Forgetful and Heedful contracts

:: migratory typing, higher-order contracts, by Ben Greenman

Forgetful and heedful are two methods for space-efficient contracts developed by Michael Greenberg in 2014. These methods were born in the shadow of a third method, eidetic, with stronger theoretic properties. Since then, however, the forgetful method has been re-invented at least twice. Both deserve a second look.


Contracts are a tool for specifying and dynamically-enforcing the behavior of a program. In a language with contracts, a programmer can annotate an API with code that documents the intended use for other readers. When client code interacts with such an API, the annotations ensure that the actual behavior matches the expected. If there is a mismatch, the contract annotations can report an issue in terms of three parties: the API code, the client code, and the contract between them.

For example, a Racket module that exports a sorting function can use a contract to describe the kind of input it expects. If a client module sends invalid input, the contract blames the client module for the error, assuming that the contract is bug-free:

  #lang racket/base

  (module sort racket
    (provide
      (contract-out
        [quicksort
          (-> (vectorof point/c) void?)]))

    (define point/c (vectorof integer?))

    (define (quicksort points)
      ....))

  (module client racket
    (require (submod ".." sort))
    (quicksort '()))

  (require 'client)
quicksort: contract violation;
 expected a vector
  given: '()
  in: the 1st argument of
      (-> (vectorof (vectorof integer?)) void?)
  contract from: 
      (file.rkt sort)
  blaming: (file.rkt client)
   (assuming the contract is correct)

That covers the basics. For an extended introduction to contracts, visit The Racket Guide.

The quicksort example and the related figures are from the paper Collapsible Contracts: Fixing a Pathology of Gradual Typing

Classic contracts and “Space Efficiency”

The (vectorof point/c) contract used above describes a possibly-mutable array whose elements match the point/c contract. Since the array can be mutated, this contract has implications for two parties:

  1. the client module must supply a good array, and
  2. the sorting module must not insert a bad element.

To enforce the second condition, the vectorof contract wraps incoming vectors in a proxy that checks future writes. Suppose the client sends a vector with four points:

(quicksort (vector (vector 4 4)
                   (vector 2 2)
                   (vector 1 1)
                   (vector 3 3)))

After applying the contract, the vector is wrapped in a proxy that checks incoming writes and outgoing reads. The following picture illustrates the wrapper with a solid blue bar for the write checks against the sort module and a striped blue bar for the read checks against the client.

A wrapped vector

In a straightforward implementation, these wrappers can stack up if multiple contracts are applied to the same value. For our quicksort in particular, the elements of the vector are mutable vectors and may accumulate wrappers as the vector is sorted — because every write and read applies a contract to the element.

Layers of element wrappers

On the bright side, these wrappers enforce the contracts and help the programmer understand the source of the error if any contract is violated.

Unfortunately, the wrappers also affect the performance of the program. There are prices to pay for: (1) checking values against the contracts, (2) allocating new wrappers, (3) and “indirecting” future writes/reads through wrappers. These space and time costs can add up.

“on a randomly ordered vector of 1,000 points, a call to quicksort can wrap the inner vectors an average of 21 times” — Collapsible Contracts

To fix the problem, researchers have been exploring space-efficient implementations of contracts that attach a bounded number of wrappers to any value. Michael Greenberg is one of these researchers, and eidetic, forgetful, and heedful are his names for three implementations.

(Although the goal of this post is to promote forgetful and heedful, we will review all three.)

Eidetic space-efficiency

The eidetic method introduces a data structure to represent higher-order contracts. The structure supports a merge operation; when two contracts meet, they are merged in a way that avoids duplication. Eidetic contracts have the same behavior as normal “wrapping” contracts and their size is bounded by the number (and height) of source-code contracts in the program.

An eidetic contract is an N-ary tree (for N > 0):

  • each node represents a higher-order contract combinator, such as vectorof
  • the N children of a node represent the different interactions that the value supports
  • each leaf is a list of non-higher-order, or flat, contracts

For example, the (vectorof point/c) source-code contract describes an eidetic tree with 3 nodes and 4 singleton-list leaves. Section 3.1 of the Collapsible Contracts paper has an illustration. Each tree node represents a vectorof contract; these nodes have N=2 children because vectors support reads and writes.

A successful merge combines two trees of the same shape by re-using half the nodes and appending the leaf lists. Re-using nodes saves some space, and helps reduce the overhead of trees relative to simple wrapping contracts. The main savings comes from filtering the leaf lists — if an implementation comes with a contract-stronger? predicate that tests whether one flat contract accepts fewer values than a second, then it can remove leaf-list contracts that are preceded by stronger ones. Trees make this filtering possible.

Suffice to say, eidetic is an ideal solution in theory but comes with practical challenges. Are trees more expensive than wrappers in the common case? Can the leaf-lists in a tree share elements? Should contract-stronger? try to solve problems that lack polynomial-time solutions?

Thankfully, there are at least two “compromise” alternatives.

Forgetful space-efficiency

“Forgetful is an interesting middle ground: if contracts exist to make partial operations safe (and not abstraction or information hiding), forgetfulness may be a good strategy.” — Space-Efficient Manifest Contracts

The forgetful method is exceptionally simple. When applying a new contract to a value, first check whether it is wrapped in a similar contract. If so, then replace the existing wrapper with one that combines:

  1. the client obligations from the old contract, and
  2. the server obligations from the new contract

If not, proceed as usual — by wrapping (an unwrapped value) or raising an error. Every value receives at most one wrapper; this wrapper changes as the value flows to different clients.

Forgetful is safe in the sense that every piece of code can trust the top-level shape of the values it receives. Suppose module A exports a function f with contract (-> T1 T2) to module B, and suppose module B shares this function with a few other client modules using different contracts. As f flows to a new client, it keeps the T1 domain check and gets a replacement for the T2 codomain check.

  • Keeping T1 ensures that the code inside the function (defined by module A) receives input that matches its expectation.
  • Replacing T2 ensures that each new client receives output that it expects.

Unfortunately, replacing T2 also means that clients of module B cannot trust the T2 contract. This contract is not checked, and so forgetful contracts miss some errors that would be caught by standard contracts. For the same reason, a bug in module B may go undetected by its clients — even if a later contract reports an issue, the contract system has no memory that B was partly-responsible.

Despite these changes in behavior, forgetful is a straightforward method for saving space and time relative to classic contracts.

Heedful space-efficiency

A heedful contract is a set of classic higher-order contracts. When applying a new contract to a value, check whether the new contract is in the set. If so, ignore the new contract. If not, add the new contract to the set — or raise an error. Every value gets at most one set-wrapper, and each member of a set-wrapper represents a new constraint.

To check a value against a set, for example when reading from a vector, check each of the elements in any order. If an element raises an error, report it.* Alternatively, an implementation can check all the elements and report all that disagree with the value.

The heedful method is a compromise between forgetful and eidetic.

  • Unlike forgetful, heedful uses a new data structure to represent contacts and requires some kind of contract-stronger? predicate. Heedful also remembers (some of) the history of a value and catches the same errors as classic and eidetic contracts.

  • Unlike eidetic, heedful uses a simpler data structure with no need to keep duplicate flat contracts depending on the order they are encountered. Heedful cannot, however, uniquely identify the two parties involved in a contract error. In general, there are multiple contracts that a programmer must inspect to find the source of a mismatch.

For details, see the extended version of Michael’s POPL 2015 paper. Don’t bother searching the conference version — aside from one remark in Appendix B, heedful and forgetful are nowhere to be found.

* If an implementation promises to report one mismatch, instead of all mismatches, then it does not need to keep the full set of contracts. Thanks to Michael Ballantyne for explaining this to me.

Priorities and Appearances

The extended version of Space-Efficient Manifest Contracts introduces the forgetful and heedful methods with extreme modesty. It’s tempting to skip past them and focus on the eidetic method.

“Since eidetic and classic contracts behave the same, why bother with forgetful and heedful? First and foremost, the calculi offer insights into the semantics of contracts: the soundness of forgetful depends on a certain philosophy of contracts; heedful relates to threesomes without blame [Siek and Wadler 2010]. Second, we offer them as alternative points in the design space. Finally and perhaps cynically, they are strawmen—warm up exercises for eidetic.” — Space-Efficient Manifest Contracts

And yet, at least two other research papers rely on these “strawmen” — or rather, the ideas behind the names.

Gradual Typing with Union and Intersection Types, at ICFP 2017, demonstrates one technique for adding two varieties of types to a gradual language. The semantics in the paper is forgetful; if a higher-order value crosses multiple type boundaries, the intermediate server obligations disappear.

“if a lambda abstraction is preceded by multiple casts, then the rule erases all of them, except for the last one” — Gradual Typing with Union and Intersection Types

This forgetfulness was a deliberate choice. A classic semantics would satisfy the same type soundness theorem, but the authors picked forgetful for its simplicity and performance implications.

“removing these casts preserves the soundness of the evaluation while reducing the number of them”

“while this choice makes the calculus simpler without hindering soundness, it yields a formalism unfit to finger culprits” — Gradual Typing with Union and Intersection Types

Big Types in Little Runtime, at POPL 2017, presents a gradual typing system that avoids the use of wrappers. Instead, their transient semantics rewrites typed code ahead of time to mimic the checks that forgetful contracts would perform. These checks suffice for a shallow type soundness theorem.

That paper also introduces a heedful-like strategy for improving the error messages produced by a forgetful check. The strategy adds a global map to the semantics; keys in the map are unique identifiers for values (heap addresses), and values are sets of types. When a value meets a compatible type, the type is added to the value’s set. When a mismatch occurs, the semantics tries to report every type in the set that relates to the mismatch.

And so, forgetful and heedful were edged out of POPL 2015 but managed to sneak in to POPL 2017. Since then, forgetful appeared in ICFP 2017 and, briefly, in ICFP 2018. Where will we see them next?