677 lines
31 KiB
C#
677 lines
31 KiB
C#
//
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// Polynomial.cs
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//
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// Authors:
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// Alexander Chebaturkin (chebaturkin@gmail.com)
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//
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// Copyright (C) 2012 Alexander Chebaturkin
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//
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// Permission is hereby granted, free of charge, to any person obtaining
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// a copy of this software and associated documentation files (the
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// "Software"), to deal in the Software without restriction, including
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// without limitation the rights to use, copy, modify, merge, publish,
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// distribute, sublicense, and/or sell copies of the Software, and to
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// permit persons to whom the Software is furnished to do so, subject to
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// the following conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
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// LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
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// OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
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// WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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//
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using System;
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using System.Collections.Generic;
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using System.Diagnostics;
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using System.Linq;
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using System.Text;
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using System.Threading;
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using Mono.CodeContracts.Static.DataStructures;
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namespace Mono.CodeContracts.Static.Analysis.Numerical {
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struct Polynomial<TVar, TExpr>
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where TVar : IEquatable<TVar> {
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readonly Monomial<TVar>[] left;
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readonly ExpressionOperator? relation;
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readonly Monomial<TVar>[] right;
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IImmutableSet<TVar> cached_variables;
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public Polynomial (Monomial<TVar> monome)
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{
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relation = null;
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left = new[] {monome};
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right = null;
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cached_variables = null;
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}
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public Polynomial (Monomial<TVar>[] monomes)
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{
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relation = null;
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left = monomes;
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right = null;
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cached_variables = null;
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}
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public Polynomial (ExpressionOperator op, Monomial<TVar>[] left, Monomial<TVar>[] right)
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{
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relation = op;
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this.left = left;
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this.right = right;
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cached_variables = null;
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}
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public Polynomial (ExpressionOperator op, Monomial<TVar>[] left, Monomial<TVar> right)
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{
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relation = op;
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this.left = left;
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this.right = new[] {right};
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cached_variables = null;
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}
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public Polynomial (ExpressionOperator op, Polynomial<TVar, TExpr> left, Polynomial<TVar, TExpr> right)
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{
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relation = op;
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this.left = left.left;
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this.right = right.left;
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cached_variables = null;
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}
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public Polynomial (Polynomial<TVar, TExpr> that)
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{
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relation = that.relation;
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left = DuplicateFrom (that.left);
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right = that.right != null ? DuplicateFrom (that.right) : null;
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cached_variables = null;
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}
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public ExpressionOperator? Relation { get { return relation; } }
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public IImmutableSet<TVar> Variables
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{
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get
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{
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if (cached_variables == null) {
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cached_variables = ImmutableSet<TVar>.Empty ();
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foreach (var monome in left)
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cached_variables = cached_variables.AddRange (monome.Variables);
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if (relation.HasValue)
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foreach (var monome in right)
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cached_variables = cached_variables.AddRange (monome.Variables);
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}
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return cached_variables;
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}
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}
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public Monomial<TVar>[] Left { get { return left; } }
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public Monomial<TVar>[] Right { get { return right; } }
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public Polynomial<TVar, TExpr> LeftAsPolynomial
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{
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get
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{
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Polynomial<TVar, TExpr> polynome;
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TryToPolynomial (left, out polynome);
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return polynome;
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}
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}
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/// <summary>
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/// Returns true if poly is in form: 'k*x < c'
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/// </summary>
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public bool IsIntervalForm
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{
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get
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{
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var op = Relation;
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if (!op.HasValue)
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return false;
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if (left.Length != 1 || right.Length != 1 || left[0].Degree != 1 || !right[0].IsConstant)
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return false;
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return op.Value == ExpressionOperator.LessEqualThan;
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}
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}
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public bool IsLinear
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{
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get
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{
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if (left.Any (m => !m.IsLinear))
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return false;
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if (Relation.HasValue && right.Any (m => !m.IsLinear))
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return false;
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return true;
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}
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}
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static Monomial<TVar>[] DuplicateFrom (Monomial<TVar>[] original)
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{
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var res = new Monomial<TVar>[original.Length];
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Array.Copy (original, res, original.Length);
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return res;
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}
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public static bool TryToPolynomial (Monomial<TVar>[] monomials, out Polynomial<TVar, TExpr> polynome)
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{
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if (monomials.Length > 1) {
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polynome = new Polynomial<TVar, TExpr> (monomials);
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return polynome.TryToCanonicalForm (out polynome);
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}
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return true.With (new Polynomial<TVar, TExpr> (monomials), out polynome);
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}
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public bool TryToCanonicalForm (out Polynomial<TVar, TExpr> polynome)
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{
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var poly = this;
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if (poly.relation.HasValue) {
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switch (poly.relation.Value) {
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case ExpressionOperator.GreaterThan:
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poly = poly.SwapOperands (ExpressionOperator.LessThan);
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break;
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case ExpressionOperator.GreaterEqualThan:
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poly = poly.SwapOperands (ExpressionOperator.LessEqualThan);
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break;
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}
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poly = poly.MoveConstantsAndMonomes ();
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Debug.Assert (poly.relation.HasValue);
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Monomial<TVar>[] left;
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Monomial<TVar>[] right;
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if (TrySimplifyMonomes (poly.left, out left) && TrySimplifyMonomes (poly.right, out right))
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return true.With (new Polynomial<TVar, TExpr> (poly.relation.Value, left, right), out polynome);
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}
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else {
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Monomial<TVar>[] monome;
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if (TrySimplifyMonomes (left, out monome))
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return true.With (new Polynomial<TVar, TExpr> (monome), out polynome);
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}
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return false.Without (out polynome);
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}
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static bool TrySimplifyMonomes (Monomial<TVar>[] monomes, out Monomial<TVar>[] result)
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{
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if (monomes.Length <= 1)
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return true.With (monomes, out result);
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var dict = new Dictionary<ListEqual<TVar>, Monomial<TVar>> ();
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foreach (var monomial in monomes) {
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var key = new ListEqual<TVar> (monomial.Degree, monomial.Variables);
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Monomial<TVar> monome;
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if (dict.TryGetValue (key, out monome)) {
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Rational coeff;
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if (!Rational.TryAdd (monome.Coeff, monomial.Coeff, out coeff))
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return false.With (monomes, out result);
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var sum = monome.With (coeff);
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dict[key] = sum;
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}
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else
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dict.Add (key, monomial);
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}
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var left = new List<Monomial<TVar>> (dict.Count);
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var right = new List<Monomial<TVar>> (dict.Count);
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Monomial<TVar>? resMonome = null;
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foreach (var pair in dict) {
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var monome = pair.Value;
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if (!monome.Coeff.IsZero) {
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if (monome.IsConstant)
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resMonome = monome;
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else if (monome.Coeff > 0L)
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left.Add (monome);
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else
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right.Add (monome);
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}
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}
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var list = new List<Monomial<TVar>> (dict.Count);
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list.AddRange (left);
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list.AddRange (right);
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if (resMonome.HasValue)
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list.Add (resMonome.Value);
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if (list.Count == 0)
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list.Add (Monomial<TVar>.From (Rational.Zero));
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return true.With (list.ToArray (), out result);
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}
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Polynomial<TVar, TExpr> MoveConstantsAndMonomes ()
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{
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if (!relation.HasValue)
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return this;
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var left = new List<Monomial<TVar>> ();
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var right = new List<Monomial<TVar>> ();
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foreach (var monomial in this.left)
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if (!monomial.IsConstant)
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left.Add (monomial);
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else
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right.Add (-monomial);
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foreach (var monomial in this.right)
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if (!monomial.IsConstant)
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left.Add (-monomial);
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else
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right.Add (monomial);
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if (left.Count == 0)
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left.Add (Monomial<TVar>.From (0L));
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if (right.Count == 0)
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right.Add (Monomial<TVar>.From (0));
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return new Polynomial<TVar, TExpr> (relation.Value, left.ToArray (), right.ToArray ());
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}
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Polynomial<TVar, TExpr> SwapOperands (ExpressionOperator newOp)
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{
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return new Polynomial<TVar, TExpr> (newOp, right, left);
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}
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public override string ToString ()
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{
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if (!relation.HasValue)
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return ListToString (left);
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return string.Format ("{0} {1} {2}", ListToString (left), relation, ListToString (right));
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}
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static string ListToString (Monomial<TVar>[] list)
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{
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if (list == null)
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return string.Empty;
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if (list.Length == 0)
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return "()";
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var sb = new StringBuilder ();
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var stringsOrdered = list.Select (x => x.ToString ()).OrderBy (x => x);
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var first = true;
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sb.Append ("(");
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foreach (var str in stringsOrdered) {
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if (first)
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first = false;
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else
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sb.Append (", ");
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sb.Append (str);
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}
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sb.Append (")");
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return sb.ToString ();
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}
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public static bool TryReduceCoefficients (Monomial<TVar>[] simplifiedLeft, Monomial<TVar>[] simplifiedRight)
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{
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if (simplifiedRight[0].Coeff.IsMinValue)
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return false;
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var abs = Rational.Abs (simplifiedRight[0].Coeff);
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if (abs.IsZero)
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return true;
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if (simplifiedLeft.Any (monome => !(monome.Coeff / abs).IsInteger))
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return true;
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simplifiedRight[0] = simplifiedRight[0].With ((c) => c / abs);
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for (var i = 0; i < simplifiedLeft.Length; i++)
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simplifiedLeft[i] = simplifiedLeft[i].With (c => c / abs);
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return true;
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}
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public static bool TryBuildFrom (TExpr expr, IExpressionDecoder<TVar, TExpr> decoder, out Polynomial<TVar, TExpr> result)
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{
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return new PolynomialBuilder (decoder).Build (expr, out result);
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}
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static bool TryMinus (Polynomial<TVar, TExpr> p, out Polynomial<TVar, TExpr> result)
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{
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if (p.Relation.HasValue)
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return false.Without (out result);
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var monomes = new Monomial<TVar>[p.left.Length];
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var cnt = 0;
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foreach (var m in p.left) {
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var k = -m.Coeff;
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if (k.IsInfinity)
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return false.Without (out result);
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var mm = m.With (k);
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monomes[cnt++] = mm;
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}
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return true.With (new Polynomial<TVar, TExpr> (monomes), out result);
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}
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bool IsIntConstant (out long constant)
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{
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if (Relation.HasValue || left.Length != 1)
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return false.Without (out constant);
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return left[0].IsIntConstant (out constant);
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}
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static Polynomial<TVar, TExpr> Concatenate (Polynomial<TVar, TExpr> left, Polynomial<TVar, TExpr> right)
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{
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if (left.Relation.HasValue || right.Relation.HasValue)
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throw new InvalidOperationException ();
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var monomes = new Monomial<TVar>[left.left.Length + right.left.Length];
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Array.Copy (left.left, monomes, left.left.Length);
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Array.Copy (right.left, 0, monomes, left.left.Length, right.left.Length);
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return new Polynomial<TVar, TExpr> (left.left);
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}
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public static bool TryToPolynomial (ExpressionOperator op, Polynomial<TVar, TExpr> left, Polynomial<TVar, TExpr> right, out Polynomial<TVar, TExpr> result)
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{
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if (op.IsBinary () && !left.Relation.HasValue) {
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try {
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var poly = new Polynomial<TVar, TExpr> (op, left, right);
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if (poly.TryToCanonicalForm (out result))
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return true;
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}
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catch (Exception) {
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}
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}
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return false.Without (out result);
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}
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#region Nested type: ConstantGatherer
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class ConstantGatherer : GenericTypeExpressionVisitor<TVar, TExpr, Dummy, Pair<Rational, bool>> {
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static readonly Pair<Rational, bool> Failure = new Pair<Rational, bool> (Rational.PlusInfinity, false);
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public ConstantGatherer (IExpressionDecoder<TVar, TExpr> decoder)
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: base (decoder)
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{
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}
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protected override Pair<Rational, bool> Default (TExpr expr, Dummy input)
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{
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if (Decoder.IsNull (expr))
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return Success (Rational.Zero);
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return Failure;
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}
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protected override Pair<Rational, bool> VisitBool (TExpr expr, Dummy input)
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{
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return Failure;
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}
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protected override Pair<Rational, bool> VisitInt32 (TExpr expr, Dummy input)
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{
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int value;
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if (Decoder.TryValueOf (expr, ExpressionType.Int32, out value))
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return Success (Rational.For (value));
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return Failure;
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}
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static Pair<Rational, bool> Success (Rational value)
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{
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return new Pair<Rational, bool> (value, true);
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}
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}
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#endregion
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#region Nested type: ListEqual
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class ListEqual<T> {
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readonly Lazy<int> cached_hash_code;
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readonly T[] elements;
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public ListEqual (int len, IEnumerable<T> seq)
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{
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elements = seq.Take (len).ToArray ();
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cached_hash_code = new Lazy<int> (() => elements.Sum (elem => elem.GetHashCode ()), LazyThreadSafetyMode.None);
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}
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public override int GetHashCode ()
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{
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return cached_hash_code.Value;
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}
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public override bool Equals (object obj)
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{
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var leq = obj as ListEqual<T>;
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if (leq == null || elements.Length != leq.elements.Length)
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return false;
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if (elements.Length == 1)
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return elements[0].Equals (leq.elements[0]);
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return elements.Intersect (leq.elements).Count () == elements.Count ();
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}
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}
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#endregion
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#region Nested type: PolynomialBuilder
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class PolynomialBuilder : GenericExpressionVisitor<Dummy, bool, TVar, TExpr> {
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Polynomial<TVar, TExpr> poly;
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public PolynomialBuilder (IExpressionDecoder<TVar, TExpr> decoder)
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: base (decoder)
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{
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}
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public bool Build (TExpr expr, out Polynomial<TVar, TExpr> result)
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{
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if (base.Visit (expr, Dummy.Value))
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return true.With (poly, out result);
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return false.Without (out result);
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}
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protected override bool Default (Dummy data)
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{
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return false;
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}
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public override bool VisitSizeOf (TExpr expr, Dummy data)
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{
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int sizeOf;
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if (!Decoder.TrySizeOf (expr, out sizeOf))
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return false;
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return true.With (new Polynomial<TVar, TExpr> (Monomial<TVar>.From (Rational.For (sizeOf))), out poly);
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}
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public override bool VisitConstant (TExpr expr, Dummy data)
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{
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var pair = new ConstantGatherer (Decoder).Visit (expr, Dummy.Value);
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if (!pair.Value || pair.Key.IsInfinity)
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return false;
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poly = new Polynomial<TVar, TExpr> (Monomial<TVar>.From (pair.Key));
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return true;
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}
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public override bool VisitAddition (TExpr left, TExpr right, TExpr original, Dummy data)
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{
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Polynomial<TVar, TExpr> polyLeft, polyRight;
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if (!Build (left, out polyLeft) || !Build (right, out polyRight))
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return false;
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long kLeft, kRight;
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object addition;
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if (polyLeft.IsIntConstant (out kLeft) && polyRight.IsIntConstant (out kRight) &&
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EvaluateArithmeticWithOverflow.TryBinary (Decoder.TypeOf (original), ExpressionOperator.Add, kLeft, kRight, out addition)) {
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var longValue = addition.ConvertToLong ();
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if (longValue.HasValue)
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return true.With (new Polynomial<TVar, TExpr> (Monomial<TVar>.From (Rational.For (longValue.Value))), out poly);
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}
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return true.With (Concatenate (polyLeft, polyRight), out poly);
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}
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public override bool VisitSubtraction (TExpr left, TExpr right, TExpr original, Dummy data)
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{
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Polynomial<TVar, TExpr> polyLeft, polyRight;
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if (!Build (left, out polyLeft) || !Build (right, out polyRight))
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return false;
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long kLeft, kRight;
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object subtraction;
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if (polyLeft.IsIntConstant (out kLeft) && polyRight.IsIntConstant (out kRight) &&
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EvaluateArithmeticWithOverflow.TryBinary (Decoder.TypeOf (original), ExpressionOperator.Sub, kLeft, kRight, out subtraction)) {
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|
var longValue = subtraction.ConvertToLong ();
|
|
if (longValue.HasValue)
|
|
return true.With (new Polynomial<TVar, TExpr> (Monomial<TVar>.From (Rational.For (longValue.Value))), out poly);
|
|
}
|
|
return TryToPolynomialHelperForSubtraction (polyLeft, polyRight, out poly);
|
|
}
|
|
|
|
public override bool VisitMultiply (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
Polynomial<TVar, TExpr> l, r;
|
|
var lBuilt = Build (left, out l);
|
|
var rBuilt = Build (right, out r);
|
|
|
|
if (lBuilt && rBuilt) {
|
|
long kLeft, kRight;
|
|
object mult;
|
|
if (l.IsIntConstant (out kLeft) && r.IsIntConstant (out kRight) &&
|
|
EvaluateArithmeticWithOverflow.TryBinary (
|
|
Decoder.TypeOf (original), ExpressionOperator.Mult, kLeft, kRight, out mult)) {
|
|
var longValue = mult.ConvertToLong ();
|
|
if (longValue.HasValue) {
|
|
var monomial = Monomial<TVar>.From (Rational.For (longValue.Value));
|
|
return true.With (new Polynomial<TVar, TExpr> (monomial), out poly);
|
|
}
|
|
}
|
|
return TryToPolynomialHelperForMultiplication (l, r, out poly);
|
|
}
|
|
return false;
|
|
}
|
|
|
|
static bool TryToPolynomialHelperForMultiplication (Polynomial<TVar, TExpr> left, Polynomial<TVar, TExpr> right, out Polynomial<TVar, TExpr> result)
|
|
{
|
|
var list = new List<Monomial<TVar>> (left.left.Length + right.left.Length);
|
|
foreach (var m in left.left)
|
|
foreach (var n in right.left) {
|
|
Rational mul;
|
|
if (!Rational.TryMultiply (m.Coeff, n.Coeff, out mul))
|
|
return false.Without (out result);
|
|
|
|
list.Add (Monomial<TVar>.From (mul, m.Variables.Concat (n.Variables)));
|
|
}
|
|
|
|
return true.With (new Polynomial<TVar, TExpr> (list.ToArray ()), out result);
|
|
}
|
|
|
|
static bool TryToPolynomialHelperForSubtraction (Polynomial<TVar, TExpr> left, Polynomial<TVar, TExpr> right, out Polynomial<TVar, TExpr> result)
|
|
{
|
|
Polynomial<TVar, TExpr> minusRight;
|
|
if (TryMinus (right, out minusRight))
|
|
return true.With (Concatenate (left, right), out result);
|
|
|
|
return false.Without (out result);
|
|
}
|
|
|
|
public override bool VisitVariable (TVar var, TExpr expr, Dummy data)
|
|
{
|
|
var monome = Monomial<TVar>.From (Rational.One, Sequence<TVar>.Singleton (Decoder.UnderlyingVariable (expr)));
|
|
|
|
return true.With (new Polynomial<TVar, TExpr> (monome), out poly);
|
|
}
|
|
|
|
public override bool VisitNotEqual (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
return DefaultRelation (left, right, original);
|
|
}
|
|
|
|
public override bool VisitLessThan (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
return DefaultRelation (left, right, original);
|
|
}
|
|
|
|
public override bool VisitLessEqualThan (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
return DefaultRelation (left, right, original);
|
|
}
|
|
|
|
public override bool VisitGreaterThan (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
return DefaultRelation (left, right, original);
|
|
}
|
|
|
|
public override bool VisitGreaterEqualThan (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
return DefaultRelation (left, right, original);
|
|
}
|
|
|
|
public override bool VisitDivision (TExpr left, TExpr right, TExpr original, Dummy data)
|
|
{
|
|
Polynomial<TVar, TExpr> l, r;
|
|
if (!Build (left, out l) || !Build (right, out r))
|
|
return false;
|
|
|
|
return HelperForDivision (l, r, out poly);
|
|
}
|
|
|
|
bool HelperForDivision (Polynomial<TVar, TExpr> left, Polynomial<TVar, TExpr> right, out Polynomial<TVar, TExpr> result)
|
|
{
|
|
if (right.left.Length == 1) {
|
|
var div = right.left[0];
|
|
|
|
if (div.IsConstant && !div.Coeff.IsZero) {
|
|
var monomes = new Monomial<TVar>[left.left.Length];
|
|
var cnt = 0;
|
|
foreach (var m in left.left) {
|
|
Rational k;
|
|
try {
|
|
k = m.Coeff / div.Coeff;
|
|
}
|
|
catch (ArithmeticException) {
|
|
return false.Without (out result);
|
|
}
|
|
|
|
monomes[cnt++] = m.With (k);
|
|
}
|
|
|
|
return true.With (new Polynomial<TVar, TExpr> (monomes), out result);
|
|
}
|
|
}
|
|
|
|
return false.Without (out result);
|
|
}
|
|
|
|
bool DefaultRelation (TExpr left, TExpr right, TExpr original)
|
|
{
|
|
Polynomial<TVar, TExpr> l, r;
|
|
if (!Build (left, out l) || !Build (right, out r) || l.Relation.HasValue || r.Relation.HasValue)
|
|
return false;
|
|
|
|
return true.With (new Polynomial<TVar, TExpr> (Decoder.OperatorFor (original), l.left, r.left), out poly);
|
|
}
|
|
}
|
|
|
|
#endregion
|
|
}
|
|
} |