Adding GCD rings and Euclidean rings, along with instances for BigInt

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

@@ -10,7 +10,7 @@ trait BigIntInstances extends cats.kernel.instances.BigIntInstances {
     new BigIntAlgebra
 }
 
-class BigIntAlgebra extends CommutativeRing[BigInt] with Serializable {
+class BigIntAlgebra extends EuclideanRing[BigInt] with Serializable {
 
   val zero: BigInt = BigInt(0)
   val one: BigInt = BigInt(1)
@@ -25,4 +25,32 @@ class BigIntAlgebra extends CommutativeRing[BigInt] with Serializable {
 
   override def fromInt(n: Int): BigInt = BigInt(n)
   override def fromBigInt(n: BigInt): BigInt = n
+
+  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
+}

laws/shared/src/main/scala/algebra/laws/RingLaws.scala

@@ -201,6 +201,57 @@ trait RingLaws[A] extends GroupLaws[A] { self =>
     parents = Seq(ring, commutativeRig, commutativeRng)
   )
 
+  def gcdRing(implicit A: GCDRing[A]) = RingProperties.fromParent(
+    name = "gcd domain",
+    parent = commutativeRing,
+    "gcd/lcm" -> forAll { (x: A, y: A) =>
+      val d = A.gcd(x, y)
+      val m = A.lcm(x, y)
+      A.times(x, y) ?== A.times(d, m)
+    },
+    "gcd is commutative" -> forAll { (x: A, y: A) =>
+      A.gcd(x, y) ?== A.gcd(y, x)
+    },
+    "lcm is commutative" -> forAll { (x: A, y: A) =>
+      A.lcm(x, y) ?== A.lcm(y, x)
+    },
+    "gcd(0, 0)" -> (A.gcd(A.zero, A.zero) ?== A.zero),
+    "lcm(0, 0) === 0" -> (A.lcm(A.zero, A.zero) ?== A.zero),
+    "lcm(x, 0) === 0" -> forAll { (x: A) => A.lcm(x, A.zero) ?== A.zero }
+  )
+
+  def euclideanRing(implicit A: EuclideanRing[A]) = RingProperties.fromParent(
+    name = "euclidean ring",
+    parent = gcdRing,
+    "euclidean division rule" -> forAll { (x: A, y: A) =>
+      pred(y) ==> {
+        val (q, r) = A.equotmod(x, y)
+        x ?== A.plus(A.times(y, q), r)
+      }
+    },
+    "equot" -> forAll { (x: A, y: A) =>
+      pred(y) ==> {
+        A.equotmod(x, y)._1 ?== A.equot(x, y)
+      }
+    },
+    "emod" -> forAll { (x: A, y: A) =>
+      pred(y) ==> {
+        A.equotmod(x, y)._2 ?== A.emod(x, y)
+      }
+    },
+    "euclidean function" -> forAll { (x: A, y: A) =>
+      pred(y) ==> {
+        val (_, r) = A.equotmod(x, y)
+        A.isZero(r) || (A.euclideanFunction(r) < A.euclideanFunction(y))
+      }
+    },
+    "submultiplicative function" -> forAll { (x: A, y: A) =>
+      (pred(x) && pred(y)) ==> {
+        A.euclideanFunction(x) <= A.euclideanFunction(A.times(x, y))
+      }
+    }
+  )
+
   // boolean rings
 
   def boolRng(implicit A: BoolRng[A]) = RingProperties.fromParent(
@@ -227,6 +278,31 @@ trait RingLaws[A] extends GroupLaws[A] { self =>
   // zero * x == x * zero hold.
   // Luckily, these follow from the other field and group axioms.
   def field(implicit A: Field[A]) = new RingProperties(
+    name = "field",
+    al = additiveCommutativeGroup,
+    ml = multiplicativeCommutativeGroup,
+    parents = Seq(euclideanRing),
+    "fromDouble" -> forAll { (n: Double) =>
+      if (Platform.isJvm) {
+        // TODO: BigDecimal(n) is busted in scalajs, so we skip this test.
+        val bd = new java.math.BigDecimal(n)
+        val unscaledValue = new BigInt(bd.unscaledValue)
+        val expected =
+          if (bd.scale > 0) {
+            A.div(A.fromBigInt(unscaledValue), A.fromBigInt(BigInt(10).pow(bd.scale)))
+          } else {
+            A.fromBigInt(unscaledValue * BigInt(10).pow(-bd.scale))
+          }
+        Field.fromDouble[A](n) ?== expected
+      } else {
+        Prop(true)
+      }
+    }
+  )
+
+  // Approximate fields such a Float or Double, even through filtered using FPFilter, do not work well with
+  // Euclidean ring tests
+  def approxField(implicit A: Field[A]) = new RingProperties(
     name = "field",
     al = additiveCommutativeGroup,
     ml = multiplicativeCommutativeGroup,

laws/shared/src/test/scala/algebra/laws/FPApprox.scala

@@ -17,6 +17,8 @@ import algebra.ring._
  * equal to 0.1, then it's plausible they could be equal to each other, so we
  * return true. On the other hand, if the error bound is less than 0.1, then we
  * can definitely say they cannot be equal to each other.
+ *
+ * Based on https://dl.acm.org/doi/10.1145/276884.276904
  */
 case class FPApprox[A](approx: A, mes: A, ind: BigInt) {
   import FPApprox.{abs, Epsilon}
@@ -142,4 +144,5 @@ class FPApproxAlgebra[A: Order: FPApprox.Epsilon](implicit ev: Field[A]) extends
   override def fromInt(x: Int): FPApprox[A] = FPApprox.approx(ev.fromInt(x))
   override def fromBigInt(x: BigInt): FPApprox[A] = FPApprox.approx(ev.fromBigInt(x))
   override def fromDouble(x: Double): FPApprox[A] = FPApprox.approx(ev.fromDouble(x))
+
 }

laws/shared/src/test/scala/algebra/laws/LawTests.scala

@@ -123,10 +123,10 @@ class LawTests extends munit.DisciplineSuite {
   checkAll("Long", RingLaws[Long].commutativeRing)
   checkAll("Long", LatticeLaws[Long].boundedDistributiveLattice)
 
-  checkAll("BigInt", RingLaws[BigInt].commutativeRing)
+  checkAll("BigInt", RingLaws[BigInt].euclideanRing)
 
-  checkAll("FPApprox[Float]", RingLaws[FPApprox[Float]].field)
-  checkAll("FPApprox[Double]", RingLaws[FPApprox[Double]].field)
+  checkAll("FPApprox[Float]", RingLaws[FPApprox[Float]].approxField)
+  checkAll("FPApprox[Double]", RingLaws[FPApprox[Double]].approxField)
 
   // let's limit our BigDecimal-related tests to the JVM for now.
   if (Platform.isJvm) {