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Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
Variable binding relates three things: a variable v, a location a for that variable in an expression and a non-leaf node n of the form Q(v, P). Note: we define a location in an expression as a leaf node in the syntax tree. Variable binding occurs when that location is below the node n. In the lambda calculus, x is a bound variable in the term M
In mathematical logic, the de Bruijn index is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. [1] Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as that for syntactic ...
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 purpose of β-reduction is to calculate a value. A value in lambda calculus is a function. So β-reduction continues until the expression looks like a function abstraction. A lambda expression that cannot be reduced further, by either β-redex, or η-redex is in normal form. Note that alpha-conversion may convert functions.
Mogensen extends Scott encoding to encode any untyped lambda term as data. This allows a lambda term to be represented as data, within a Lambda calculus meta program. The meta function mse converts a lambda term into the corresponding data representation of the lambda term; [] =,,. [ ] =,,. [] [] [.
The lambda cube. Direction of each arrow is direction of inclusion. In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt [1] to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus.
The expressions to be typed are exactly those of the lambda calculus extended with a let-expression as shown in the adjacent table. Parentheses can be used to disambiguate an expression. The application is left-binding and binds stronger than abstraction or the let-in construct. Types are syntactically split into two groups, monotypes and ...