From 047c54e1f6f3a35c4a208e3c24f3a8ed3593c27b Mon Sep 17 00:00:00 2001 From: Denis Rosset Date: Fri, 14 May 2021 16:43:44 +0200 Subject: [PATCH] Added Signed and TruncatedDivision typeclasses, with instances for BigInt --- .../main/scala/algebra/instances/bigInt.scala | 16 +- core/src/main/scala/algebra/ring/Signed.scala | 141 ++++++++++++++++++ .../algebra/ring/TruncatedDivision.scala | 87 +++++++++++ .../main/scala/algebra/laws/OrderLaws.scala | 71 +++++++++ .../test/scala/algebra/laws/LawTests.scala | 2 + 5 files changed, 315 insertions(+), 2 deletions(-) create mode 100644 core/src/main/scala/algebra/ring/Signed.scala create mode 100644 core/src/main/scala/algebra/ring/TruncatedDivision.scala diff --git a/core/src/main/scala/algebra/instances/bigInt.scala b/core/src/main/scala/algebra/instances/bigInt.scala index d6facbb7..663c715e 100644 --- a/core/src/main/scala/algebra/instances/bigInt.scala +++ b/core/src/main/scala/algebra/instances/bigInt.scala @@ -2,15 +2,23 @@ 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 = + private val instance: TruncatedDivision[BigInt] with CommutativeRing[BigInt] = new BigIntAlgebra + + implicit def bigIntAlgebra: CommutativeRing[BigInt] = instance + + implicit def bigIntTruncatedDivision: TruncatedDivision[BigInt] = instance } -class BigIntAlgebra extends CommutativeRing[BigInt] with Serializable { +class BigIntAlgebra extends CommutativeRing[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 +33,8 @@ 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 } diff --git a/core/src/main/scala/algebra/ring/Signed.scala b/core/src/main/scala/algebra/ring/Signed.scala new file mode 100644 index 00000000..54f550b3 --- /dev/null +++ b/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 + } + +} diff --git a/core/src/main/scala/algebra/ring/TruncatedDivision.scala b/core/src/main/scala/algebra/ring/TruncatedDivision.scala new file mode 100644 index 00000000..8438952d --- /dev/null +++ b/core/src/main/scala/algebra/ring/TruncatedDivision.scala @@ -0,0 +1,87 @@ +package algebra +package ring + +import scala.{specialized => sp} + +/** + * Division and modulus for computer scientists + * taken from https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/divmodnote-letter.pdf + * + * For two numbers x (dividend) and y (divisor) on an ordered ring with y != 0, + * there exists a pair of numbers q (quotient) and r (remainder) + * such that these laws are satisfied: + * + * (1) q is an integer + * (2) x = y * q + r (division rule) + * (3) |r| < |y|, + * (4t) r = 0 or sign(r) = sign(x), + * (4f) r = 0 or sign(r) = sign(y). + * + * where sign is the sign function, and the absolute value + * function |x| is defined as |x| = x if x >=0, and |x| = -x otherwise. + * + * We define functions tmod and tquot such that: + * q = tquot(x, y) and r = tmod(x, y) obey rule (4t), + * (which truncates effectively towards zero) + * and functions fmod and fquot such that: + * q = fquot(x, y) and r = fmod(x, y) obey rule (4f) + * (which floors the quotient and effectively rounds towards negative infinity). + * + * Law (4t) corresponds to ISO C99 and Haskell's quot/rem. + * Law (4f) is described by Knuth and used by Haskell, + * and fmod corresponds to the REM function of the IEEE floating-point standard. + */ +trait TruncatedDivision[@sp(Byte, Short, Int, Long, Float, Double) A] extends Any with Signed[A] { + def tquot(x: A, y: A): A + def tmod(x: A, y: A): A + def tquotmod(x: A, y: A): (A, A) = (tquot(x, y), tmod(x, y)) + + def fquot(x: A, y: A): A + def fmod(x: A, y: A): A + def fquotmod(x: A, y: A): (A, A) = (fquot(x, y), fmod(x, y)) +} + +trait TruncatedDivisionFunctions[S[T] <: TruncatedDivision[T]] extends SignedFunctions[S] { + def tquot[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: TruncatedDivision[A]): A = + ev.tquot(x, y) + def tmod[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: TruncatedDivision[A]): A = + ev.tmod(x, y) + def tquotmod[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: TruncatedDivision[A]): (A, A) = + ev.tquotmod(x, y) + def fquot[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: TruncatedDivision[A]): A = + ev.fquot(x, y) + def fmod[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: TruncatedDivision[A]): A = + ev.fmod(x, y) + def fquotmod[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: TruncatedDivision[A]): (A, A) = + ev.fquotmod(x, y) +} + +object TruncatedDivision extends TruncatedDivisionFunctions[TruncatedDivision] { + trait forCommutativeRing[@sp(Byte, Short, Int, Long, Float, Double) A] + extends Any + with TruncatedDivision[A] + with Signed.forAdditiveCommutativeGroup[A] + with CommutativeRing[A] { self => + + def fmod(x: A, y: A): A = { + val tm = tmod(x, y) + if (signum(tm) == -signum(y)) plus(tm, y) else tm + } + + def fquot(x: A, y: A): A = { + val (tq, tm) = tquotmod(x, y) + if (signum(tm) == -signum(y)) minus(tq, one) else tq + } + + override def fquotmod(x: A, y: A): (A, A) = { + val (tq, tm) = tquotmod(x, y) + val signsDiffer = signum(tm) == -signum(y) + val fq = if (signsDiffer) minus(tq, one) else tq + val fm = if (signsDiffer) plus(tm, y) else tm + (fq, fm) + } + + } + + def apply[A](implicit ev: TruncatedDivision[A]): TruncatedDivision[A] = ev +} diff --git a/laws/shared/src/main/scala/algebra/laws/OrderLaws.scala b/laws/shared/src/main/scala/algebra/laws/OrderLaws.scala index ad86191c..3e2dee0f 100644 --- a/laws/shared/src/main/scala/algebra/laws/OrderLaws.scala +++ b/laws/shared/src/main/scala/algebra/laws/OrderLaws.scala @@ -9,6 +9,8 @@ import org.scalacheck.Prop._ import cats.kernel.instances.all._ +import algebra.ring.{CommutativeRing, Signed, TruncatedDivision} + object OrderLaws { def apply[A: Eq: Arbitrary: Cogen]: OrderLaws[A] = new OrderLaws[A] { @@ -112,6 +114,75 @@ trait OrderLaws[A] extends Laws { } ) + def signed(implicit A: Signed[A]) = new OrderProperties( + name = "signed", + parent = Some(order), + "abs non-negative" -> forAll((x: A) => A.sign(A.abs(x)) != Signed.Negative), + "signum returns -1/0/1" -> forAll((x: A) => A.signum(A.abs(x)) <= 1), + "signum is sign.toInt" -> forAll((x: A) => A.signum(x) == A.sign(x).toInt) + ) + + def truncatedDivision(implicit ring: CommutativeRing[A], A: TruncatedDivision[A]) = new DefaultRuleSet( + name = "truncatedDivision", + parent = Some(signed), + "division rule (tquotmod)" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + val (q, r) = A.tquotmod(x, y) + x ?== ring.plus(ring.times(y, q), r) + } + }, + "division rule (fquotmod)" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + val (q, r) = A.fquotmod(x, y) + x ?== ring.plus(ring.times(y, q), r) + } + }, + "|r| < |y| (tmod)" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + val r = A.tmod(x, y) + A.lt(A.abs(r), A.abs(y)) + } + }, + "|r| < |y| (fmod)" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + val r = A.fmod(x, y) + A.lt(A.abs(r), A.abs(y)) + } + }, + "r = 0 or sign(r) = sign(x) (tmod)" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + val r = A.tmod(x, y) + A.isSignZero(r) || (A.sign(r) ?== A.sign(x)) + } + }, + "r = 0 or sign(r) = sign(y) (fmod)" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + val r = A.fmod(x, y) + A.isSignZero(r) || (A.sign(r) ?== A.sign(y)) + } + }, + "tquot" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + A.tquotmod(x, y)._1 ?== A.tquot(x, y) + } + }, + "tmod" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + A.tquotmod(x, y)._2 ?== A.tmod(x, y) + } + }, + "fquot" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + A.fquotmod(x, y)._1 ?== A.fquot(x, y) + } + }, + "fmod" -> forAll { (x: A, y: A) => + A.isSignNonZero(y) ==> { + A.fquotmod(x, y)._2 ?== A.fmod(x, y) + } + } + ) + class OrderProperties( name: String, parent: Option[RuleSet], diff --git a/laws/shared/src/test/scala/algebra/laws/LawTests.scala b/laws/shared/src/test/scala/algebra/laws/LawTests.scala index 41c520a6..2d5fbfd1 100644 --- a/laws/shared/src/test/scala/algebra/laws/LawTests.scala +++ b/laws/shared/src/test/scala/algebra/laws/LawTests.scala @@ -123,6 +123,8 @@ class LawTests extends munit.DisciplineSuite { checkAll("Long", RingLaws[Long].commutativeRing) checkAll("Long", LatticeLaws[Long].boundedDistributiveLattice) +// catsKernelStdOrderForBigInt + checkAll("BigInt", OrderLaws[BigInt].truncatedDivision) checkAll("BigInt", RingLaws[BigInt].commutativeRing) checkAll("FPApprox[Float]", RingLaws[FPApprox[Float]].field)