FuzzySet.cs

using FuzzyMath;
using System.Collections.Frozen;
using System.Numerics;

namespace Algorithm;

/// <summary>
/// <see href="https://en.wikipedia.org/wiki/Fuzzy_set" />
/// <seealso href="https://en.wikipedia.org/wiki/Fuzzy_set"/>
/// <![CDATA[<sender>John Smith</sender>]]>
/// </summary>
/// <typeparam name="T1"></typeparam>
/// <typeparam name="T2"></typeparam>
public record FuzzySet<T1, T2> where T1 : INumber<T1>, IMultiplyOperators<T1, T2, T1>
    where T2 : struct, INumber<T2>, IMultiplyOperators<T2, T1, T1>
{
    public required FrozenSet<T1> BaseSet { get; init; }
    /// <summary>
    /// T -> [0,1]
    /// </summary>
    public required Func<T1, T2> MemberShip { get; init; } = (element) => T2.Zero;
    public required Func<T2, T2> Negator { get; set; } = (m) => T2.One - m;
    public required Func<T2, T2, T2> TNorm { get; init; }
    public FuzzySet<T1, T2> ToComplement => this with
    {
        MemberShip = element => Negator(MemberShip(element))
    };
    public FuzzySet<T1, T2> ToIntersection(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => TNorm(MemberShip(element), set.MemberShip(element))
    };
    public FuzzySet<T1, T2> ToUnion(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => SNorm(MemberShip(element), set.MemberShip(element))
    };
    public FuzzySet<T1, T2> ToPower(T2 power)
    {
        if (power <= T2.Zero) throw new NotDefinedException();
        throw new NotImplementedException();
    }
    public FuzzySet<T1, T2> ToConcentration() => this with
    {
        MemberShip = element => MemberShip(element) * MemberShip(element)
    };
    public FuzzySet<T1, T2> ToDifference1(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => TNorm(MemberShip(element), Negator(set.MemberShip(element)))
    };
    public FuzzySet<T1, T2> ToDifference2(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => T2.Min(MemberShip(element), T2.One - set.MemberShip(element))
    };
    public FuzzySet<T1, T2> ToDifference3(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => MemberShip(element) - TNorm(MemberShip(element), set.MemberShip(element))
    };
    public FuzzySet<T1, T2> ToSymmetric1(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => T2.Abs(MemberShip(element) - set.MemberShip(element))
    };
    public FuzzySet<T1, T2> ToSymmetric2(FuzzySet<T1, T2> set) => this with
    {
        MemberShip = element => T2.Max(T2.Min(MemberShip(element), T2.One - set.MemberShip(element)), T2.Min(set.MemberShip(element), T2.One - MemberShip(element)))
    };

    public Func<T1, T2> NormalizedMemberShip => Height != T2.Zero ? e => MemberShip(e) / Height : throw new NotDefinedException();
    /// <summary>
    /// T-conorm
    /// https://en.wikipedia.org/wiki/T-norm#T-conorms
    /// </summary>
    public Func<T2, T2, T2> SNorm => (m1, m2) => T2.One - TNorm(T2.One - m1, T2.One - m2);
    //public T2 Cardinality() => BaseSet.Sum(e=> MemberShip(e))
    public T2 Cardinality(Func<T1, T2>? memberShip = null)
    {
        T2 sum = T2.AdditiveIdentity;
        foreach (T1 element in BaseSet)
        {
            sum += (memberShip ??= MemberShip)(element);
        }
        return sum;
    }
    public T2 RelativeCardinality() => Cardinality() / T2.CreateChecked(BaseSet.Count);
    public T2 RelativeCardinality(FuzzySet<T1, T2> set) => Cardinality(element => TNorm(MemberShip(element), set.MemberShip(element))) / set.Cardinality();
    public T2 Grade(T1 element)
    {
        CheckBaseSetMemberShip(element);
        return MemberShip(element);
    }
    public T2 Height => BaseSet.Max(e => MemberShip(e));
    /// <summary>
    /// Cannot guaranteed membership in C#
    /// </summary>
    /// <param name="element"></param>
    /// <returns></returns>
    public bool IsNotIncluded(T1 element) => CheckBaseSetMemberShip(element) && MemberShip(element) == T2.Zero;
    public bool IsFullyIncluded(T1 element) => CheckBaseSetMemberShip(element) && MemberShip(element) == T2.One;
    public bool IsPartiallyIncluded(T1 element) => CheckBaseSetMemberShip(element) && MemberShip(element) > T2.Zero && MemberShip(element) < T2.One;
    public bool IsCrossOverPoint(T1 element) => CheckBaseSetMemberShip(element) && MemberShip(element) == T2.CreateChecked(0.5);
    /// <summary>
    /// Crisp Sets
    /// </summary>
    /// <param name="cut"></param>
    /// <returns></returns>
    public IEnumerable<T1> AsAlphaCut(T2 cut) => BaseSet.Where(e => MemberShip(e) >= cut);
    public IEnumerable<T1> AsStrongAlphaCut(T2 cut) => BaseSet.Where(e => MemberShip(e) > cut);
    public IEnumerable<T1> AsSupport() => BaseSet.Where(e => MemberShip(e) > T2.Zero);
    public IEnumerable<T1> AsKernel() => BaseSet.Where(e => MemberShip(e) == T2.Zero);
    public IEnumerable<T1> AsCore() => BaseSet.Where(e => MemberShip(e) == T2.Zero);
    /// <summary>
    /// Calculate distinct cuts, actual definition is harder to compute
    /// </summary>
    /// <returns></returns>
    public IEnumerable<T2> AsLevel => BaseSet.Select(e => MemberShip(e));
    public bool IsEmpty => BaseSet.All(e => MemberShip(e) == T2.Zero);
    public bool IsSetEqual(FuzzySet<T1, T2> set) => BaseSet.All(e => MemberShip(e) == set.MemberShip(e));
    public bool IsIncludedIn(FuzzySet<T1, T2> set) => BaseSet.All(e => MemberShip(e) <= set.MemberShip(e));
    public bool IsLevel(T2 cut) => BaseSet.Where(e => MemberShip(e) == cut).Any();
    public bool IsNormalized() => Height == T2.One;
    public bool IsDisjoint(FuzzySet<T1, T2> set) => BaseSet.All(e => T2.Min(MemberShip(e), set.MemberShip(e)) == T2.Zero);
    private bool CheckBaseSetMemberShip(T1 element)
    {
        if (!BaseSet.Contains(element)) throw new NotDefinedException("The element is not found in the base set");
        return true;
    }

    /// <summary>
    /// Possible not works for these. The set needs to be measureable for these
    /// </summary>
    public T1? Width()
    {
        T1? max = AsSupport().Max();
        T1? min = AsSupport().Min();
        //return AsSupport().Max() ?? default - AsSupport().Min() ?? default;
        if (max is not null && min is not null)
        {
            return max - min;
        }
        return default;
    }

    public bool CheckConvexity(T1 element1, T1 element2, T1 step)
    {
        if (T1.Abs(step) > T1.One) throw new ArgumentOutOfRangeException();
        return CheckBaseSetMemberShip(element1) && CheckBaseSetMemberShip(element2) &&
            MemberShip(step * element1 + (T1.One - step) * element2) >= T2.Min(MemberShip(element1), MemberShip(element2));
    }
    public bool CheckConvexity(T1 element1, T1 element2, T2 step)
    {
        if (T2.Abs(step) > T2.One) throw new ArgumentOutOfRangeException();
        return CheckBaseSetMemberShip(element1) && CheckBaseSetMemberShip(element2) &&
            MemberShip(step * element1 + (T2.One - step) * element2) >= T2.Min(MemberShip(element1), MemberShip(element2));
    }

    //public bool IsConvex()
    //{
    //    //CheckConvexity for all (element1, element2, 0 <= step <= 1)
    //    return (BaseSet is IList<T1> U && (U.AsReadOnly() is not [_, var x, _] && U.AsReadOnly() is not [_, var y, _] && x != y);
    //}

    /// <summary>
    /// Check if the set is convex
    /// </summary>
    /// <returns></returns>
    /// <exception cref="NotImplementedException"></exception>
    public bool IsConvex()
    {
        if (System.Environment.Is64BitOperatingSystem)
        {
            for (T2 step = T2.One; step <= T2.One; step += T2.CreateChecked<System.Double>(System.Double.Epsilon))
            {
                foreach (T1 element1 in BaseSet)
                {
                    foreach (T1 element2 in BaseSet)
                    {
                        if (!CheckConvexity(element1, element2, T1.CreateChecked(step)))
                        {
                            return false;
                        }
                    }
                }
            }
        }
        else
        {
            for (T2 step = T2.One; step <= T2.One; step += T2.CreateChecked<System.Single>(System.Single.Epsilon))
            {
                foreach (T1 element1 in BaseSet)
                {
                    foreach (T1 element2 in BaseSet)
                    {
                        if (!CheckConvexity(element1, element2, T1.CreateChecked(step)))
                        {
                            return false;
                        }
                    }
                }
            }
        }
        return true;
    }

    public T2 Distance(FuzzySet<T1, T2> fuzzySet)
        => BaseSet.Select(e => T2.Abs(MemberShip(e) - fuzzySet.MemberShip(e))).Max();
}

//public static record Valid();

public class MathExtensions
{
    /// <summary>
    /// https://zh.wikipedia.org/zh-tw/武卡谢维奇范数
    /// "無窮多值武卡謝維奇邏輯是實數值邏輯，其中來自命題演算的句子被指派上在 0 到 1 之間的任意精度的真值。"
    /// </summary>
    /// <typeparam name="T1"></typeparam>
    /// <typeparam name="T2"></typeparam>
    /// <param name="m1"></param>
    /// <param name="m2"></param>
    /// <returns></returns>
    public static T2 TukasiewiczTNorm<T1, T2>(T2 m1, T2 m2)
        where T1 : INumber<T1>, IMultiplyOperators<T1, T2, T1>
        where T2 : struct, INumber<T2>, IMultiplyOperators<T2, T1, T1>
    {
        return T2.Max(T2.Zero, m1 + m2 - T2.One);
    }

    /// <summary>
    /// https://en.wikipedia.org/wiki/T-norm#:~:text=Graph%20of%20the%20drastic%20t,and%200%20%3C%20y%20%3D%201.&text=The%20name%20reflects%20the%20fact,%2Dcontinuous%20Archimedean%20t%2Dnorm.
    /// </summary>
    /// <typeparam name="T1"></typeparam>
    /// <typeparam name="T2"></typeparam>
    /// <param name="m1"></param>
    /// <param name="m2"></param>
    /// <returns></returns>
    public static T2 DrasticTNorm<T1, T2>(T2 m1, T2 m2)
        where T1 : INumber<T1>, IMultiplyOperators<T1, T2, T1>
        where T2 : struct, INumber<T2>, IMultiplyOperators<T2, T1, T1>
    {
        if (m1 == T2.One) return m2;
        if (m2 == T2.One) return m1;
        return T2.Zero;
    }

    public static T2 Indicator<T1, T2>(T1 m)
        where T1 : INumber<T1>, IMultiplyOperators<T1, T2, T1>
        where T2 : struct, INumber<T2>, IMultiplyOperators<T2, T1, T1>
    {
        return T2.One;
    }

    public static T2 Characteristic<T1, T2>(T1 m)
        where T1 : INumber<T1>, IMultiplyOperators<T1, T2, T1>
        where T2 : struct, INumber<T2>, IMultiplyOperators<T2, T1, T1>
    {
        return T2.One;
    }
}



publc class NotDefinedException : Exception
{
    public NotDefinedException() : base() { }
    public NotDefinedException(string? message) : base(message) { }
    public NotDefinedException(string? message, Exception? innerException) : base(message, innerException) { }
}