System f out of the blue official video
System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974).
Contents
- System f out of the blue official video
- System f cry official video
- Logic and predicates
- System F structures
- Use in programming languages
- System F
- References
Whereas simply typed lambda calculus has variables ranging over functions, and binders for them, System F additionally 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→ A would be formalized in System F as the judgment
where
As a term rewriting system, System F is strongly normalizing. However, type inference in System F (without explicit type annotations) is undecidable. Under the Curry–Howard isomorphism, System F corresponds to the fragment of second-order intuitionistic logic that uses only universal quantification. System F can be seen as part of the lambda cube, together with even more expressive typed lambda calculi, including those with dependent types.
System f cry official video
Logic and predicates
The
The following two definitions for the boolean values
(Note that the above two functions require three — not two — parameters. The latter two should be lambda expressions, but the first one should be a type. This fact is reflected in the fact that the type of these expressions is
Then, with these two
As in Church encodings, there is no need for an IFTHENELSE function as one can just use raw
will do. A predicate is a function which returns a
System 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
For instance, the natural numbers can be defined as an inductive datatype
The System F type corresponding to this structure is
If we reverse the order of the curried arguments (i.e.,
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-checking straightforward. 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.
Wells' result implies that type inference for 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 statically typed functional programming languages such 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, GHC, a Haskell compiler, goes beyond HM, and now uses System F extended with non-syntactic type equality.
System Fω
While System F corresponds to the first axis of the Barendregt's lambda cube, System Fω or the higher-order polymorphic lambda calculus combines the first axis (polymorphism) with the second axis (type operators); it is a different, more complex system.
System Fω can be defined inductively on a family of systems, where induction is based on the kinds permitted in each system:
In the limit, we can define system
That is, Fω is the system which allows functions from types to types where the argument (and result) may be of any order.
Note that although Fω 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 Fω does not permit mappings from values to types (Dependent types), though it does permit mappings from values to values (