- System F
System F, also known as the polymorphic lambda calculus or the second-order lambda calculus, is a
typed lambda calculus. It was discovered independently by the logician Jean-Yves Girardand the computer scientist John C. Reynolds. System F formalizes the notion of parametric polymorphism in programming languages.
Just as the
lambda calculushas variables ranging over functions, and binders for them,the second-order lambda calculus has variables ranging over "types", and binders for them.
As an example, the fact that the identity function can have any type of the form A→ Awould be formalized in System F as the judgement
where α is a
Curry-Howard isomorphism, System F corresponds to a second-order logic.
System F, together with even more expressive lambda calculi, can be seen as part of the
Logic and predicates
The "Boolean" type is defined as:, where α is a
type variable.This produces the following two definitions for the boolean values TRUE and FALSE:
: TRUE := : FALSE :=
Then, with these two λ-terms, we can define some logic operators:: AND := : OR := : NOT :=
There really is no need for a "IFTHENELSE" function as one can just use raw Boolean typed terms as decision functions. However, if one is requested:: IFTHENELSE := will do.A "predicate" is a function which returns a boolean value. The most fundamental predicate is ISZERO which returns TRUE if and only if its argument is the Church numeral 0:: ISZERO := λ "n". "n" (λ "x". FALSE) TRUE
ystem F Structures
System F allows recursive constructions to be embedded in a natural manner, related to that in
Martin-Löf's type theory. Abstract structures (S) are created using "constructors". These are functions typed as::.
Recursivity is manifested when itself appears within one of the types . If you have of these constructors, you can define the type of as::
For instance, the natural numbers can be defined as an inductive datatype with constructors::The System F type corresponding to this structure is. The terms of this type comprise a typed version of the
Church numerals, the first few of which are:: 0 := : 1 := : 2 := : 3 :=
If we reverse the order of the curried arguments ("i.e.," ), then the Church numeral for is a function that takes a function "f" as argument and returns the power of "f". That is to say, a Church numeral is a
higher-order function-- it takes a single-argument function "f", and returns another single-argument function.
Use in programming languages
The version of System F used in this article is as an explicitly-typed, or Church-style, calculus. The typing information contained in λ-terms makes
type-checkingstraightforward. Joe Wells(1994) settled an "embarrassing open problem" by proving that type checking is undecidable for a Curry-style variant of System F, that is, one that lacks explicit typing annotations. [http://www.macs.hw.ac.uk/~jbw/research-summary.html] [http://www.church-project.org/reports/Wells:APAL-1999-v98-no-note.html]
Wells' result implies that
type inferencefor System F is impossible.A restriction of System F known as " Hindley-Milner", or simply "HM", does have an easy type inference algorithm and is used for many strongly typed functional programming languagessuch as Haskell 98 and ML. Over time, as the restrictions of HM-style type systems have become apparent, languages have steadily moved to more expressive logics for their type systems. As of 2008, the Haskell compiler, GHC, goes beyond HM, and now uses System F extended with non-syntactic type equality, for example.
System is the
simply-typed lambda calculus; it includes no mappings from types to types. The System F described in this article is technically System ; that is, the system wherein all mappings from types to types take arguments which are strictly first-order (not functions themselves).
In general, there is a family of systems defined inductively by the "kinds" permitted in each system:
* does not permit any kinds (only types)
* permits kinds:
** (the kind of types) and
** where and (the kind of functions from types to types, where the argument type is of a lower order)
In the limit, we can define system to be
That is, is the system which allows functions from types to types where the argument (and result) may be of any order.
Note that although places no restrictions on the "order" of the arguments in these mappings, it does restrict the "universe" of the arguments for these mappings -- they must be types rather than values. System does not permit mappings from values to types (
Dependent types), though it does permit mappings from values to values ( abstraction), mappings from types to values ( abstraction, sometimes written ) and mappings from types to types ( abstraction at the level of types)
*Girard, Lafont and Taylor, [http://www.PaulTaylor.EU/stable/Proofs%2BTypes.html "Proofs and Types"] . Cambridge University Press, 1989, ISBN 0 521 37181 3.
*J. B. Wells. "Typability and type checking in the second-order lambda-calculus are equivalent and undecidable." In "Proceedings of the 9th Annual
IEEESymposium on Logic in Computer Science (LICS)," pages 176-185, 1994. [http://www.macs.hw.ac.uk/~jbw/papers/Wells:Typability-and-Type-Checking-in-the-Second-Order-Lambda-Calculus-Are-Equivalent-and-Undecidable:LICS-1994.ps.gz]
* [http://www.site.uottawa.ca/~fbinard/Intuitionism/TypeTheory/SystemF/ Summary of System F] by Franck Binard.
Wikimedia Foundation. 2010.