool isBottom)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
isBottom = false;
var result = new Dictionary<TVar, Sequence<TInterval>> ();
if (IsFloat (left, decoder) || IsFloat (right, decoder))
return result;
var leftIntv = env.Eval (left);
var rightIntv = env.Eval (right);
var leftVar = decoder.UnderlyingVariable (left);
var rightVar = decoder.UnderlyingVariable (right);
TInterval refinedIntv;
if (TryRefineLessEqualThan<TEnv, TVar, TExpr, TInterval> (leftIntv, rightVar, env, out refinedIntv))
AddToResult (result, rightVar, refinedIntv);
if (TryRefineLeftLessEqualThanK<TEnv, TVar, TExpr, TInterval> (leftVar, rightIntv, env, out refinedIntv))
AddToResult (result, leftVar, refinedIntv);
Polynomial<TVar, TExpr> poly;
Polynomial<TVar, TExpr> leftPoly;
Polynomial<TVar, TExpr> rightPoly;
if (Polynomial<TVar, TExpr>.TryBuildFrom (left, decoder, out leftPoly) &&
Polynomial<TVar, TExpr>.TryBuildFrom (right, decoder, out rightPoly) &&
Polynomial<TVar, TExpr>.TryToPolynomial (ExpressionOperator.LessEqualThan, leftPoly, rightPoly, out poly) &&
poly.IsLinear) {
if (poly.Left.Length == 1)
return TestTrueLessEqualThan_AxLeqK (poly, env, result, out isBottom);
if (poly.Left.Length == 2)
return TestTrueLessEqualThan_AxByLeqK (poly, env, result, out isBottom);
}
return result;
}
public static IDictionary<TVar, Sequence<TInterval>> LessThan<TEnv, TVar, TExpr, TInterval>
(TExpr left, TExpr right, IExpressionDecoder<TVar, TExpr> decoder, TEnv env, out bool isBottom)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
isBottom = false;
var result = new Dictionary<TVar, Sequence<TInterval>> ();
var leftIntv = env.Eval (left);
var rightIntv = env.Eval (right);
var rightVar = decoder.UnderlyingVariable (right);
var successor = IsFloat (left, decoder) || IsFloat (right, decoder) ? Rational.Zero : Rational.One;
TInterval refinedIntv;
if (TryRefineKLessThanRight<TEnv, TVar, TExpr, TInterval> (leftIntv, rightVar, successor, env, out refinedIntv) && !refinedIntv.IsSinglePoint)
AddToResult (result, rightVar, refinedIntv);
if (successor.IsZero)
return result;
var leftVar = decoder.UnderlyingVariable (left);
if (TryRefineLessThan<TEnv, TVar, TExpr, TInterval> (leftVar, rightIntv, env, out refinedIntv) && !refinedIntv.IsSinglePoint)
AddToResult (result, leftVar, refinedIntv);
Polynomial<TVar, TExpr> poly;
Polynomial<TVar, TExpr> leftPoly;
Polynomial<TVar, TExpr> rightPoly;
if (Polynomial<TVar, TExpr>.TryBuildFrom (left, decoder, out leftPoly) &&
Polynomial<TVar, TExpr>.TryBuildFrom (right, decoder, out rightPoly) &&
Polynomial<TVar, TExpr>.TryToPolynomial (ExpressionOperator.LessThan, leftPoly, rightPoly, out poly) &&
poly.IsLinear) {
if (poly.Left.Length == 1)
return TestTrueLessEqualThan_AxLtK (poly, env, result, out isBottom);
if (poly.Left.Length == 2)
return TestTrueLessEqualThan_AxByLtK (poly, env, result, out isBottom);
}
return result;
}
/// <summary>
/// Get interval for 'left' in inequation 'left <= k'.
/// </summary>
public static bool TryRefineLeftLessEqualThanK<TEnv, TVar, TExpr, TInterval> (TVar left, TInterval k, TEnv env, out TInterval refined)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TInterval : IntervalBase<TInterval, Rational>
{
if (!k.IsNormal ())
return false.Without (out refined);
var interval = env.Context.For (Rational.MinusInfinity, k.UpperBound);
TInterval leftIntv;
if (env.TryGetValue (left, out leftIntv))
interval = interval.Meet (leftIntv);
return true.With (interval, out refined);
}
/// <summary>
/// Get interval for 'left' in inequation 'left < k'.
/// </summary>
public static bool TryRefineLessThan<TEnv, TVar, TExpr, TInterval> (TVar left, TInterval k, TEnv env, out TInterval refined)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TInterval : IntervalBase<TInterval, Rational>
{
if (!k.IsNormal ())
return false.Without (out refined);
var interval = env.Context.For (Rational.MinusInfinity, k.UpperBound.IsInteger ? k.UpperBound - 1L : k.UpperBound);
TInterval leftIntv;
if (env.TryGetValue (left, out leftIntv))
interval = interval.Meet (leftIntv);
return true.With (interval, out refined);
}
/// <summary>
/// Get interval for 'right' in inequation 'k <= right'.
/// </summary>
public static bool TryRefineLessEqualThan<TEnv, TVar, TExpr, TInterval> (TInterval k, TVar right, TEnv env, out TInterval refined)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TInterval : IntervalBase<TInterval, Rational>
{
if (!k.IsNormal ())
return false.Without (out refined);
var interval = env.Context.RightOpen (k.LowerBound);
TInterval rightIntv;
if (env.TryGetValue (right, out rightIntv))
interval = interval.Meet (rightIntv);
return true.With (interval, out refined);
}
/// <summary>
/// Get interval for 'right' in inequation 'k < right'.
/// </summary>
public static bool TryRefineKLessThanRight<TEnv, TVar, TExpr, TInterval> (TInterval k, TVar right, Rational successor, TEnv env, out TInterval refined)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TInterval : IntervalBase<TInterval, Rational>
{
if (!k.IsNormal ())
return false.Without (out refined);
// [k, +oo] or (k, +oo]
var interval = env.Context.For (k.LowerBound.IsInteger ? k.LowerBound + successor : k.LowerBound, Rational.PlusInfinity);
TInterval rightIntv;
if (env.TryGetValue (right, out rightIntv))
interval = interval.Meet (rightIntv);
return true.With (interval, out refined);
}
public static void NotEqual<TEnv, TVar, TExpr, TInterval>
(TExpr left, TExpr right, IExpressionDecoder<TVar, TExpr> decoder, TEnv env, out InferenceResult<TVar, TInterval> resultLeft,
out InferenceResult<TVar, TInterval> resultRight) where TInterval : IntervalBase<TInterval, Rational>
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
{
resultLeft = InferenceResult<TVar, TInterval>.Empty;
resultRight = InferenceResult<TVar, TInterval>.Empty;
var leftIntv = env.Eval (left);
var rightIntv = env.Eval (right);
var leftVar = decoder.UnderlyingVariable (left);
var rightVar = decoder.UnderlyingVariable (right);
var successor = IsFloat (left, decoder) || IsFloat (right, decoder) ? Rational.Zero : Rational.One;
// l != r <==> l < r && r < l
LessThanRefinement<TEnv, TVar, TExpr, TInterval> (successor, env, leftIntv, rightIntv, leftVar, rightVar, ref resultLeft);
LessThanRefinement<TEnv, TVar, TExpr, TInterval> (successor, env, rightIntv, leftIntv, rightVar, leftVar, ref resultRight);
}
static void LessThanRefinement<TEnv, TVar, TExpr, TInterval>
(Rational successor, TEnv env,
TInterval leftIntv, TInterval rightIntv,
TVar leftVar, TVar rightVar,
ref InferenceResult<TVar, TInterval> result)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TInterval : IntervalBase<TInterval, Rational>
where TVar : IEquatable<TVar>
{
TInterval refined;
if (TryRefineKLessThanRight<TEnv, TVar, TExpr, TInterval> (leftIntv, rightVar, successor, env, out refined))
result = result.AddConstraintFor (rightVar, refined);
if (TryRefineLessThan<TEnv, TVar, TExpr, TInterval> (leftVar, rightIntv, env, out refined))
result = result.AddConstraintFor (leftVar, refined);
}
static IDictionary<TVar, Sequence<TInterval>> TestTrueLessEqualThan_AxByLeqK<TEnv, TVar, TExpr, TInterval>
(Polynomial<TVar, TExpr> poly, TEnv env, IDictionary<TVar, Sequence<TInterval>> result, out bool isBottom)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
// ax + by <= k
var ax = poly.Left[0];
var by = poly.Left[1];
TVar x, y;
ax.IsSingleVariable (out x);
by.IsSingleVariable (out y);
var k = poly.Right[0].Coeff;
var a = ax.Coeff;
var b = by.Coeff;
var aInterval = env.Context.For (a);
var bInterval = env.Context.For (b);
var kInterval = env.Context.For (k);
var xInterval = env.Eval (x);
var yInterval = env.Eval (y);
IntervalContextBase<TInterval, Rational> ctx = env.Context;
// x <= (k - (b * y)) / a;
var boundingInterval = ctx.Div (ctx.Sub (kInterval, ctx.Mul (bInterval, yInterval)), aInterval);
Func<Rational, TInterval> upperBounded = (i) => ctx.For (Rational.MinusInfinity, i);
Func<Rational, TInterval> lowerBounded = (i) => ctx.For (i, Rational.PlusInfinity);
if (BoundVariable (ctx, a, xInterval, boundingInterval, x, result, out isBottom, upperBounded, lowerBounded))
return result;
// y <= (k - (a * x)) / b;
boundingInterval = ctx.Div (ctx.Sub (kInterval, ctx.Mul (aInterval, xInterval)), bInterval);
if (BoundVariable (ctx, b, yInterval, boundingInterval, y, result, out isBottom, upperBounded, lowerBounded))
return result;
return result;
}
static IDictionary<TVar, Sequence<TInterval>> TestTrueLessEqualThan_AxByLtK<TEnv, TVar, TExpr, TInterval>
(Polynomial<TVar, TExpr> poly, TEnv env, IDictionary<TVar, Sequence<TInterval>> result, out bool isBottom)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
// ax + by <= k
var ax = poly.Left[0];
var by = poly.Left[1];
TVar x, y;
ax.IsSingleVariable (out x);
by.IsSingleVariable (out y);
var k = poly.Right[0].Coeff;
var a = ax.Coeff;
var b = by.Coeff;
var aInterval = env.Context.For (a);
var bInterval = env.Context.For (b);
var kInterval = env.Context.For (k);
var xInterval = env.Eval (x);
var yInterval = env.Eval (y);
IntervalContextBase<TInterval, Rational> ctx = env.Context;
Func<Rational, TInterval> upperBounded =
(i) => ctx.For (Rational.MinusInfinity, !i.IsInteger ? i.PreviousInt32 : i - 1L);
Func<Rational, TInterval> lowerBounded =
(i) => ctx.For (!i.IsInteger ? i.NextInt32 : i + 1L, Rational.PlusInfinity);
// x <= (k - (b * y)) / a;
var boundingInterval = ctx.Div (ctx.Sub (kInterval, ctx.Mul (bInterval, yInterval)), aInterval);
if (BoundVariable (ctx, a, xInterval, boundingInterval, x, result, out isBottom, upperBounded, lowerBounded))
return result;
// y <= (k - (a * x)) / b;
boundingInterval = ctx.Div (ctx.Sub (kInterval, ctx.Mul (aInterval, xInterval)), bInterval);
if (BoundVariable (ctx, b, yInterval, boundingInterval, y, result, out isBottom, upperBounded, lowerBounded))
return result;
return result;
}
/// <summary>
/// Get constraints for variables from polynome in form 'a*x <= k'
/// </summary>
/// <param name="poly">Polynome in canonical form. Two monomes involved. </param>
static IDictionary<TVar, Sequence<TInterval>> TestTrueLessEqualThan_AxLeqK<TEnv, TVar, TExpr, TInterval>
(Polynomial<TVar, TExpr> poly, TEnv env, IDictionary<TVar, Sequence<TInterval>> result, out bool isBottom)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
isBottom = false;
var ax = poly.Left[0];
var k = poly.Right[1];
if (ax.IsConstant) {
if (ax.Coeff > k.Coeff) {
isBottom = true;
return result;
}
}
else {
TVar x;
ax.IsSingleVariable (out x);
Rational div;
if (Rational.TryDivide (k.Coeff, ax.Coeff, out div)) {
var intv = env.Eval (x);
var refined = ax.Coeff.Sign < 1
? intv.Meet (env.Context.For (div, Rational.PlusInfinity))
: intv.Meet (env.Context.For (Rational.MinusInfinity, div));
AddToResult (result, x, refined);
}
}
return result;
}
/// <summary>
/// Get constraints for variables from polynome in form 'a*x <= k'
/// </summary>
/// <param name="poly">Polynome in canonical form. Two monomes involved. </param>
static IDictionary<TVar, Sequence<TInterval>> TestTrueLessEqualThan_AxLtK<TEnv, TVar, TExpr, TInterval>
(Polynomial<TVar, TExpr> poly, TEnv env, IDictionary<TVar, Sequence<TInterval>> result, out bool isBottom)
where TEnv : IIntervalEnvironment<TVar, TExpr, TInterval, Rational>
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
isBottom = false;
var ax = poly.Left[0];
var k = poly.Right[1];
if (ax.IsConstant) {
if (ax.Coeff >= k.Coeff) {
isBottom = true;
return result;
}
}
else {
TVar x;
ax.IsSingleVariable (out x);
Rational div;
if (Rational.TryDivide (k.Coeff, ax.Coeff, out div)) {
var intv = env.Eval (x);
var boundByDivPlus = !div.IsInteger
? env.Context.For (div.NextInt32, Rational.PlusInfinity)
: env.Context.For (div + 1L, Rational.PlusInfinity);
var boundByDivMinus = !div.IsInteger
? env.Context.For (Rational.MinusInfinity, div.NextInt32 - 1L)
: env.Context.For (Rational.MinusInfinity, div - 1L);
var refined = intv.Meet (ax.Coeff.Sign < 1 ? boundByDivPlus : boundByDivMinus);
if (refined.IsBottom) {
isBottom = true;
return result;
}
AddToResult (result, x, refined);
}
}
return result;
}
static bool BoundVariable<TVar, TInterval>
(IntervalContextBase<TInterval, Rational> ctx, Rational a, TInterval xIntervalOld, TInterval boundingInterval, TVar x,
IDictionary<TVar, Sequence<TInterval>> result, out bool isBottom,
Func<Rational, TInterval> upperBounded, Func<Rational, TInterval> lowerBounded)
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
isBottom = false;
if (a.IsZero) {
TInterval boundingForVariable;
if (a.Sign > 0L)
boundingForVariable = upperBounded (boundingInterval.UpperBound);
else
boundingForVariable = lowerBounded (boundingInterval.LowerBound);
var refined = xIntervalOld.Meet (boundingForVariable);
if (refined.IsBottom) {
isBottom = true;
return true;
}
AddToResult (result, x, refined);
}
return false;
}
static bool IsFloat<TVar, TExpr> (TExpr expr, IExpressionDecoder<TVar, TExpr> decoder)
{
if (decoder == null)
return false;
var type = decoder.TypeOf (expr);
return type == ExpressionType.Float32 || type == ExpressionType.Float64;
}
static void AddToResult<TVar, TInterval> (IDictionary<TVar, Sequence<TInterval>> result, TVar variable, TInterval intv)
where TVar : IEquatable<TVar>
where TInterval : IntervalBase<TInterval, Rational>
{
Sequence<TInterval> value;
result.TryGetValue (variable, out value);
result[variable] = value.Cons (intv);
}
}
public struct InferenceResult<TVar, TInterval>
where TInterval : IAbstractDomain<TInterval>
where TVar : IEquatable<TVar> {
public static readonly InferenceResult<TVar, TInterval> Empty =
new InferenceResult<TVar, TInterval> (false, ImmutableMap<TVar, Sequence<TInterval>>.Empty);
public readonly bool IsBottom;
public readonly IImmutableMap<TVar, Sequence<TInterval>> Constraints;
InferenceResult (bool isbottom, IImmutableMap<TVar, Sequence<TInterval>> map)
{
Constraints = map;
IsBottom = isbottom;
}
public InferenceResult<TVar, TInterval> AddConstraintFor (TVar rightVar, TInterval refined)
{
Sequence<TInterval> seq;
Constraints.TryGetValue (rightVar, out seq);
return new InferenceResult<TVar, TInterval> (IsBottom, Constraints.Add (rightVar, seq.Cons (refined)));
}
public InferenceResult<TVar, TInterval> Join (InferenceResult<TVar, TInterval> that)
{
if (IsBottom)
return that;
if (that.IsBottom)
return this;
var result = Empty;
foreach (var var in Constraints.Keys) {
var leftContraints = Constraints[var];
if (leftContraints == null)
continue;
// tops are not included
Sequence<TInterval> rightConstraints;
if (that.Constraints.TryGetValue (var, out rightConstraints) && rightConstraints != null) {
var intv = leftContraints.Head.Join (rightConstraints.Head);
if (!intv.IsTop)
result.AddConstraintFor (var, intv);
}
}
return result;
}
}
}
}
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