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The "lambda term" is represented as a tagged union with three cases: Constructor a - a variable (arity 1, not recursive) Constructor b - function application (arity 2, recursive in both arguments), Constructor c - lambda-abstraction (arity 1, recursive). For example,
For example, in simply typed lambda calculus, it is a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms need not terminate (see below). One reason there are many different typed lambda calculi has been the desire to do more (of what the untyped calculus can do ...
In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator), [1]: p.26 is a higher-order function (i.e. a function which takes a function as argument) that returns some fixed point (a value that is mapped to itself) of its argument function, if one exists.
In the Lambda lifting section, a meta function for first lifting and then converting the resulting lambda expression into recursive equation was described. The Lambda Drop meta function performs the reverse by first converting recursive equations to lambda abstractions, and then dropping the resulting lambda expression, into the smallest scope ...
Scott encoding can be done in untyped lambda calculus, whereas its use with types requires a type system with recursion and type polymorphism. A list with element type E in this representation that is used to compute values of type C would have the following recursive type definition, where '=>' denotes function type:
In the 1930s Alonzo Church sought to use the logistic method: [a] his lambda calculus, as a formal language based on symbolic expressions, consisted of a denumerably infinite series of axioms and variables, [b] but also a finite set of primitive symbols, [c] denoting abstraction and scope, as well as four constants: negation, disjunction, universal quantification, and selection respectively ...
The technique was formalized in 1989 as "F-bounded quantification."[2] The name "CRTP" was independently coined by Jim Coplien in 1995, [3] who had observed it in some of the earliest C++ template code as well as in code examples that Timothy Budd created in his multiparadigm language Leda. [4]
System F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages, thus forming a theoretical basis for languages such as Haskell and ML