Combination of Signed/TruncatedDivision/GCDRing/EuclideanRing

core/src/main/scala/algebra/instances/bigInt.scala

@@ -2,15 +2,22 @@ package algebra
 package instances
 
 import algebra.ring._
+import cats.kernel.instances.BigIntOrder
+import cats.kernel.{Hash, UnboundedEnumerable}
 
 package object bigInt extends BigIntInstances
 
 trait BigIntInstances extends cats.kernel.instances.BigIntInstances {
-  implicit val bigIntAlgebra: BigIntAlgebra =
-    new BigIntAlgebra
+  private val instance: BigIntAlgebra = new BigIntAlgebra
+
+  implicit def bigIntAlgebra: EuclideanRing[BigInt] = instance
+
+  implicit def bigIntTruncatedDivision: TruncatedDivision[BigInt] = instance
 }
 
-class BigIntAlgebra extends CommutativeRing[BigInt] with Serializable {
+class BigIntAlgebra extends EuclideanRing[BigInt] with TruncatedDivision.forCommutativeRing[BigInt] with Serializable {
+
+  override def compare(x: BigInt, y: BigInt): Int = x.compare(y)
 
   val zero: BigInt = BigInt(0)
   val one: BigInt = BigInt(1)
@@ -25,4 +32,36 @@ class BigIntAlgebra extends CommutativeRing[BigInt] with Serializable {
 
   override def fromInt(n: Int): BigInt = BigInt(n)
   override def fromBigInt(n: BigInt): BigInt = n
+
+  def tquot(x: BigInt, y: BigInt): BigInt = x / y
+  def tmod(x: BigInt, y: BigInt): BigInt = x % y
+  override def tquotmod(x: BigInt, y: BigInt): (BigInt, BigInt) = x /% y
+
+  override def lcm(a: BigInt, b: BigInt)(implicit ev: Eq[BigInt]): BigInt =
+    if (a.signum == 0 || b.signum == 0) zero else (a / a.gcd(b)) * b
+  override def gcd(a: BigInt, b: BigInt)(implicit ev: Eq[BigInt]): BigInt = a.gcd(b)
+
+  def euclideanFunction(a: BigInt): BigInt = a.abs
+
+  override def equotmod(a: BigInt, b: BigInt): (BigInt, BigInt) = {
+    val (qt, rt) = a /% b // truncated quotient and remainder
+    if (rt.signum >= 0) (qt, rt)
+    else if (b.signum > 0) (qt - 1, rt + b)
+    else (qt + 1, rt - b)
+  }
+
+  def equot(a: BigInt, b: BigInt): BigInt = {
+    val (qt, rt) = a /% b // truncated quotient and remainder
+    if (rt.signum >= 0) qt
+    else if (b.signum > 0) qt - 1
+    else qt + 1
+  }
+
+  def emod(a: BigInt, b: BigInt): BigInt = {
+    val rt = a % b // truncated remainder
+    if (rt.signum >= 0) rt
+    else if (b > 0) rt + b
+    else rt - b
+  }
+
 }

core/src/main/scala/algebra/ring/DivisionRing.scala

@@ -0,0 +1,22 @@
+package algebra
+package ring
+
+import scala.{specialized => sp}
+
+trait DivisionRing[@sp(Byte, Short, Int, Long, Float, Double) A] extends Any with Ring[A] with MultiplicativeGroup[A] {
+  self =>
+
+  def fromDouble(a: Double): A = Field.defaultFromDouble[A](a)(self, self)
+
+}
+
+trait DivisionRingFunctions[F[T] <: DivisionRing[T]] extends RingFunctions[F] with MultiplicativeGroupFunctions[F] {
+  def fromDouble[@sp(Int, Long, Float, Double) A](n: Double)(implicit ev: F[A]): A =
+    ev.fromDouble(n)
+}
+
+object DivisionRing extends DivisionRingFunctions[DivisionRing] {
+
+  @inline final def apply[A](implicit f: DivisionRing[A]): DivisionRing[A] = f
+
+}

core/src/main/scala/algebra/ring/EuclideanRing.scala

@@ -0,0 +1,59 @@
+package algebra
+package ring
+
+import scala.annotation.tailrec
+import scala.{specialized => sp}
+
+/**
+ * EuclideanRing implements a Euclidean domain.
+ *
+ * The formal definition says that every euclidean domain A has (at
+ * least one) euclidean function f: A -> N (the natural numbers) where:
+ *
+ *   (for every x and non-zero y) x = yq + r, and r = 0 or f(r) < f(y).
+ *
+ * This generalizes the Euclidean division of integers, where f represents
+ * a measure of length (or absolute value), and the previous equation
+ * represents finding the quotient and remainder of x and y. So:
+ *
+ *   quot(x, y) = q
+ *   mod(x, y) = r
+ */
+trait EuclideanRing[@sp(Int, Long, Float, Double) A] extends Any with GCDRing[A] { self =>
+  def euclideanFunction(a: A): BigInt
+  def equot(a: A, b: A): A
+  def emod(a: A, b: A): A
+  def equotmod(a: A, b: A): (A, A) = (equot(a, b), emod(a, b))
+  def gcd(a: A, b: A)(implicit ev: Eq[A]): A =
+    EuclideanRing.euclid(a, b)(ev, self)
+  def lcm(a: A, b: A)(implicit ev: Eq[A]): A =
+    if (isZero(a) || isZero(b)) zero else times(equot(a, gcd(a, b)), b)
+//  def xgcd(a: A, b: A)(implicit ev: Eq[A]): (A, A, A) =
+}
+
+trait EuclideanRingFunctions[R[T] <: EuclideanRing[T]] extends GCDRingFunctions[R] {
+  def euclideanFunction[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: R[A]): BigInt =
+    ev.euclideanFunction(a)
+  def equot[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A]): A =
+    ev.equot(a, b)
+  def emod[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A]): A =
+    ev.emod(a, b)
+  def equotmod[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A]): (A, A) =
+    ev.equotmod(a, b)
+}
+
+object EuclideanRing extends EuclideanRingFunctions[EuclideanRing] {
+
+  @inline final def apply[A](implicit e: EuclideanRing[A]): EuclideanRing[A] = e
+
+  /**
+   * Simple implementation of Euclid's algorithm for gcd
+   */
+  @tailrec final def euclid[@sp(Int, Long, Float, Double) A: Eq: EuclideanRing](a: A, b: A): A = {
+    if (EuclideanRing[A].isZero(b)) a else euclid(b, EuclideanRing[A].emod(a, b))
+  }
+
+/*  @tailrec final def extendedEuclid[@sp(Int, Long, Float, Double) A: Eq: EuclideanRing](a: A, b: A): (A, A, A) = {
+    if (EuclideanRing[A].isZero(b)) a else euclid(b, EuclideanRing[A].emod(a, b))*/
+
+}

core/src/main/scala/algebra/ring/Field.scala

@@ -3,7 +3,20 @@ package ring
 
 import scala.{ specialized => sp }
 
-trait Field[@sp(Int, Long, Float, Double) A] extends Any with CommutativeRing[A] with MultiplicativeCommutativeGroup[A] { self =>
+trait Field[@sp(Int, Long, Float, Double) A] extends Any with EuclideanRing[A] with MultiplicativeCommutativeGroup[A] { self =>
+
+  // default implementations for GCD
+
+  override def gcd(a: A, b: A)(implicit eqA: Eq[A]): A =
+    if (isZero(a) && isZero(b)) zero else one
+  override def lcm(a: A, b: A)(implicit eqA: Eq[A]): A = times(a, b)
+
+  // default implementations for Euclidean division in a field (as every nonzero element is a unit!)
+
+  def euclideanFunction(a: A): BigInt = BigInt(0)
+  def equot(a: A, b: A): A = div(a, b)
+  def emod(a: A, b: A): A = zero
+  override def equotmod(a: A, b: A): (A, A) = (div(a, b), zero)
 
   /**
    * This is implemented in terms of basic Field ops. However, this is
@@ -17,7 +30,7 @@ trait Field[@sp(Int, Long, Float, Double) A] extends Any with CommutativeRing[A]
 
 }
 
-trait FieldFunctions[F[T] <: Field[T]] extends RingFunctions[F] with MultiplicativeGroupFunctions[F] {
+trait FieldFunctions[F[T] <: Field[T]] extends EuclideanRingFunctions[F] with MultiplicativeGroupFunctions[F] {
   def fromDouble[@sp(Int, Long, Float, Double) A](n: Double)(implicit ev: F[A]): A =
     ev.fromDouble(n)
 }

core/src/main/scala/algebra/ring/GCDRing.scala

@@ -0,0 +1,41 @@
+package algebra
+package ring
+
+import scala.{specialized => sp}
+
+/**
+ * GCDRing implements a GCD ring.
+ *
+ * For two elements x and y in a GCD ring, we can choose two elements d and m
+ * such that:
+ *
+ * d = gcd(x, y)
+ * m = lcm(x, y)
+ *
+ * d * m = x * y
+ *
+ * Additionally, we require:
+ *
+ * gcd(0, 0) = 0
+ * lcm(x, 0) = lcm(0, x) = 0
+ *
+ * and commutativity:
+ *
+ * gcd(x, y) = gcd(y, x)
+ * lcm(x, y) = lcm(y, x)
+ */
+trait GCDRing[@sp(Int, Long, Float, Double) A] extends Any with CommutativeRing[A] {
+  def gcd(a: A, b: A)(implicit ev: Eq[A]): A
+  def lcm(a: A, b: A)(implicit ev: Eq[A]): A
+}
+
+trait GCDRingFunctions[R[T] <: GCDRing[T]] extends RingFunctions[R] {
+  def gcd[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A], eqA: Eq[A]): A =
+    ev.gcd(a, b)(eqA)
+  def lcm[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A], eqA: Eq[A]): A =
+    ev.lcm(a, b)(eqA)
+}
+
+object GCDRing extends GCDRingFunctions[GCDRing] {
+  @inline final def apply[A](implicit ev: GCDRing[A]): GCDRing[A] = ev
+}

core/src/main/scala/algebra/ring/Signed.scala

@@ -0,0 +1,141 @@
+package algebra
+package ring
+
+import scala.{specialized => sp}
+
+/**
+ * A trait that expresses the existence of signs and absolute values on linearly ordered additive commutative monoids
+ * (i.e. types with addition and a zero).
+ *
+ * The following laws holds:
+ *
+ * (1) if `a <= b` then `a + c <= b + c` (linear order),
+ * (2) `signum(x) = -1` if `x < 0`, `signum(x) = 1` if `x > 0`, `signum(x) = 0` otherwise,
+ *
+ * Negative elements only appear when the scalar is taken from a additive abelian group. Then:
+ *
+ * (3) `abs(x) = -x` if `x < 0`, or `x` otherwise,
+ *
+ * Laws (1) and (2) lead to the triange inequality:
+ *
+ * (4) `abs(a + b) <= abs(a) + abs(b)`
+ *
+ * Signed should never be extended in implementations, rather the [[Signed.forAdditiveCommutativeMonoid]] and
+ * [[Signed.forAdditiveCommutativeGroup subtraits]].
+ *
+ * It's better to have the Eq/PartialOrder/Order/Signed hierarchy separate from the Ring hierarchy, so that
+ * we do not end up with duplicate implicits. At the same time, we cannot use self-types to express
+ * the constraint that Signed must be an [[AdditiveCommutativeMonoid]], due to interaction with specialization.
+ */
+trait Signed[@sp(Byte, Short, Int, Long, Float, Double) A] extends Any with Order[A] {
+
+  /**
+   * Returns Zero if `a` is 0, Positive if `a` is positive, and Negative is `a` is negative.
+   */
+  def sign(a: A): Signed.Sign = Signed.Sign(signum(a))
+
+  /**
+   * Returns 0 if `a` is 0, 1 if `a` is positive, and -1 is `a` is negative.
+   */
+  def signum(a: A): Int
+
+  /**
+   * An idempotent function that ensures an object has a non-negative sign.
+   */
+  def abs(a: A): A
+
+  def isSignZero(a: A): Boolean = signum(a) == 0
+  def isSignPositive(a: A): Boolean = signum(a) > 0
+  def isSignNegative(a: A): Boolean = signum(a) < 0
+
+  def isSignNonZero(a: A): Boolean = signum(a) != 0
+  def isSignNonPositive(a: A): Boolean = signum(a) <= 0
+  def isSignNonNegative(a: A): Boolean = signum(a) >= 0
+}
+
+trait SignedFunctions[S[T] <: Signed[T]] extends cats.kernel.OrderFunctions[S] {
+  def sign[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Signed.Sign =
+    ev.sign(a)
+  def signum[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Int =
+    ev.signum(a)
+  def abs[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): A =
+    ev.abs(a)
+  def isSignZero[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Boolean =
+    ev.isSignZero(a)
+  def isSignPositive[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Boolean =
+    ev.isSignPositive(a)
+  def isSignNegative[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Boolean =
+    ev.isSignNegative(a)
+  def isSignNonZero[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Boolean =
+    ev.isSignNonZero(a)
+  def isSignNonPositive[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Boolean =
+    ev.isSignNonPositive(a)
+  def isSignNonNegative[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: S[A]): Boolean =
+    ev.isSignNonNegative(a)
+}
+
+object Signed extends SignedFunctions[Signed] {
+
+  /** Signed implementation for additive commutative monoids */
+  trait forAdditiveCommutativeMonoid[A] extends Any with Signed[A] with AdditiveCommutativeMonoid[A] {
+    def signum(a: A): Int = {
+      val c = compare(a, zero)
+      if (c < 0) -1
+      else if (c > 0) 1
+      else 0
+    }
+  }
+
+  /** Signed implementation for additive commutative groups */
+  trait forAdditiveCommutativeGroup[A] extends Any with forAdditiveCommutativeMonoid[A] with AdditiveCommutativeGroup[A] {
+    def abs(a: A): A = if (compare(a, zero) < 0) negate(a) else a
+  }
+
+  def apply[A](implicit s: Signed[A]): Signed[A] = s
+
+  /**
+   * A simple ADT representing the `Sign` of an object.
+   */
+  sealed abstract class Sign(val toInt: Int) {
+    def unary_- : Sign = this match {
+      case Positive => Negative
+      case Negative => Positive
+      case Zero     => Zero
+    }
+
+    def *(that: Sign): Sign = Sign(this.toInt * that.toInt)
+
+    def **(that: Int): Sign = this match {
+      case Positive => Positive
+      case Zero if that == 0 => Positive
+      case Zero => Zero
+      case Negative if (that % 2) == 0 => Positive
+      case Negative => Negative
+    }
+  }
+
+  case object Zero extends Sign(0)
+  case object Positive extends Sign(1)
+  case object Negative extends Sign(-1)
+
+  object Sign {
+    implicit def sign2int(s: Sign): Int = s.toInt
+
+    def apply(i: Int): Sign =
+      if (i == 0) Zero else if (i > 0) Positive else Negative
+
+    private val instance: CommutativeMonoid[Sign] with MultiplicativeCommutativeMonoid[Sign] with Eq[Sign] =
+    new CommutativeMonoid[Sign] with MultiplicativeCommutativeMonoid[Sign] with Eq[Sign] {
+      def eqv(x: Sign, y: Sign): Boolean = x == y
+      def empty: Sign = Positive
+      def combine(x: Sign, y: Sign): Sign = x*y
+      def one: Sign = Positive
+      def times(x: Sign, y: Sign): Sign = x*y
+    }
+
+    implicit final def signMultiplicativeMonoid: MultiplicativeCommutativeMonoid[Sign] = instance
+    implicit final def signMonoid: CommutativeMonoid[Sign] = instance
+    implicit final def signEq: Eq[Sign] = instance
+  }
+
+}