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Monoid.java
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1124 lines (966 loc) · 27.2 KB
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package fj;
import static fj.F1Functions.dimap;
import fj.data.Array;
import fj.data.DList;
import fj.data.List;
import fj.data.IO;
import fj.data.Natural;
import fj.data.Option;
import fj.data.Set;
import fj.data.Stream;
import static fj.Function.*;
import static fj.Semigroup.semigroupDef;
import static fj.Unit.unit;
import static fj.data.List.nil;
import static fj.data.Natural.natural;
import static fj.data.Option.none;
import static fj.data.Option.some;
import static fj.data.Stream.iterableStream;
import java.math.BigInteger;
import java.math.BigDecimal;
/**
* A monoid abstraction to be defined across types of the given type argument. Implementations must
* follow the monoidal laws:
* <ul>
* <li><em>Left Identity</em>; forall x. sum(zero(), x) == x</li>
* <li><em>Right Identity</em>; forall x. sum(x, zero()) == x</li>
* <li><em>Associativity</em>; forall x y z. sum(sum(x, y), z) == sum(x, sum(y, z))</li>
* </ul>
*
* @version %build.number%
*/
public final class Monoid<A> {
private final Definition<A> def;
/**
* Primitives functions of Monoid: minimal definition and overridable methods.
*/
public interface Definition<A> extends Semigroup.Definition<A> {
A empty();
default A sum(F0<Stream<A>> as) {
return as.f().foldLeft(this::append, empty());
}
@Override
default A sum(A a, F0<Stream<A>> as) {
return sum(() -> Stream.cons(a, as));
}
default A multiply(int n, A a) {
return (n <= 0)
? empty()
: Semigroup.Definition.super.multiply1p(n - 1, a);
}
@Override
default A multiply1p(int n, A a) {
return n == Integer.MAX_VALUE
? append(a, multiply(n, a))
: multiply(n + 1, a);
}
default Definition<A> dual() {
return new Definition<A>(){
@Override
public A empty() {
return Definition.this.empty();
}
@Override
public A append(A a1, A a2) {
return Definition.this.append(a2, a1);
}
@Override
public A multiply(int n, A a) {
return Definition.this.multiply(n, a);
}
@Override
public Definition<A> dual() {
return Definition.this;
}
};
}
}
/**
* Primitives functions of Monoid: alternative minimal definition and overridable methods.
*/
public interface AltDefinition<A> extends Definition<A> {
@Override
F<A, A> prepend(A a);
@Override
default A append(A a1, A a2) {
return prepend(a1).f(a2);
}
}
private Monoid(Definition<A> def) {
this.def = def;
}
/**
* Composes this monoid with another.
*/
public <B> Monoid<P2<A,B>>compose(Monoid<B> m) {
return compose(m, P2.__1(), P2.__2(), P::p);
}
/**
* Returns a semigroup projection of this monoid.
*
* @return A semigroup projection of this monoid.
*/
public Semigroup<A> semigroup() {
return semigroupDef(def);
}
/**
* Maps the given functions across this monoid as an invariant functor.
*
* @param f The covariant map.
* @param g The contra-variant map.
* @return A new monoid.
*/
public <B> Monoid<B> xmap(final F<A, B> f, final F<B, A> g) {
Monoid.Definition<A> def = this.def;
B zero = f.f(def.empty());
return monoidDef(new Definition<B>() {
@Override
public B empty() {
return zero;
}
@Override
public B append(B a1, B a2) {
return f.f(def.append(g.f(a1), g.f(a2)));
}
@Override
public F<B, B> prepend(B b) {
return dimap(def.prepend(g.f(b)), g, f);
}
@Override
public B multiply(int n, B b) {
return f.f(def.multiply(n , g.f(b)));
}
@Override
public B sum(F0<Stream<B>> as) {
return f.f(def.sum(() -> as.f().map(g)));
}
});
}
public <B, C> Monoid<C> compose(Monoid<B> mb, final F<C, A> a, final F<C, B> b, final F2<A, B, C> c) {
Definition<A> maDef = this.def;
Definition<B> mbDef = mb.def;
C empty = c.f(maDef.empty(), mbDef.empty());
return monoidDef(new Definition<C>() {
@Override
public C empty() {
return empty;
}
@Override
public C append(C c1, C c2) {
return c.f(maDef.append(a.f(c1), a.f(c2)), mbDef.append(b.f(c1), b.f(c2)));
}
@Override
public F<C, C> prepend(C c1) {
F<A, A> prependA = maDef.prepend(a.f(c1));
F<B, B> prependB = mbDef.prepend(b.f(c1));
return c2 -> c.f(prependA.f(a.f(c2)), prependB.f(b.f(c2)));
}
@Override
public C multiply(int n, C c1) {
return c.f(maDef.multiply(n, a.f(c1)), mbDef.multiply(n, b.f(c1)));
}
@Override
public C sum(F0<Stream<C>> cs) {
return c.f(maDef.sum(() -> cs.f().map(a)), mbDef.sum(() -> cs.f().map(b)));
}
});
}
/**
* Sums the two given arguments.
*
* @param a1 A value to sum with another.
* @param a2 A value to sum with another.
* @return The of the two given arguments.
*/
public A sum(final A a1, final A a2) {
return def.append(a1, a2);
}
/**
* Returns a function that sums the given value according to this monoid.
*
* @param a1 The value to sum.
* @return A function that sums the given value according to this monoid.
*/
public F<A, A> sum(final A a1) {
return def.prepend(a1);
}
/**
* Returns a function that sums according to this monoid.
*
* @return A function that sums according to this monoid.
*/
public F<A, F<A, A>> sum() {
return def::prepend;
}
/**
* The zero value for this monoid.
*
* @return The zero value for this monoid.
*/
public A zero() {
return def.empty();
}
/**
* Returns a value summed <code>n</code> times (<code>a + a + ... + a</code>).
* The default definition uses peasant multiplication, exploiting
* associativity to only require {@code O(log n)} uses of
* {@link #sum(Object, Object)}.
*
* @param n multiplier
* @param a the value to be reapeatly summed
* @return {@code a} summed {@code n} times. If {@code n <= 0}, returns
* {@code zero()}
*/
public A multiply(final int n, final A a) {
return def.multiply(n, a);
}
/**
* Sums the given values with right-fold.
*
* @param as The values to sum.
* @return The sum of the given values.
*/
public A sumRight(final List<A> as) {
return as.foldRight(def::append, def.empty());
}
/**
* Sums the given values with right-fold.
*
* @param as The values to sum.
* @return The sum of the given values.
*/
public A sumRight(final Stream<A> as) {
return as.foldRight1(def::append, def.empty());
}
/**
* Sums the given values with left-fold.
*
* @param as The values to sum.
* @return The sum of the given values.
*/
public A sumLeft(final List<A> as) {
return as.foldLeft(def::append, def.empty());
}
/**
* Sums the given values with left-fold.
*
* @param as The values to sum.
* @return The sum of the given values.
*/
public A sumLeft(final Stream<A> as) {
return def.sum(() -> as);
}
/**
* Returns a function that sums the given values with left-fold.
*
* @return a function that sums the given values with left-fold.
*/
public F<List<A>, A> sumLeft() {
return this::sumLeft;
}
/**
* Returns a function that sums the given values with right-fold.
*
* @return a function that sums the given values with right-fold.
*/
public F<List<A>, A> sumRight() {
return this::sumRight;
}
/**
* Returns a function that sums the given values with left-fold.
*
* @return a function that sums the given values with left-fold.
*/
public F<Stream<A>, A> sumLeftS() {
return this::sumLeft;
}
/**
* Intersperses the given value between each two elements of the iterable, and sums the result.
*
* @param as An iterable of values to sum.
* @param a The value to intersperse between values of the given iterable.
* @return The sum of the given values and the interspersed value.
*/
public A join(final Iterable<A> as, final A a) {
final Stream<A> s = iterableStream(as);
F<A, A> prependA = def.prepend(a);
return s.isEmpty()
? def.empty()
: s.foldLeft1((a1, a2) -> def.append(a1, prependA.f(a2)));
}
/**
* Swaps the arguments when summing.
*/
public Monoid<A> dual() {
return monoidDef(def.dual());
}
/**
* Constructs a monoid from the given definition, which must follow the monoidal
* laws.
*
* @param def The definition for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
public static <A> Monoid<A> monoidDef(Definition<A> def) {
return new Monoid<>(def);
}
/**
* Constructs a monoid from the given definition, which must follow the monoidal
* laws.
*
* @param def The definition for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
public static <A> Monoid<A> monoidDef(AltDefinition<A> def) {
return new Monoid<>(def);
}
/**
* Constructs a monoid from the given semigroup definition and zero value, which must follow the monoidal laws.
*
* @param s The semigroup definition for the monoid.
* @param zero The zero for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
public static <A> Monoid<A> monoidDef(final Semigroup.Definition<A> s, final A zero) {
return new Monoid<>(new Monoid.Definition<A>() {
@Override
public A empty() {
return zero;
}
@Override
public A sum(F0<Stream<A>> as) {
return s.sum(zero, as);
}
@Override
public A sum(A a, F0<Stream<A>> as) {
return s.sum(a, as);
}
@Override
public A multiply(int n, A a) {
return (n <= 0)
? zero
: s.multiply1p(n - 1, a);
}
@Override
public A multiply1p(int n, A a) {
return s.multiply1p(n, a);
}
@Override
public A append(A a1, A a2) {
return s.append(a1, a2);
}
@Override
public F<A, A> prepend(A a) {
return s.prepend(a);
}
});
}
/**
* Constructs a monoid from the given semigroup definition and zero value, which must follow the monoidal laws.
*
* @param s The semigroup definition for the monoid.
* @param zero The zero for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
public static <A> Monoid<A> monoidDef(final Semigroup.AltDefinition<A> s, final A zero) {
return monoidDef((Semigroup.Definition<A>) s, zero);
}
/**
* Constructs a monoid from the given sum function and zero value, which must follow the monoidal
* laws.
* Java 8+ users: use {@link #monoidDef(Semigroup.Definition, Object)} instead.
*
* @param sum The sum function for the monoid.
* @param zero The zero for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
public static <A> Monoid<A> monoid(final F<A, F<A, A>> sum, final A zero) {
return new Monoid<>(new AltDefinition<A>() {
@Override
public F<A, A> prepend(A a) {
return sum.f(a);
}
@Override
public A empty() {
return zero;
}
});
}
/**
* Constructs a monoid from the given sum function and zero value, which must follow the monoidal
* laws.
*
* Java 8+ users: use {@link #monoidDef(Semigroup.Definition, Object)} instead.
*
* @param sum The sum function for the monoid.
* @param zero The zero for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
public static <A> Monoid<A> monoid(final F2<A, A, A> sum, final A zero) {
return new Monoid<>(new Definition<A>() {
@Override
public A empty() {
return zero;
}
@Override
public A append(A a1, A a2) {
return sum.f(a1, a2);
}
});
}
/**
* Constructs a monoid from the given semigroup and zero value, which must follow the monoidal laws.
* @deprecated since 4.7. Use {@link #monoidDef(Semigroup.Definition, Object)} or {@link Semigroup#monoid(Object)} instead.
*
* @param s The semigroup for the monoid.
* @param zero The zero for the monoid.
* @return A monoid instance that uses the given sun function and zero value.
*/
@Deprecated
public static <A> Monoid<A> monoid(final Semigroup<A> s, final A zero) {
return s.monoid(zero);
}
/**
* A monoid that adds integers.
*/
public static final Monoid<Integer> intAdditionMonoid = monoidDef(new Definition<Integer>() {
@Override
public Integer empty() {
return 0;
}
@Override
public Integer append(Integer a1, Integer a2) {
return a1 + a2;
}
@Override
public Integer multiply(int n, Integer i) {
return n <= 0 ? 0 : n * i;
}
});
/**
* A monoid that multiplies integers.
*/
public static final Monoid<Integer> intMultiplicationMonoid = monoidDef(new Definition<Integer>() {
@Override
public Integer empty() {
return 1;
}
@Override
public Integer append(Integer i1, Integer i2) {
return i1 * i2;
}
@Override
public Integer sum(F0<Stream<Integer>> as) {
int x = 1;
for (Stream<Integer> xs = as.f(); x != 0 && !xs.isEmpty(); xs = xs.tail()._1()) {
x *= xs.head();
}
return x;
}
@Override
public Integer multiply(int n, Integer integer) {
return n <= 0 ? 1 : (int) StrictMath.pow(integer.doubleValue(), n);
}
});
/**
* @deprecated Since 4.7. Due to rounding errors, addition of doubles does not comply with monoid laws
*/
@Deprecated
public static final Monoid<Double> doubleAdditionMonoid = monoidDef((d1, d2) -> d1 + d2, 0.0);
/**
* @deprecated Since 4.7. Due to rounding errors, multiplication of doubles does not comply with monoid laws
*/
@Deprecated
public static final Monoid<Double> doubleMultiplicationMonoid = monoidDef((d1, d2) -> d1 * d2, 1.0);
/**
* A monoid that adds big integers.
*/
public static final Monoid<BigInteger> bigintAdditionMonoid = monoidDef(new Definition<BigInteger>() {
@Override
public BigInteger empty() {
return BigInteger.ZERO;
}
@Override
public BigInteger append(BigInteger a1, BigInteger a2) {
return a1.add(a2);
}
@Override
public BigInteger multiply(int n, BigInteger a) {
return n <= 0 ? BigInteger.ZERO : a.multiply(BigInteger.valueOf(n));
}
});
/**
* A monoid that multiplies big integers.
*/
public static final Monoid<BigInteger> bigintMultiplicationMonoid = monoidDef(new Definition<BigInteger>() {
@Override
public BigInteger empty() {
return BigInteger.ONE;
}
@Override
public BigInteger append(BigInteger a1, BigInteger a2) {
return a1.multiply(a2);
}
@Override
public BigInteger multiply(int n, BigInteger a) {
return n <= 0 ? BigInteger.ONE : a.pow(n);
}
});
/**
* A monoid that adds big decimals.
*/
public static final Monoid<BigDecimal> bigdecimalAdditionMonoid =
monoidDef(new Definition<BigDecimal>() {
@Override
public BigDecimal empty() {
return BigDecimal.ZERO;
}
@Override
public BigDecimal append(BigDecimal a1, BigDecimal a2) {
return a1.add(a2);
}
@Override
public BigDecimal multiply(int n, BigDecimal a) {
return n <= 0 ? BigDecimal.ZERO : a.multiply(BigDecimal.valueOf(n));
}
});
/**
* A monoid that multiplies big decimals.
*/
public static final Monoid<BigDecimal> bigdecimalMultiplicationMonoid =
monoidDef(new Definition<BigDecimal>() {
@Override
public BigDecimal empty() {
return BigDecimal.ONE;
}
@Override
public BigDecimal append(BigDecimal a1, BigDecimal a2) {
return a1.multiply(a2);
}
@Override
public BigDecimal multiply(int n, BigDecimal decimal) {
return n <= 0 ? BigDecimal.ONE : decimal.pow(n);
}
});
/**
* A monoid that adds natural numbers.
*/
public static final Monoid<Natural> naturalAdditionMonoid =
monoidDef(new Definition<Natural>() {
@Override
public Natural empty() {
return Natural.ZERO;
}
@Override
public Natural append(Natural a1, Natural a2) {
return a1.add(a2);
}
@Override
public Natural multiply(int n, Natural a) {
return natural(n).map(positiveN -> a.multiply(positiveN)).orSome(Natural.ZERO);
}
});
/**
* A monoid that multiplies natural numbers.
*/
public static final Monoid<Natural> naturalMultiplicationMonoid =
monoidDef(new Definition<Natural>() {
@Override
public Natural empty() {
return Natural.ONE;
}
@Override
public Natural append(Natural a1, Natural a2) {
return a1.multiply(a2);
}
});
/**
* A monoid that adds longs.
*/
public static final Monoid<Long> longAdditionMonoid = monoidDef(new Definition<Long>() {
@Override
public Long empty() {
return 0L;
}
@Override
public Long append(Long a1, Long a2) {
return a1 + a2;
}
@Override
public Long multiply(int n, Long a) {
return n <= 0 ? 0L : n * a;
}
});
/**
* A monoid that multiplies longs.
*/
public static final Monoid<Long> longMultiplicationMonoid = monoidDef(new Definition<Long>() {
@Override
public Long empty() {
return 1L;
}
@Override
public Long append(Long i1, Long i2) {
return i1 * i2;
}
@Override
public Long sum(F0<Stream<Long>> as) {
long x = 1L;
for (Stream<Long> xs = as.f(); x != 0L && !xs.isEmpty(); xs = xs.tail()._1()) {
x *= xs.head();
}
return x;
}
@Override
public Long multiply(int n, Long l) {
return n <= 0 ? 1L : (long) StrictMath.pow(l.doubleValue(), n);
}
});
/**
* A monoid that ORs booleans.
*/
public static final Monoid<Boolean> disjunctionMonoid = monoidDef(new Definition<Boolean>() {
@Override
public Boolean empty() {
return false;
}
@Override
public Boolean append(Boolean a1, Boolean a2) {
return a1 | a2;
}
@Override
public Boolean sum(F0<Stream<Boolean>> as) {
return as.f().filter(identity()).isNotEmpty();
}
@Override
public Boolean multiply(int n, Boolean a) {
return n <= 0 ? false : a;
}
});
/**
* A monoid that XORs booleans.
*/
public static final Monoid<Boolean> exclusiveDisjunctionMonoid = monoidDef(new Definition<Boolean>() {
@Override
public Boolean empty() {
return false;
}
@Override
public Boolean append(Boolean a1, Boolean a2) {
return a1 ^ a2;
}
@Override
public Boolean multiply(int n, Boolean a) {
return a && (n == 1);
}
});
/**
* A monoid that ANDs booleans.
*/
public static final Monoid<Boolean> conjunctionMonoid = monoidDef(new Definition<Boolean>() {
@Override
public Boolean empty() {
return true;
}
@Override
public Boolean append(Boolean a1, Boolean a2) {
return a1 & a2;
}
@Override
public Boolean multiply(int n, Boolean a) {
return a;
}
@Override
public Boolean sum(F0<Stream<Boolean>> as) {
return as.f().filter(a -> !a).isEmpty();
}
});
/**
* A monoid that appends strings.
*/
public static final Monoid<String> stringMonoid = monoidDef(new Definition<String>() {
@Override
public String empty() {
return "";
}
@Override
public String append(String a1, String a2) {
return a1.concat(a2);
}
@Override
public String sum(F0<Stream<String>> as) {
StringBuilder sb = new StringBuilder();
as.f().foreachDoEffect(sb::append);
return sb.toString();
}
});
/**
* A monoid that appends string buffers.
*/
public static final Monoid<StringBuffer> stringBufferMonoid = monoidDef((s1, s2) -> new StringBuffer(s1).append(s2), new StringBuffer(0));
/**
* A monoid that appends string builders.
*/
public static final Monoid<StringBuilder> stringBuilderMonoid = monoidDef((s1, s2) -> new StringBuilder(s1).append(s2), new StringBuilder(0));
/**
* A monoid for functions.
*
* @param mb The monoid for the function codomain.
* @return A monoid for functions.
*/
public static <A, B> Monoid<F<A, B>> functionMonoid(final Monoid<B> mb) {
Definition<B> mbDef = mb.def;
return monoidDef(new Definition<F<A, B>>() {
@Override
public F<A, B> empty() {
return __ -> mbDef.empty();
}
@Override
public F<A, B> append(F<A, B> a1, F<A, B> a2) {
return a -> mbDef.append(a1.f(a), a2.f(a));
}
});
}
/**
* A monoid for lists.
*
* @return A monoid for lists.
*/
public static <A> Monoid<List<A>> listMonoid() {
return monoidDef(new Definition<List<A>>() {
@Override
public List<A> empty() {
return nil();
}
@Override
public List<A> append(List<A> a1, List<A> a2) {
return a1.append(a2);
}
@Override
public List<A> sum(F0<Stream<List<A>>> as) {
return as.f().map(DList::listDList).foldLeft(DList::append, DList.<A>nil()).run();
}
});
}
/**
* Lift a {@code Semigroup<A>} for A to a {@code Monoid<Option<A>>}, using Option.none() as zero.
*
* @return A monoid for option.
*/
public static <A> Monoid<Option<A>> optionMonoid(Semigroup<A> aSemigroup) {
return monoidDef(new Monoid.Definition<Option<A>>() {
@Override
public Option<A> empty() {
return none();
}
@Override
public Option<A> append(Option<A> a1, Option<A> a2) {
return a1.liftM2(a2, aSemigroup::sum).orElse(a1).orElse(a2);
}
@Override
public Option<A> multiply(int n, Option<A> oa) {
return n > 0 ? oa.map(a -> aSemigroup.multiply1p(n - 1, a)) : none();
}
@Override
public Option<A> sum(F0<Stream<Option<A>>> oas) {
Stream<A> as = oas.f().bind(Option::toStream);
return as.uncons(none(), h -> tail -> some(aSemigroup.sumStream(h, tail::_1)));
}
});
}
/**
* A monoid for options.
* @deprecated since 4.7. Use {@link #firstOptionMonoid()}.
*
* @return A monoid for options.
*/
@Deprecated
public static <A> Monoid<Option<A>> optionMonoid() {
return firstOptionMonoid();
}
/**
* A monoid for options that take the first available value.
*
* @return A monoid for options that take the first available value.
*/
public static <A> Monoid<Option<A>> firstOptionMonoid() {
return monoidDef(new Definition<Option<A>>() {
@Override
public Option<A> empty() {
return none();
}
@Override
public Option<A> append(Option<A> a1, Option<A> a2) {
return a1.orElse(a2);
}
@Override
public F<Option<A>, Option<A>> prepend(Option<A> a1) {
return a1.isSome() ? __ -> a1 : identity();
}
@Override
public Option<A> multiply(int n, Option<A> as) {
return as;
}
@Override
public Option<A> sum(F0<Stream<Option<A>>> as) {
return as.f().filter(Option.isSome_()).orHead(Option::none);
}
});
}
/**
* A monoid for options that take the last available value.
*
* @return A monoid for options that take the last available value.
*/
public static <A> Monoid<Option<A>> lastOptionMonoid() {
return monoidDef(new Definition<Option<A>>() {
@Override
public Option<A> empty() {
return none();
}
@Override
public Option<A> append(Option<A> a1, Option<A> a2) {
return a2.orElse(a1);
}
@Override
public F<Option<A>, Option<A>> prepend(Option<A> a1) {
return a1.isNone() ? identity() : a2 -> a2.orElse(a1);
}
@Override
public Option<A> multiply(int n, Option<A> as) {
return as;
}
});
}
/**
* A monoid for streams.
*
* @return A monoid for streams.
*/
public static <A> Monoid<Stream<A>> streamMonoid() {
return monoidDef(new Definition<Stream<A>>() {
@Override
public Stream<A> empty() {
return Stream.nil();
}
@Override
public Stream<A> append(Stream<A> a1, Stream<A> a2) {
return a1.append(a2);
}
@Override
public Stream<A> sum(F0<Stream<Stream<A>>> as) {
return Stream.join(as.f());
}
});
}
/**
* A monoid for arrays.