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The purpose of this project is to explore the theoretical foundation as well as conceptual possibilities of capability-based systems.

A main trait of the calculi under study is the introduction of stoic functions. Compared to normal functions which can capture anything from the environment, stoic functions don't capture capabilities or non-stoic functions from the environment. In this sense, they are capability-disciplined.

There are two major applications of the calculus:

  • capability-based effect systems
  • capability-based security

Capability-based Effect System

The core idea is that capabilities are required to produce side effects, thus by tracking capabilities in the type system it's possible to track side effects in the type system. To make sure capabilities are passed by function parameters instead of being captured from the environment, we need to resort to stoic functions.

Compared to existing type-and-effect systems(Gifford and Lucassen, 1986, 1988) and monad-based effect systems, capability-based effect systems have the advantage of more succinct syntax and lower cognitive load in handling of effect polymorphism. Thus capability-based effect systems stand a better chance to be adopted by programmers.

For example, in the following example, the type system would report an error on foo, as it's not allowed to capture any capability variables in the environment:

def map(xs: List[Int], f: Int => Int): List[Int]
def pmap(xs: List[Int], f: Int -> Int): List[Int]                //  -> means f is stoic function type
def print(x: Any)(implicit c: IO): ()

def bar(xs: List[Int])(implicit c: IO) = {
    map(xs, { x => print(x); x })

def foo(xs: List[Int])(implicit c: IO) = {
    pmap(xs, { x => print(x); x })                              // Error, can't capture c:IO

In the code above, map allows the passed in function f to have any side effects, while pmap forbids any side effects.

If the designer of pmap wants the passed function f to only have input/output side effects, he can simply adapt the signature of pmap as follows:

def pmap(xs: List[Int], f: Int -> IO -> Int)(implicit c: IO): List[Int]

def foo(xs: List[Int])(implicit c: IO) = {
    pmap(xs, { x => c => print(x)(c); x })                       // capability c is passed in by pmap

This way, the designers of APIs can strictly control what side effects a passed in function can have.

Capability-base Security

Stoic functions lend itself as a powerful conceptual tool in designing and reasoning security protection in capability-based systems.

For example, KeyKos is the first capability-based operating system that implements confinement. Roughly speaking, confinement means a user is able to use a potentially malicious program from an unknown source to process his sensitive data without worrying that the malicious program will leak the data.

While KeyKos is a very powerful and secure system (UNIX/Linux/Windows/OSX will never match), the concepts in KeyKos are not easy to understand, for instance it has concepts like factory, discreet and sensory capabilities, to name just a few. Stoic functions help clarify the concepts and reason the design of the system. It is our hypothesis that any capability-based operating system has to implement a primitive similar to stoic functions in order to enable useful security patterns.

We also believe stoic functions will be useful in designing security protection in open ecosystems, like Caja, cloud-based computing platforms and app security in mobile phones.


A capability is a term of type E. The base type is represented as B.

A pure environment is a subset of the ordinary typing environment, which particularly excludes variables of ordinary function types, variables of capability types and so on. Different systems may differ in details about what types can be kept in the pure environment, though they must all be effect-safe. Note that the pure environment may contain variables of uninhabited types, such as All X.X, B -> E and so on. These uninhabited types doesn't pose a problem, as from absurdity everything follows (ex falso quodlibet).

A stoic function is a function that can be typed in pure environment. Its type is represented by A -> B.

A free function is a function which can capture anything in the lexical scope. Its type is represented by A => B.

A type T is inhabited if there exists a value v which can be typed as T under the environment {x:B, y:E} (note: variables are values).

An inhabited environment is an environment with only variables of inhabitable types.

A capability-based system is cpability-safe if a stoic function can only use explicitly or implicitly (via free functions) given capabilities. It cannot use any other capabilities directly or indirectly.

This guarantee is the same as that in a pure and inhabited environment (1) it’s impossible to construct a term of capability type E; (2) it's impossible to construct an application where the first term is not stoic.

A capsafe environment is a construct to help prove effect safety. The intended relation between "pure", "inhabited" and "capsafe" environment is as follows:

A pure and inhabited environment is also a capsafe (and pure) environment.


  1. A Study of Capability-Based Effect Systems, F. Liu, S. Stucki (dir), N. Amin(dir), master thesis, EPFL, 2016


  1. Spores Heather Miller et al, 2014
  2. Software Foundations Benjamin C. Pierce et al
  3. Types and programming languages Benjamin C. Pierce
  4. Certified Programming with Dependent Types Adam Chlipala
  5. Locally Nameless Arthur Charguéraud
  6. TLC Arthur Charguéraud
  7. Integrating functional and imperative programming D. K. Gifford & J. M. Lucassen, 1986
  8. Polymorphic effect systems J. M. Lucassen & D. K. Gifford, 1988
  9. Witnessing Purity, Constancy and Mutability Ben Lippmeier, 2009
  10. The marriage of effects and monads Philip Wadler, 1999
  11. Type and Effect Systems Flemming Nielson & Hanne Riis Nielson, 1999


a capability-based system



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