From 2dcac5e6ee177147ac96220ce8a6aa0f0096da60 Mon Sep 17 00:00:00 2001 From: Denis Rosset Date: Fri, 14 May 2021 15:26:57 +0200 Subject: [PATCH] Adding GCD rings and Euclidean rings, along with instances for BigInt --- .../main/scala/algebra/instances/bigInt.scala | 30 +++++++- .../scala/algebra/ring/DivisionRing.scala | 22 ++++++ .../scala/algebra/ring/EuclideanRing.scala | 59 ++++++++++++++ core/src/main/scala/algebra/ring/Field.scala | 17 ++++- .../src/main/scala/algebra/ring/GCDRing.scala | 41 ++++++++++ .../main/scala/algebra/laws/RingLaws.scala | 76 +++++++++++++++++++ .../test/scala/algebra/laws/FPApprox.scala | 3 + .../test/scala/algebra/laws/LawTests.scala | 6 +- 8 files changed, 248 insertions(+), 6 deletions(-) create mode 100644 core/src/main/scala/algebra/ring/DivisionRing.scala create mode 100644 core/src/main/scala/algebra/ring/EuclideanRing.scala create mode 100644 core/src/main/scala/algebra/ring/GCDRing.scala diff --git a/core/src/main/scala/algebra/instances/bigInt.scala b/core/src/main/scala/algebra/instances/bigInt.scala index d6facbb7..27ef4623 100644 --- a/core/src/main/scala/algebra/instances/bigInt.scala +++ b/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 + } + } diff --git a/core/src/main/scala/algebra/ring/DivisionRing.scala b/core/src/main/scala/algebra/ring/DivisionRing.scala new file mode 100644 index 00000000..69de6743 --- /dev/null +++ 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 + +} diff --git a/core/src/main/scala/algebra/ring/EuclideanRing.scala b/core/src/main/scala/algebra/ring/EuclideanRing.scala new file mode 100644 index 00000000..dbf0c9b8 --- /dev/null +++ b/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))*/ + +} diff --git a/core/src/main/scala/algebra/ring/Field.scala b/core/src/main/scala/algebra/ring/Field.scala index 9e51c450..d06b609b 100644 --- a/core/src/main/scala/algebra/ring/Field.scala +++ 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) } diff --git a/core/src/main/scala/algebra/ring/GCDRing.scala b/core/src/main/scala/algebra/ring/GCDRing.scala new file mode 100644 index 00000000..a2031f36 --- /dev/null +++ b/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 +} diff --git a/laws/shared/src/main/scala/algebra/laws/RingLaws.scala b/laws/shared/src/main/scala/algebra/laws/RingLaws.scala index 16dfcc07..b5511915 100644 --- a/laws/shared/src/main/scala/algebra/laws/RingLaws.scala +++ b/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, diff --git a/laws/shared/src/test/scala/algebra/laws/FPApprox.scala b/laws/shared/src/test/scala/algebra/laws/FPApprox.scala index 18b9e3ba..b9a076ab 100644 --- a/laws/shared/src/test/scala/algebra/laws/FPApprox.scala +++ b/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)) + } diff --git a/laws/shared/src/test/scala/algebra/laws/LawTests.scala b/laws/shared/src/test/scala/algebra/laws/LawTests.scala index 41c520a6..158bd18e 100644 --- a/laws/shared/src/test/scala/algebra/laws/LawTests.scala +++ b/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) {