Union and intersection contracts are hard, actually

28 April 2022 — by Yann Hamdaoui

Nickel, a configuration language that we are developing at Tweag, strives to provide first-class data validation. Nickel does so thanks to its contract system (although contracts are useful beyond data validation). While the whole story is more involved, contracts can be loosely seen as validation functions. They can be nicely combined using built-in constructors: one can form record contracts, array contracts, function contracts, and more.

A natural and useful addition that we soon considered are the boolean combinators or and and: at first sight, implementing them for boolean predicates looks trivial. But contracts are not exactly predicates, and this subtle difference makes the implementation of general unions (or) and intersections (and) contracts deceptively difficult. The existing literature, while keenly aware of the fact, doesn’t really explain why. This post intends to explain and illustrate this why. You’ll find a thorough exposure in our paper Union and intersection contracts are hard, actually presented at DLS21.



The generic way to define a custom contract in Nickel is to write a function that takes the value to check (and a label that you can ignore for now), and either fails, or returns the value otherwise. It’s sufficient to define any contract we may need in principle, but it’s not always very ergonomic. Let’s write such a contract for an array whose elements are records with a field foo. The value of each foo must also be a number greater than 10.

let Foos = fun label value =>
  if builtin.is_array value then
    |> (fun elem =>
      if builtin.is_record elem && record.fields elem == ["foo"] then
        if builtin.is_num && >= 10 then
            "a foo field is not a number greater than 10"
        contract.blame_with "an element is not a record with a foo field" label)

    contract.blame_with "not an array" label in

[{foo = 20}, {foo = "30"}] | Foos

The pipe operator | is used to apply a contract to a value. Now compare the previous definition with a version using the built-in constructors:

let GreaterThan = fun bound =>
  contract.from_predicate (fun value => builtin.is_num value && value >= bound) in
let Foos = Array {foo | GreaterThan 10} in

[{foo = 20}, {foo = "30"}] | Foos

We’ve used the from_predicate, Array and record constructors to assemble the same contract in a more concise, more modular and clearer way. Because such constructors have built-in support, the error messages are also better localized1.

Currently, Nickel features contract constructors corresponding to native values:

  • primitive contracts (numbers, strings, booleans)
  • record contracts
  • dictionary contracts
  • array contracts
  • function contracts

Unions and intersections

The existing constructors can get us quite far already, but some common contracts are still out of reach. Sometimes, we wish to apply several conditions to the same value. For example, ensuring that a field is not only a valid port number, but also a non-reserved port number (greater than 1023). There is currently no combinator to build this contract out of Port and GreaterThan. If we had intersections (written /\ thereafter), we could2:

let NonReservedPort = Port /\ GreaterThan 1023 in
  port | NonReservedPort

Unions would be very useful too. An ubiquitous example is nullable contracts, accepting a value that either satisfies some contract A, or can be null. This would simply be A \/ Null.

Beyond nullable values, we may allow a field to accept several alternative representations. For example, a date contract which accepts both a structured record or a valid ISO-8601 string. Once again, with unions, this contract is straightforward to write (assuming prior definitions of Iso8601Date, Day, and so on):

let Data = {day | Day, month | Month, year | Year} \/ Iso8601Date

Unions and intersections are hard

However appealing union and intersection contracts may be, they happen — perhaps surprisingly — to break fundamental properties of the core Nickel language. In the following, I only mention unions for simplicity, but all of the points made have a counterpart for the dual case of intersections.


The crux of the issue concerns lazy contracts. By lazy, I mean that the checks embodied by such contracts don’t, and often can’t, fire right when the contract is first evaluated.

Eager contracts

Primitive contracts (Num, Bool and Str), and more generally any contract defined as a boolean predicate (e.g. using contract.from_predicate), are eager. When evaluating exp | Bool, the Bool contract checks the nature of exp and either fails immediately, or returns the boolean value of exp unchanged. The contract won’t interfere with the evaluation anymore.

Union of predicates can be trivially defined as the pointwise or operator ||:

P1 \/ ... \/ PN := fun value => P1 value || ... || PN value

Data contracts

On the other hand, datatype contracts like arrays and records are lazy. To understand why, consider, for example, Nixpkgs: it is a dictionary mapping packages to build recipes. That is, a massive, over-50 000-key-value-pair wide dictionary. It is absolutely out of the question to evaluate the entirety of this dictionary every time one needs to install 10 new packages: this would result in a painfully slow experience. Outside of Nix, one may want to query just a field or a subset of a configuration, without having to evaluate the whole thing.

To cope with such use-cases, Nickel has been made lazy. Expressions are only evaluated when needed, including the content of arrays and records. To preserve this capability, array and record contracts must be lazy as well. For if they were simple eager predicates, applying a top-level contract like nixpkgs | Packages would require the full evaluation of nixpkgs, in spite of the language’s lazyness.

Concretely, a record contract {foo | Str, bar | Num} will:

  • check that the value is a record with fields foo and bar. This part happens immediately.

  • lazily maps contracts Str and Num onto the inner fields. That is:

    {foo = 1 + 1, bar = 2} | {foo | Str, bar | Num}`
    # evaluates to
    {foo = 1 + 1 | Str, bar = 2 | Num}

Here, the Str contract violation for 1 + 1 will only cause an error once foo is used or serialized to a configuration, but not when the contract is first evaluated. If foo is never used, the contract won’t fail the execution. Lazy contracts return the original value, but with delayed checks buried inside.

Union contracts as a side-effect

The problem with lazy contracts is that union contracts can’t know right away which branch of the union to take. This implies that the implementation of union requires a form of backtracking and exception-like control flow, making them effectful. Take the following example (assume Pos and Neg are contracts for positive and negative numbers):

let Contract = {foo | Pos, bar | Pos } \/ {foo | Neg, bar | Neg} in

let data | Contract = {
  foo = 0 + 5,
  bar = 0 - 7,

This contract should fail at some point, because foo and bar are neither both positive nor both negatives. However, because of lazyness, the union contract can’t evaluate foo or bar right away. Hence, it doesn’t know which branch of the union to try yet.

If we use later in the program, the union contract will evaluate the Neg contract, acknowledge its failure, and rule out the second branch {foo | Neg, bar | Neg}. We still can’t figure if bar satisfies Pos yet.

Symmetrically, if we rather use alone, we can only rule out the first branch of the union, because fails Pos.

Detecting the violation of the original contract is possible only once we have used both and in the same program. Now, imagine that data is defined in a library data.ncl. We import the library in two different files foo.ncl and bar.ncl, each using only one of the fields:

# foo.ncl
let data = import "data.ncl" in

# bar.ncl
let data = import "data.ncl" in

They run totally fine in isolation. Now, if in a third program we import and use both foo.ncl and bar.ncl:

let foo = import "foo.ncl" in
let bar = import "bar.ncl" in
foo + bar

The interpreter now reports a contract violation pointing to one of the imports! This is a spooky action at a distance, or side-effect.

  • For the programmer, side-effects are hard to reason about because they prevent local reasoning.
  • For the interpreter, side-effects inhibit many optimizations and program transformations. Nickel being lazy and pure, a lot of program optimizations can be applied unconditionally. Not so much once we add unions.

Unions are also complex to implement efficiently. Here, the union contract needs to maintain shared mutable state between all the use points of data. At each contract violation on or, we need to update the shared state, and use it to decide if we should actually raise an error. This also implies that not all contract failures immediately stop the execution anymore, which requires to turn the simple bail-out semantics of contract.blame into a recoverable exception-like mechanism.

Lazy data contracts may look quite specific to Nickel. Alas, we can recast the same arguments and examples for function contracts, even in a strict (non-lazy) language. This is what we did in the paper. As first-class functions are a founding principle of functional programming, a contract system for functional languages without function contracts would be seriously impaired. Thus, any functional language with contracts faces difficulties when adding unions to the mix.

A way out

We’ve seen through an example why adding general union and intersection contracts to any contract system with lazy data contract or function contracts incurs a prohibitive cost in complexity.

Those issues arise when one tries to implement unions that must work with arbitrary contracts. We already observed that a number of contracts are not lazy, such as predicates. And indeed, the union of predicates is trivial to implement. If Nickel could distinguish the contracts built from contract.from_predicate and remember their boolean definition, we could form the union of an arbitrary number of predicates together with one arbitrary contract. Just apply each predicate in order, and if one succeeds, return the value. If they all fail, apply the last contract.

Even for lazy contracts like records, a lot of unions are actually workable. Take for example {foo | Num} \/ {foo | Num, bar | Str}. Just looking at the shape of the records, we see that the right one has an additional field bar. Nickel could systematically generate a discriminating predicate based on the structure of the operands. Here, that would be record.has_field "bar". We can then implement the union easily, because we can decide right away which branch to try: if the predicate returns true, apply the right contract, otherwise apply the left one. Such a restricted union constructor would bail out on harder cases such as {foo | A} \/ {foo | B}, where there is no obvious discriminating predicate.

A similar analysis and strategy is implemented for unions of function contracts in the Racket language. Having such a theoretically restricted but practically useful union constructor is probably the road we are going to take for Nickel.

  1. Contracts constructors are also helpful in the interaction with the static side of the gradual type system of Nickel, but this is orthogonal to the issues explored in this post.
  2. One can actually attach several contract to a value, and this particular example is already possible to write as {port | Port | GreaterThan 1023}. However, this is only works for the and combinator, and such an and isn’t first class: we can’t write a contract for an array of non reserved ports in a simple way.

About the author

Yann Hamdaoui

Yann is working at Tweag on the Nickel programming language, a next-generation configuration language to manage the growing complexity of Infrastructure-as-Code and a candidate successor for the language Nix.

If you enjoyed this article, you might be interested in joining the Tweag team.

This article is licensed under a Creative Commons Attribution 4.0 International license.


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