- Typed lambda calculus
A typed
lambda calculus is a typedformalism that uses the lambda-symbol (lambda) to denote anonymous function abstraction. Typed lambda calculi are foundationalprogramming languages and are the base of typedfunctional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. They are closely related tomathematical logic andproof theory via theCurry-Howard isomorphism and they can be considered as the internal language of classes of categories, e.g. the simply typed lambda calculus is the language of Cartesian closed categories (CCCs).From a certain point of view, typed lambda calculi can be seen as refinements of the
untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory anduntyped lambda calculus a special case with only one type.Various typed lambda calculi have been studied: The types of the
simply typed lambda calculus are only base types (or type variables) and function types sigma o au. System T extends the simply typed lambda calculus with a type of natural numbers and higher order primitive recursion; in this system all functions provably recursive inPeano arithmetic are definable.System F allows polymorphism by using universal quantification over all types; from a logical perspective it can describe all functions which are provably total insecond-order logic . Lambda calculi withdependent types are the base ofintuitionistic type theory , thecalculus of constructions and the logical framework (LF), a pure lambda calculus with dependent types. Based on work by Berardi,Barendregt proposed theLambda cube to systematize the relations of pure typed lambda calculi (including simply typed lambda calculus, System F, LF and the calculus of constructions).Some typed lambda calculi introduce a notion of "subtyping", i.e. if A is a subtype of B, then all terms of type A also have type B. Typed lambda calculi with subtyping are the simply typed lambda calculus with conjunctive types and F^leq (F-sub).
All the systems mentioned so far, with the exception of the untyped lambda calculus, are "strongly normalizing": all computations terminate. As a consequence they are consistent as a logic, i.e. there are uninhabited types. There exist, however, typed lambda calculi that are not strongly normalizing. For example the dependently typed lambda calculus with a type of all types (Type : Type) is not normalizing due to
Girard's paradox . This system is also the simplestPure type system , a formalism which generalizes theLambda cube . Systems with explicit recursion combinators, such as Plotkin's PCF, are not normalizing, but they are not intended to be interpreted as a logic. Indeed, PCF (for Programming language for Computable Functions) is a prototypical, typed functional programming language, where types are used to ensure that programs are well-behaved but not necessarily terminating.Typed lambda calculi play an important role in the design of new type systems for programming languages; here typability usually captures desirable properties of the program, e.g. the program will not cause a memory access violation.
In
programming , the routines (functions, procedures, methods) ofstrongly-typed programming languages closely correspond to typed lambda expressions. Eiffel has a notion of "inline agent" that makes it possible to define and manipulate typed lambda expressions directly, through such expressions as agent (p: PERSON): STRING do Result := p.spouse.name end, denoting an object that represents a function which returns a person's spouse's name.French computer scientist
Gérard Huet gave an algorithm forunification in typed lambda calculus in 1973. [ [http://mathgate.info/cebrown/notes/huet75.php "A Unification Algorithm for Typed Lambda-Calculus", Gerard P. Huet, Theoretical Computer Science 1 (1975), 27-57] ]References
* Henk Barendregt, [ftp://ftp.cs.ru.nl/pub/CompMath.Found/HBK.ps Lambda Calculi with Types] , Handbook of Logic in Computer Science, Volume II, Oxford University Press.
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