Java Sequence

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In this article I would like to explore a sequence function and its implementation in Java.

Applicative<Traversable<T>> sequence(Traversable<Applicative<T>> values)

You can think about Traversable<T> as an interface that describes a container of values T, something like Iterable<T>. I am going to use T[] and List<T> as an example.

Applicative<T> is sort of context for values T, and it allows to join several such values in contexts together. There is no similar interface in Java but there are several types that have similar behavior.

So sequence is a function that for a given container of wrapped values produces a wrapped container of values.

Let me walk you through some common Java types with Applicative semantics and explain sequencing by example.


Optional<List<T>> sequence(Optional<T> ... optionals)

Optional represents a value that can be absent. Joining together several potentially absent values produces a potentially absent result.

So a sequence of optional values is going to produce an optional list, and it will be present as long as all of optionals are present.

Consider a function parse that extracts a number from a string, if it represents a valid integer.

Optional<Integer> parse(String number)

We can use sequence to group individual parsed numbers into parsed list.

sequence(parse("1"), parse("2"), parse("3"))
// Optional[[1,2,3]]

sequence(parse("1"), parse("X"), parse("3"))
// Optional.empty

Such semantics is useful when we cannot just ignore empty values and need to invalidate the whole list as soon as one of items is empty.


CompletableFuture<List<T>> sequence(CompletableFuture<T> ... futures)

CompletableFuture is a representation of asynchronous computation that will provide a value in the future.

If we sequence a list of futures it should produce an asynchronous list that is going to be available later, after completion of all futures.

It also has all or nothing semantics - result will be available only after all of futures are completed.

CompletableFuture<T> async(T i) // produces later
CompletableFuture<T> failed()   // finishes exceptionally
sequence(async(1), async(2), async(3))
// CompletableFuture[[1,2,3]]

sequence(async(1), failed(), async(3))
// CompletableFuture.failed

It allows to build continuations without blocking on individual results

sequence(async(1), async(2), async(3)).thenApply(this::sum)
// CompletableFuture[[6]]


List<List<T>> sequence(List<T> ... lists)

Previously I’ve used List as an example for container. But it can also be treated as an Applicative.

List represents a choice between zero to many possible values. Joining several choices together leads to multiplication of possibilities.

In this sense sequence of lists produces their Cartesian product.

sequence(asList(1, 2), asList(10, 20), asList(100))
// [[1, 10, 100], [1, 20, 100], [2, 10, 100], [2, 20, 100]]

sequence(asList(1, 2), emptyList(), asList(10, 20))
// []

Usually Cartesian product is used to generate combinations of values

sequence(asList("J", "Q", "K", "A"), 
         asList("Clubs", "Diamonds", "Hearts", "Spades"))
// [[J, Clubs], [J, Diamonds], [J, Hearts], [J, Spades],
// [Q, Clubs], [Q, Diamonds], [Q, Hearts], [Q, Spades],
// [K, Clubs], [K, Diamonds], [K, Hearts], [K, Spades],
// [A, Clubs], [A, Diamonds], [A, Hearts], [A, Spades]]


Function<A, List<T>> sequence(Function<A, T> ... functions)
BiFunction<A, B, List<T>> sequence(BiFunction<A, B, T> ... functions)

Function can be also viewed as a context for value, a value that will be computed from some key.

Sequence of functions is a function that for a given input computes a list of values.

sequence(Person::name, Person::surname).apply(person("John", "Doe"))  
// [John, Doe]

sequence(Integer::sum, Integer::max, Integer::min).apply(100, 200)
// [300, 200, 100]


The most common approach to implement sequence is to fold over items. Java’s analogy would be a reduction of Stream

<T> Optional<List<T>> sequence(List<Optional<T>> optionals) {
// Initial value Optional<List<T>>
    Optional.of(new ArrayList<>()),

// Accumulator BiFunction<Optional<List<T>>, Optional<T>, Optional<List<T>>>
    (result, optional) -> result.flatMap(list -> -> {
      return list;

// Combiner BinaryOperator<Optional<List<T>>  
    (result, chunk) -> result.flatMap(left -> -> {
      List<T> r = new ArrayList<>(left);
      return r;

Identity value in reduction is an option of empty list (which is going to be a result if stream is empty). Accumulator joins together previously accumulated optional list and current optional value. Finally combiner takes two optional lists that have been produced in parallel and joins them together.

Note that both accumulator and combiner will produce an empty Optional if at least one of arguments is empty.

Accumulator and combiner has the same structure - if both optionals are present then function is applied to their arguments. Lets exploit this pattern and make some refactoring

<T> Optional<List<T>> sequence(List<Optional<T>> optionals) {

<A, B, C> BiFunction<Optional<A>, Optional<B>, Optional<C>>
lift(BiFunction<A, B, C> f) {
  return (oa, ob) -> oa.flatMap(a -> -> f.apply(a, b)));

<A> BinaryOperator<Optional<A>> lift(BinaryOperator<A> f) {
  return (oa, ob) -> oa.flatMap(a -> -> f.apply(a, b)));

<T> Optional<T> pure(T value) {
  return Optional.of(value);

<T> BiFunction<List<T>, T, List<T>> add() {
  return (ts, t) -> {
    ArrayList<T> result = new ArrayList<>(ts);
    return result;

<T> BinaryOperator<List<T>> addAll() {
  return (ts, ts2) -> {
    ArrayList<T> result = new ArrayList<>(ts);
    return result;

Note 2 functions:

  • lift transforms a function to work on Optional arguments
  • pure wraps a value into Optional

These functions are part of Applicative, an interface which satisfy all previous example types. It means that for all of them reduction will look the same, as soon as we manage to provide lift and pure implementations.

Let’s do it for CompletableFuture

<T> CompletableFuture<List<T>> 
sequence(List<CompletableFuture<T>> futures) {

<T> CompletableFuture<T> pure(T t) {
  return completedFuture(t);

<A, B, C> BiFunction<CompletableFuture<A>, CompletableFuture<B>, CompletableFuture<C>>
lift(BiFunction<A, B, C> f) {
  return (fa, fb) -> fa.thenCombine(fb, f);

<A> BinaryOperator<CompletableFuture<A>> lift(BinaryOperator<A> f) {
  return (fa, fb) -> fa.thenCombine(fb, f);

Bodies of sequence implementations are identical, that is a place for generalization. Unfortunately Java’s type system is not powerful enough to represent generic type with generic parameter, i.e. Generics of higher kind.

So we cannot extract type safe notion of Applicative for sequence function

<T, A extends Applicative> A<List<T>> sequence(A<T> ... applicatives)


Anyway we have a space for reuse. In my daily work Streams become a tool for composition of operations over some collection of items. A situation when I need to implement similar reduction are not unique.

I have two options:

  • collect result to list and use sequence function over it
  • reimplement the same reduction as above

Collections of list to stream usually is a little bit premature. So we need to extract reduction. The way to reuse reduction functionality in Stream API is to create a Collector.

<T> Collector<Optional<T>, ?, Optional<List<T>>> optionals()

Optional<List<Integer>> result =
  Stream.of(parse("1"), parse("2"), parse("3")).collect(optionals())
//Optional[1, 2, 3]

We can go one step further and generalize resulting container, by using composition of collectors.

<T, A, R> Collector<Optional<T>, ?, Optional<R>>
optionals(Collector<T, A, R> downstream) {
  return collector(

It takes a collector of values and lifts it into Optional context such that it collects optionals with semantics of sequence operation.

Its implementation is a little bit more involved but has the same approach - it lifts each part of downstream collector into Optional context and constructs a new collector.

The full implementation of collectors for all previous types is in this repo

Now we can use it as a last step of stream processing

Stream.of(parse("1"), parse("2")).collect(optionals(toList()))
// Optional[[1, 2]]

Stream.of(parse("1"), parse("X")).collect(optionals(toList()))
// Optional.empty

Composition of collectors gives us useful flexibility - we can reuse existing collectors from JDK and external libraries.

Stream.of(async("1"), async("2"), async("3")).collect(futures(joining(":")))
// CompletableFuture["1:2:3"]

Optional<Map<Boolean, List<Integer>>> result =
Stream.of(parse("13"), parse("12"), parse("11"))
  .collect(optionals(groupingBy(i -> i % 2 == 0)))
// Optional[{false=[11, 13], true=[12]}]

Also we can compose sequencing collectors. Consider a list of futures that will complete with optional result.

list = asList(async(parse("1")), async(parse("2")), async(parse("3")))

Using composition of collectors we can decide how much structure should be extracted from list

result =
// CompletableFuture[[Optional[1], Optional[2], Optional[3]]]

result =
// CompletableFuture[Optional[[1, 2, 3]]]

We are not limited here by number of layers that can be composed.

Lets go crazy with functions

List<Function<String, CompletableFuture<Optional<Integer>>>>
list = asList(s -> async(parse(s)), s -> async(read(s)))

Function<String, List<CompletableFuture<Optional<Integer>>>>
result =

Function<String, CompletableFuture<List<Optional<Integer>>>>
result =

Function<String, CompletableFuture<Optional<List<Integer>>>>
result =

each composed sequence pushes List deeper and deeper inside a stack of contexts.


I hope that today you’ve learned about Applicative and an operation that it enables - sequence.

Also we have learned that Java type system is not the most powerful but definitely have an API that enables composition.

You can checkout code examples from this repo

Have a nice hack ;)